Bell Theorem [Archive] - Physics Forums

JesseM

Jan7-09, 11:46 AM

Yes, mn4j's analogy shows a misunderstanding of Bell's inequality. Here's a more accurate analogy I came up with a while ago:
Suppose we have a machine that generates pairs of scratch lotto cards, each of which has three boxes that, when scratched, can reveal either a cherry or a lemon. We give one card to Alice and one to Bob, and each scratches only one of the three boxes. When we repeat this many times, we find that whenever they both pick the same box to scratch, they always get opposite results--if Bob scratches box A and finds a cherry, and Alice scratches box A on her card, she's guaranteed to find a lemon.

Classically, we might explain this by supposing that there is definitely either a cherry or a lemon in each box, even though we don't reveal it until we scratch it, and that the machine prints pairs of cards in such a way that the "hidden" fruit in a given box of one card is always the opposite of the hidden fruit in the same box of the other card. If we represent cherries as + and lemons as -, so that a B+ card would represent one where box B's hidden fruit is a cherry, then the classical assumption is that each card's +'s and -'s are the opposite of the other--if the first card was created with hidden fruits A+,B+,C-, then the other card must have been created with the hidden fruits A-,B-,C+.

The problem is that if this were true, it would force you to the conclusion that on those trials where Alice and Bob picked different boxes to scratch, they should find opposite fruits on at least 1/3 of the trials. For example, if we imagine Bob's card has the hidden fruits A+,B-,C+ and Alice's card has the hidden fruits A-,B+,C-, then we can look at each possible way that Alice and Bob can randomly choose different boxes to scratch, and what the results would be:

Bob picks A, Alice picks B: same result (Bob gets a cherry, Alice gets a cherry)

Bob picks A, Alice picks C: opposite results (Bob gets a cherry, Alice gets a lemon)

Bob picks B, Alice picks A: same result (Bob gets a lemon, Alice gets a lemon)

Bob picks B, Alice picks C: same result (Bob gets a lemon, Alice gets a lemon)

Bob picks C, Alice picks A: opposite results (Bob gets a cherry, Alice gets a lemon)

Bob picks C, Alice picks picks B: same result (Bob gets a cherry, Alice gets a cherry)

In this case, you can see that in 1/3 of trials where they pick different boxes, they should get opposite results. You'd get the same answer if you assumed any other preexisting state where there are two fruits of one type and one of the other, like A+,B+,C-/A-,B-,C+ or A+,B-,C-/A-,B+,C+. On the other hand, if you assume a state where each card has the same fruit behind all three boxes, like A+,B+,C+/A-,B-,C-, then of course even if Alice and Bob pick different boxes to scratch they're guaranteed to get opposite fruits with probability 1. So if you imagine that when multiple pairs of cards are generated by the machine, some fraction of pairs are created in inhomogoneous preexisting states like A+,B-,C-/A-,B+,C+ while other pairs are created in homogoneous preexisting states like A+,B+,C+/A-,B-,C-, then the probability of getting opposite fruits when you scratch different boxes should be somewhere between 1/3 and 1. 1/3 is the lower bound, though--even if 100% of all the pairs were created in inhomogoneous preexisting states, it wouldn't make sense for you to get opposite answers in less than 1/3 of trials where you scratch different boxes, provided you assume that each card has such a preexisting state with "hidden fruits" in each box.

But now suppose Alice and Bob look at all the trials where they picked different boxes, and found that they only got opposite fruits 1/4 of the time! That would be the violation of Bell's inequality, and something equivalent actually can happen when you measure the spin of entangled photons along one of three different possible axes. So in this example, it seems we can't resolve the mystery by just assuming the machine creates two cards with definite "hidden fruits" behind each box, such that the two cards always have opposite fruits in a given box.
And you can modify this example to show some different Bell inequalities, see post #8 of this thread (http://www.physicsforums.com/showthread.php?t=210793) if you're interested.

mn4j

Jan7-09, 02:10 PM

I bet it's a pretty complicated answer, so if you guys aren't up to writing it all down here, it'll be great if someone could post a link to another page which explains it more fully.
thanks

Your questions are answered by the following article:
Jaynes, E. T., 1989, "Clearing up Mysteries - The Original Goal ," in Maximum-Entropy and Bayesian Methods, J. Skilling (ed.), Kluwer, Dordrecht, p. 1 (http://bayes.wustl.edu/etj/articles/cmystery.pdf)

Specifically, it treats the Bell inequalities from p.10. It also shows how the QM expectation value is consistent with probability theory and explains very clearly the mistakes Bell made that caused the conflict.

If you have access to Bell's original paper (NOT the multitude of third-party proofs on the web) you can read and follow alongside and see exactly where Bell introduced the conflict (equation 2) in Bell's original paper.

JesseM

Jan7-09, 03:06 PM

Specifically, it treats the Bell inequalities from p.10. It also shows how the QM expectation value is consistent with probability theory and explains very clearly the mistakes Bell made that caused the conflict.
It's not consistent with probability theory applied to a local hidden variables scenario. If you disagree, please look at the example I gave and explain how any local hidden variables scenario (i.e. a scenario where each lotto card has a preexisting fruit under each box, and scratching merely reveals the fruit that was already underneath) could be consistent with the statistics given.

The article you posted is confused on many points--for example, they claim that "as his words show above, Bell took it for granted that a conditional probability P(X|Y) expresses a physical causal influence, exerted by Y on X". But if you look at Bell's words that they're quoting, this is a misinterpretation. What Bell said was "It would be very remarkable if b proved to be a causal factor for A, or a for B; i.e. if P(A | a \lambda) depended on b or P(B | b \lambda) depended on a." Nowhere is Bell saying that in general P(X|Y) being different from P(X) implies that Y was causally influencing X, he's just saying that in this particular case the only sensible way that P(A | a \lambda) could depend on b would be if b exerted an influence on the hidden variables \lambda. And the reason for that has to do with the specific meaning of the terms--here a represents one experimenter's choice of what variable to measure, and b represents the other experimenter's choice of what variable to measure. It is assumed that these choices happen at a spacelike separation from one another so there can be no direct causal influence from one to another in a local hidden variables theory, and it is assumed that the choices are spontaneous and random, so that there were no factors in their common past light cone which preconditioned both the experimenter's choices and the hidden variables \lambda for each particle in such a way that they could be correlated. In physics there are two ways that events at different locations can be correlated, either by a direct causal influence from one to another, or by some factor in their common past light cone which influenced both; the monkey/urn example is simply an example of the latter, since both the probability of a given ball being found in one hand and the probability of a given ball being found in another are determined by the selection procedure from the urn, with the combination of balls in the urn at the time the second ball was selected having been influenced by the event of the first ball being selected. In the case of the experimenter's choices, Bell was simply passing over the possibility of some factor in the common past light cone of both measurements somehow predetermined both their choices of what to measure and the hidden variables in such a way as to produce a correlation, because he was assuming the choices were really free; once this possibility is eliminated, the only remaining possibility for a dependence between P(A | a \lambda) and b would be if the experimenter's choice of measurement b somehow exerted a faster-than-light influence on the hidden variables \lambda of the other particle. In later writings Bell and other physicists have explicitly recognized this loophole of the experimenter's choices being predetermined by factors in their common past light cone, see for example near the end of this article from the Stanford Encyclopedia of Philosophy (http://plato.stanford.edu/entries/bell-theorem/) where they write:
The last resort of a dedicated adherent of local realistic theories, influenced perhaps by Einstein's advocacy of this point of view, is to conjecture that apparent violations of locality are the result of conspiracy plotted in the overlap of the backward light cones of the analysis-detection events in the 1 and 2 arms of the experiment. These backward light cones always do overlap in the Einstein-Minkowski space-time of Special Relativity Theory (SRT) — a framework which can accommodate infinitely many processes besides the standard ones of relativistic field theory. Elimination of any finite set of concrete scenarios to account for the conspiracy leaves infinitely many more as unexamined and indeed unarticulated possibilities. What attitude should a reasonable scientist take towards these infinite riches of possible scenarios? We should certainly be warned by the power of Hume's skepticism concerning induction not to expect a solution that would be as convincing as a deductive demonstration and not to expect that the inductive support of induction itself can fill the gap formed by the shortcoming of a deductive justification of induction (Hume 1748, Sect. 4). One solution to this problem is a Bayesian strategy that attempts to navigate between dogmatism and excessive skepticism (Shimony 1993, Shimony 1994). To avoid the latter one should keep the mind open to a concrete and testable proposal regarding the mechanism of the suspected conspiracy in the overlap of the backward light cones, giving such a proposal a high enough prior probability to allow the possibility that its posterior probability after testing will warrant acceptance. To avoid the former one should not give the broad and unspecific proposal that a conspiracy exists such high prior probability that the concrete hypothesis of the correctness of Quantum Mechanics is debarred effectively from acquiring a sufficiently high posterior probability to warrant acceptance. This strategy actually is implicit in ordinary scientific method. It does not guarantee that in any investigation the scientific method is sure to find a good approximation to the truth, but it is a procedure for following the great methodological maxim: “Do not block the way of inquiry” (Peirce 1931).[5]
In terms of my lotto card analogy, it would be as if the machine printing the cards could know ahead of time whether me and my partner would both scratch the same box on our respective cards or different boxes (because there was some causal factor in the past light cone of the scratching events which determined in advance which boxes we would scratch, and the machine could examine this causal factor to know ahead of time what the choices would be), and the machine would vary the statistics of what fruits were behind each box depending on what our future choices were going to be on each trial.

mn4j

Jan7-09, 03:35 PM

The article you posted is confused on many points--for example, they claim that "as his words show above, Bell took it for granted that a conditional probability P(X|Y) expresses a physical causal influence, exerted by Y on X". But if you look at Bell's words that they're quoting, this is a misinterpretation. What Bell said was "It would be very remarkable if b proved to be a causal factor for A, or a for B; i.e. if P(A | a \lambda) depended on b or P(B | b \lambda) depended on a." Nowhere is Bell saying that in general P(X|Y) being different from P(X) implies that Y was causally influencing X, he's just saying that in this particular case the only sensible way that P(A | a \lambda) could depend on b would be if b exerted an influence on the hidden variables \lambda.

You have not understood it. Bell is thinking that events at A should not influence events at B. Physically that is correct. However he imposes that physical condition on the probabilities. So his equations are wrong. Logically, the probability of an event at A can influence the probability of an event at B even if there is no physical dependence. That is precisely the reason why he makes the fatal error. You only need to look at the example of the monkey pulling two balls to see this. Physically, the second ball can have no influence on the first. But if you impose this condition on the probability, you end up with a probability of 1/2 for the first ball even after you have seen the second ball which is wrong. If you know that the second ball is white, the probability that the first one picked was red is 1 not 1/2.

Following the correct probability rules, QM is in line with probability theory.

JesseM

Jan7-09, 04:06 PM

You have not understood it. Bell is thinking that events at A should not influence events at B.
He is also saying the experimenter's choices of what variable to measure a and b (i.e. the spin axes they are measuring on a given trial), which are distinct from the outcomes they get A and B, should not influence the outcomes at the other experimenter's detector, or even be correlated with the outcomes at the other experimenter's detector, because their choices are freely-made.
Physically that is correct. However he imposes that physical condition on the probabilities.
And what physical condition would those be? Are you talking about the one in the quote "It would be very remarkable if b proved to be a causal factor for A, or a for B; i.e. if P(A | a \lambda) depended on b or P(B | b \lambda) depended on a"? If not, please tell me what specific probability condition you're talking about. If you are talking about that quote--which seems to be the one the authors are focusing on with their urn analogy--then my analysis is correct, Bell's statement depended on his specific understanding of what a meant physically (a choice of what variable to measure by the experimenters), he wasn't saying that in general P(X|Y) can only depend on Z if Z exerts a causal influence on X.
Logically, the probability of an event at A can influence the probability of an event at B even if there is no physical dependence.
In a local realist universe, the only way this can work is if their are causal factors in the past of both A and B that influenced both and caused them to be correlated--this is exactly what's going on in the urn example. Do you think there's a third option for how A and B can be correlated that's not based on A and B exerting a causal influence on one another, and is also not based on some set of factors C in the overlap between the past light cones of A and B which causally influenced both? If so, please explain it.
You only need to look at the example of the monkey pulling two balls to see this. Physically, the second ball can have no influence on the first.
No, but the past event of the monkey pulling a ball from the urn is in the common past of both events (the event of our looking to see what color the monkey chose on his first pick, and the event of our looking to see what color the monkey chose on his second), and this shared past is what explains the correlation, obviously. Did you even bother to read my explanation???
Following the correct probability rules, QM is in line with probability theory.
Nope, it's not compatible with both the condition of local hidden variables and the condition that the experimenter's choices of what variable to measure were free ones which were not causally determined by some past factors which also determined the hidden variables associated with the particles (the 'no-conspiracy' assumption discussed in the quote from the Stanford Encyclopedia). The authors of the paper you mention have only rediscovered the conspiracy loophole, although they don't seem to have realized that the only way their observations about probabilities are relevant to Bell's theorem is if there was a conspiracy in the initial conditions which caused the experimenter's choices to be causally correlated with the hidden variables.

mn4j

Jan7-09, 04:27 PM

And what physical condition would those be? Are you talking about the one in the quote "It would be very remarkable if b proved to be a causal factor for A, or a for B; i.e. if P(A | a \lambda) depended on b or P(B | b \lambda) depended on a"? If not, please tell me what specific probability condition you're talking about. If you are talking about that quote--which seems to be the one the authors are focusing on with their urn analogy--then my analysis is correct, Bell's statement depended on his specific understanding of what a meant physically (a choice of what variable to measure by the experimenters), he wasn't saying that in general P(X|Y) can only depend on Z if Z exerts a causal influence on X.

The rules of probability demand that you calculate probabilities in specific ways. Bell used equation (12) when he should have used equation (13) and equation (14) when he should have used equation (15) [numbers according to Jaynes article]. The reason he did this was because he thought there should be no physical causal relationship. However, lack of a physical causal relationship does not mean lack of logical causality. This is explained in more detail in Jaynes' book " http://www.amazon.com/Probability-Theory-Logic-Science-Vol/dp/0521592712

No, but the past event of the monkey pulling a ball from the urn is in the common past of both events (the event of our looking to see what color the monkey chose on his first pick, and the event of our looking to see what color the monkey chose on his second), and this shared past is what explains the correlation, obviously. Did you even bother to read my explanation???
The point is that by knowing the outcome of the second pick, the calculated probability for the first pick changes from what it was if the second outcome was unknown. The calculation of probabilites is completely after the fact of the experiment. This tells you that there is a logical link between the two in the way the calculation should be done even though we know that the second ball can not cause the first one to be different.

JesseM

Jan7-09, 05:02 PM

The rules of probability demand that you calculate probabilities in specific ways. Bell used equation (12) when he should have used equation (13)
Apparently you missed (or misunderstood) the part immediately after (13) where the authors say: But if we grant that knowledge of the experimenter's free choices (a,b) would give us no information about \lambda: P(\lambda | ab) = p(\lambda ) (and in this verbiage we too are being carefully epistemological) then Bell's interpretation lies in the factorization

P(AB | ab\lambda) = P(A | a \lambda ) P(B | b \lambda ) (14)

whereas the fundamentally correct factorization would read:

P(AB | ab \lambda ) = P(A | Bab\lambda ) P(B | ab\lambda) = P(A | ab\lambda ) P(B | ab\lambda ) (15)

in which both a, b appear as conditioning statements
In other words, they're saying that if you grant the assumption that "knowledge of the experimenter's free choices (a,b) would give us no information about \lambda", which is exactly the same "no-conspiracy" condition I have been discussing, then the left hand of (14) reduces to the right hand, and their equation (13) reduces to the equation (12) that Bell used. It's only if you drop this this assumption that you're left with the "fundamentally correct" equations (13) and (15) in which no assumption about (a,b)'s relation to other events has been made.
The reason he did this was because he thought there should be no physical causal relationship.
No, the explanation for the assumption is not just that there is no direct causal link between an experimenter's choice of what to measure at one detector b and the outcome at the opposite detector A. What is also being assumed is that in a local realist universe, there should be no causal factor in the common past of these two events that would lead to a correlation between b and A, because the experimenter's choices are assumed to be "free". Again, in a local realist universe there are two ways of explaining a correlation between two events X and Y; #1 is that either X is exerting a direct causal influence on Y or vice versa, but #2 is that some event(s) Z in their common past exerted a causal influence on both in such a way as to create the correlation. For example, if I put a black ball in one box and a white ball in another, and send one box to Alice and another to Bob, then if Alice opens her box and sees a white ball she'll know Bob must have gotten the black ball; this isn't because the event of Alice opening her box exerted a causal influence on the event of Bob opening his box or vice versa, but it is because both events were determined by the past events of my putting a black ball in one box and a white ball in the other. In fact this is exactly the idea behind the \lambda in a local hidden variables theory, it represents some factors which both particles share because these factors were determined at the past event of both particles being created at the same location.
However, lack of a physical causal relationship does not mean lack of logical causality.
Please name an example of "logical causality" between two events A and B that does not reduce to either a direct causal link between A and B or some event C in the common past of both A and B that conditioned both of them and caused the correlation.
No, but the past event of the monkey pulling a ball from the urn is in the common past of both events (the event of our looking to see what color the monkey chose on his first pick, and the event of our looking to see what color the monkey chose on his second), and this shared past is what explains the correlation, obviously. Did you even bother to read my explanation???
The point is that by knowing the outcome of the second pick, the calculated probability for the first pick changes from what it was if the second outcome was unknown.
Yes, and the explanation for that is a common causal factor in the past of both picks. In a local realist universe, when you have consistent correlated outcomes of pairs of experiments, the explanation is always either 1) one outcome exerting a direct causal influence on the other, or 2) a causal factor in the past which influenced both later outcomes. Do you agree or disagree with this?

mn4j

Jan7-09, 06:31 PM

Again, in a local realist universe there are two ways of explaining a correlation between two events X and Y; #1 is that either X is exerting a direct causal influence on Y or vice versa, but #2 is that some event(s) Z in their common past exerted a causal influence on both in such a way as to create the correlation.

...

Please name an example of "logical causality" between two events A and B that does not reduce to either a direct causal link between A and B or some event C in the common past of both A and B that conditioned both of them and caused the correlation.

You fail, just like Bell, to appreciate the difference between ontological correlations, which are results of real experimental data, and logical correlations which are epistemological in nature. Bell was not analysing real experimental results, he was constructing a mental picture of what we might see if we measured it. This means he was bound by the rules of probability to do the calculations differently than he did.

You asked for an example of "logical causality". I have given one already. The one of the urn. Knowledge of the second ball being red, causes the probability of the first ball to change from 1/2 to 1 even though there is no physical causation happening. To say that the result is only because of a common physical cause in the past does not work because the calculation of probabilities is completely after the monkey had picked the balls, yet calculating the probability of the first ball before and after the revealing of the second ball you get a different value. There is no way any new knowledge of the second ball can physically cause a change in the first ball is there?

Another example:
Consider the the following proposition
- "A implies B", logically, this implies "not B implies not A".
Now pick any A and B of your choice for which there is a physical causality such that A causes B. It would still hold logically that "not B implies not A", yet it would make no sense to talk of "not B" causing "not A" physically.

The point therefore is that lack of physical causality is not a good enough excuse to assume lack of logical causality in your equations as bell did. You must still use the correct rules of probability to calculate.

JesseM

Jan7-09, 07:18 PM

You fail, just like Bell, to appreciate the difference between ontological correlations, which are results of real experimental data, and logical correlations which are epistemological in nature. Bell was not analysing real experimental results, he was constructing a mental picture of what we might see if we measured it.
But Bell was assuming a realist picture in which any hidden variables that determine results of measurements already have well-defined values before measurement. You can imagine God continuously measuring \lambda as the particles propogate, until after each one is measured by the experimenters, if it helps.
You asked for an example of "logical causality". I have given one already. The one of the urn.
I asked for an example of logical causality where the measured correlation could not also be physically explained either in terms of one measurement influencing the other directly or both measurements being influenced by events in the past. The correlation between the colors of the two balls is completely based on the fact that they were both selected from the same urn.
Knowledge of the second ball being red, causes the probability of the first ball to change from 1/2 to 1 even though there is no physical causation happening.
But if you take the God's eye view in which all objective facts exist before measurement, then what's happening here is the "hidden variable" of the second ball being red determines that when I look at it, the event "I see a red ball" is guaranteed to happen with probability 1. Meanwhile, the fact that the first ball picked was white, and was picked from an urn containing one red ball and one white ball, is what determines there'd be a probability 1 that the "hidden variable" associated with the second ball turned out to be red. In other words, P(I see a red ball when I look at second pick | I saw a white ball when I looked at the first pick) = 1 can be broken down into:

P(first ball picked had hidden state 'white' after being picked | I saw a white ball when I looked at the first pick) = 1
P(second ball picked had hidden state 'red' after being picked | first ball picked had hidden state 'white' after being picked) = 1
P(I see a red ball when I looked at second pick | second ball picked had hidden state 'red'' after being picked) = 1

In a realist universe, you should always be able to take such a God's eye view where the true state of all the unknown factors is part of the probability calculation (although these true states are represented as variables since we don't necessarily know them), and probability is interpreted in frequentist terms (in terms of the statistics of repeated trials) rather than Bayesian terms, and this is exactly what Bell was doing in his proof.

Meanwhile, I don't see the authors of the paper you link to arguing that we can only use epistemological Bayesian probabilities rather than objective frequentist probabilities, that appears to be your own original argument which has nothing to do with their own. The fact that they include \lambda in their probability equations, despite the fact that \lambda represents hidden variables whose value we can never actually know, shows this.
Another example:
Consider the the following proposition
- "A implies B", logically, this implies "not B implies not A".
Now pick any A and B of your choice for which there is a physical causality such that A causes B. It would still hold logically that "not B implies not A", yet it would make no sense to talk of "not B" causing "not A" physically.
Why not? "Causality" just means that one physical fact determines another physical fact according to the laws of physics, and an absence of a certain physical event is still a physical fact about the universe, there's no reason it can't be said to "cause" some other fact.

DrChinese

Jan7-09, 07:40 PM

ZapperZ, thanks for splitting out the thread.

mnj4,

It is the realism argument that is most important to Bell, and it really doesn't matter your point about conditional probabilities. This is of little importance to Bell.

The mathematical description of realism is:

1 >= P(A, B, C) >= 0

The reason is that A, B and C individually are elements of reality, because they can be predicted in advance in a Bell test. The issue is not whether the measurement somehow distorts the results, it is whether these elements of reality (EPR) exist simultaneously independently of the ACT of observation.

If you believe they do, you are supporting realism. And NO physical theory of local realism (i.e. hidden variables) can reproduce the predictions of QM. If that statement is not true, you can provide a counterexample! But as pointed out earlier, this has befuddled everyone who has tried (so far).

Your urn example makes no sense. There are NO conditionals before any measurement occurs. So if we ask what percentage of humans are men, are Democrats, and/or are college educated, I can make predictions all day long and my answers will always satisfy the realism criterion above. It doesn't matter whether the attributes are causally related somehow or not. And yet when QM is applied, you cannot get around the negative probabilities for some of the subensembles if Malus is followed. You MUST know this already, how can you not?

mn4j

Jan7-09, 07:55 PM

But Bell was assuming a realist picture in which any hidden variables that determine results of measurements already have well-defined values before measurement. You can imagine God continuously measuring \lambda as the particles propogate, until after each one is measured by the experimenters, if it helps.

I asked for an example of logical causality where the measured correlation could not also be physically explained either in terms of one measurement influencing the other directly or both measurements being influenced by events in the past. The correlation between the colors of the two balls is completely based on the fact that they were both selected from the same urn.

Exactly! Therefore any local realist theorem MUST also consider that measurements at A and B MUST be logically correlated! Bell sets out trying to determine the probability of observing certain events at A based on measurements at B. But in doing so, in he fails to incorporate logical dependence. This is the crucial point. And this is why you get the wrong answer for the urn if you calculate using bell's equation! Because there is no concept of logical dependence in it even though there must be! And the reason bell makes this mistake is because he does not clearly separate logical dependence from physical causation.

Meanwhile, I don't see the authors of the paper you link to arguing that we can only use epistemological Bayesian probabilities rather than objective frequentist probabilities, that appears to be your own original argument which has nothing to do with their own. The fact that they include \lambda in their probability equations, despite the fact that \lambda represents hidden variables whose value we can never actually know, shows this.

I am not arguing this at all. I have not even started talking about \lambda because there are other assumptions Bell makes about \lambda that are unfounded. The simplest explanation of what I am saying is that if A and B are determined by a local-realist theorem of hidden variables, then the probabilities of events at A MUST be logically dependent on those at B, even if there is no direct physical causation from A to B and vice versa. Therefore, failure to include logical dependence in his treatment is unfounded.

DrChinese

Jan7-09, 08:11 PM

You must still use the correct rules of probability to calculate.

You have it backwards, as usual. Bell is pointing out what actually should have been obvious in retrospect: that subensembles cannot respect Malus as does QM.

1. If you had a group of 100 trials of Alice and Bob at settings of 0 and 45 degrees respectively, then we would expect a coincidence rate of 50%. Now, I ask does this statement somehow violate your definition of proper probability? I certainly hope not...

2. Now, imagine that we have the 4 permutations of the above and choose to subdivide it into another group, the results (which by the realistic definition must exist and be well-defined) of measurements (let's call this Carrie) at 22.5 degrees - i.e. midway between the 0 and 45 degree mark. Surely you already know that the correlation rate between Alice and Carrie must equal the correlation rate between Bob and Carrie. Is that too difficult? Does this statement somehow violate your definition of proper probability? I certainly hope not...

And finally, you must also already know that there is no set of trials possible in which 1. is true and also 2. is true. Oh, and that also matches the prediction of QM for 22.5 degrees (for photons) which of course is about 85%.

Now, none of this is a problem if you drop the requirement of realism. If you said that Bell's argument was not well-phrased, I would probably agree with you. But there is nothing wrong with the conclusion it leads to.

JesseM

Jan7-09, 08:21 PM

Exactly! Therefore any local realist theorem MUST also consider that measurements at A and B MUST be logically correlated! Bell sets out trying to determine the probability of observing certain events at A based on measurements at B. But in doing so, in he fails to incorporate logical dependence.
But the only type of "dependence" when you do frequentist calculations is statistical dependence in the objective facts of the matter over many trials, and this statistical dependence must always be explainable in terms of physical causation. For example, when I broke down the urn/ball probabilities like this:

P(first ball picked had hidden state 'white' after being picked | I saw a white ball when I looked at the first pick) = 1
P(second ball picked had hidden state 'red' after being picked | first ball picked had hidden state 'white' after being picked) = 1
P(I see a red ball when I looked at second pick | second ball picked had hidden state 'red'' after being picked) = 1

...none of these were subjective probabilities. For example, "P(first ball picked had hidden state 'white' after being picked | I saw a white ball when I looked at the first pick)" can be interpreted in frequentist terms as meaning "if you look at a large number of trials, and then look at only the subset of trials where the event of me seeing a white ball when I looked at the first pick occurred, in what fraction of this subset was it also true that the the ball had the hidden state white at the moment it was picked"? And the answer of course, is 1, the reason being that on any trial where the ball had hidden state white after it was picked, this caused me to predictably see a white ball when I looked at the color.

So you see, when you're dealing with a frequentist notion of probabilities in a realist universe, any statistical correlations must involve physical causation, either one fact causing the other or both facts being caused by some common cause in their past. Do you disagree? If so, please name an example of a situation where we can interpret probabilities in a purely frequentist manner yet there is a statistical correlation that can not be explained in one of these two causal ways.
This is the crucial point. And this is why you get the wrong answer for the urn if you calculate using bell's equation!
What "Bell's equation" are you talking about? Please write it down, and explain what the variables are supposed to represent in terms of the urn example that you think leads the equation to be wrong.
The simplest explanation of what I am saying is that if A and B are determined by a local-realist theorem of hidden variables, then the probabilities of events at A MUST be logically dependent on those at A, even if there is no direct physical causation from A to B and vice versa.
Again, when you interpret probabilities in frequentist/realist terms, all statistical correlations (which I guess is what you mean by 'logical dependence') must be explained in terms of physical causes, though the explanation may involve a common cause in the past of two events rather than either one directly influencing the other. And Bell did not assume there'd be no statistical correlation between A and B--the whole point of including \lambda was to show there could be such a correlation in a local realist universe, as long as it was explained by the source creating both particles with correlated hidden variables (a common cause in the past), just like my example of sending balls to Alice and Bob and always making sure one was sent a black ball and the other was sent a white one, so their measurements results would always be opposite (here I play the role of the 'source' which determines the hidden variables of each box that determine the correlations between their observations when they open their respective boxes).

The place that Bell assumed "no statistical correlation" was between the choice made by one experimenter of what spin axis to perform a measurement on, and the hidden variables of the other particle being measured by the other experimenter far away (as well as the other experimenter's choice of what to measure). In a local realist universe, for this assumption to be wrong, there would have to be some common cause(s) in the past of both an experimenter's choices and the particles' hidden variables which predetermined what they would be, and ensured a statistical correlation (so the source would act like it 'knows in advance' what choice the two experimenters would make, and adjusts the probabilities of emitting particles with different hidden variables accordingly).

mn4j

Jan7-09, 09:59 PM

But the only type of "dependence" when you do frequentist calculations is statistical dependence in the objective facts of the matter over many trials,

This make no sense. If you had the objective facts, you will not need induction. It will be deductive and you will never have a probability other than 0 or 1. Did you read the article? The Bernouli urn example is treated in the article. The reason inductive reasoning is used, where probabilities can have values between 0 and 1, is because we don't have all the objective facts. Which means we are trying to make a prediction of what we might see if you make the measurement. Bell did not base his calculations on ANY objective facts. The first experiments trying to test his theorem were done years after. By objective facts I assume you mean real experimental data.

Take the example of the urn I gave earlier.

1. You know that there are two balls in the urn, one is red and one is white.
2. You know that the monkey picked one ball first and then another ball second.
3. You are asked to infer what the probability is of the first ball being red before seeing the result of the second ball. Then the second ball is shown to you and you are asked to again infer the probability that the first ball is red.

We both accept that "the second ball is red" has no physically causative effect on the state of the first ball, because it was picked after the second ball. At most, they have a single event in their past which caused them both. Yet, in calculating the probabilities in (3) above, you will not arrive at the correct result if you do not use the right equations which include logical dependence. Note that the urn example is the simplest example of a hidden variable theory. In this case, you are saying that once the balls have left the urn, the outcome of the experiment is determined and there is no superluminal communication between the monkey's right hand and left hand.

Bell's equation written for this situation is essentially,

P(AB|Z) = P(A|Z)P(B|Z) ( see Bell's equation (2) which is the same as eq 12 in Jaynes ) Remember the question is "What is the probability that both balls are red"?

Z: The premise that the urn contains two balls, one white and one red.
A: First ball is red
B: Second ball is red

Calculating based on that equation, the probability that both of those balls is red results in 0.5 * 0.5 or 0.25! Which is wrong! If you don't believe me, do the experiment, 1000000 times, you will never observe a red ball in both hands.

However, the correct equation should have been,
P(AB|Z) = P(A|B)P(B|Z) = P(B|A)P(A|Z) (Equation 15 in Jaynes).
which results in 0 * 0.5 = 0, the correct answer.

As you see, even though we accept that there is no physical causality from the second draw to the first draw, we still must include logical dependence to calculate the probabilities correctly. This means we must have a P(A|B) or P(B|A) term in our equation.
It is just like proponents of bell now point to experiments with real monkeys and 2-ball urns and say "since bell obtained 0.25 instead of 0 and real experiments obtain 0, it must mean that the experiments disprove local reality of the urn and the monkey."

You see, what is moving faster than light is not any physical influence. It is the same logical influence which caused our probability to suddenly change as soon as we knew the result of the second ball.

Again, when you interpret probabilities in frequentist/realist terms, all statistical correlations (which I guess is what you mean by 'logical dependence') must be explained in terms of physical causes, though the explanation may involve a common cause in the past of two events rather than either one directly influencing the other.
You are not reading what I write. I'll use a dramatic example.

"not Dead implies not Executed".
Do you agree with the above?

There is a logical dependence between "not Dead" and "not Executed". If a person is not dead, it MUST follow that the person is not Executed. However, you can not say "not Dead" physically causes "not Executed", otherwise nobody will ever be executed. Logical dependence is not the same as physical causation.

And Bell did not assume there'd be no statistical correlation between A and B--the whole point of including \lambda was to show there could be such a correlation
He must have. Equation (2) in his article, (12) in Jaynes, means just that. The correct way to include logical dependence is in how you set up the equations, not by introducing additional parameters. Adding numbers to the balls in the urn example above, will not change the results if you do not use the right equations.

in a local realist universe, as long as it was explained by the source creating both particles with correlated hidden variables (a common cause in the past), just like my example of sending balls to Alice and Bob and always making sure one was sent a black ball and the other was sent a white one, so their measurements results would always be opposite (here I play the role of the 'source' which determines the hidden variables of each box that determine the correlations between their observations when they open their respective boxes).
Maybe I should ask you a question. If you know the outcome at A, the settings at A and the settings at B will you be able to deduce the outcome at B? Isn't this the premise of any hidden variable theorem, that the outcome is determined by the values they left the source with and the settings at the terminals?
Now explain to me why when you calculate the probabilities, you do not include logical dependence. By this I mean that the term P(A|B) never occurs in any of bell's equations.

Just like knowing the result of the first second draw should influence the way you calculate the probability of the second draw. Shouldn't it? How is it supposed to influence it if you do not have a P(A|B) term??

The place that Bell assumed "no statistical correlation" was between the choice made by one experimenter of what spin axis to perform a measurement on, and the hidden variables of the other particle being measured by the other experimenter far away (as well as the other experimenter's choice of what to measure). In a local realist universe, for this assumption to be wrong, there would have to be some common cause(s) in the past of both an experimenter's choices and the particles' hidden variables which predetermined what they would be, and ensured a statistical correlation (so the source would act like it 'knows in advance' what choice the two experimenters would make, and adjusts the probabilities of emitting particles with different hidden variables accordingly).
Look at Bell's original article, DrChinese has it on his website. Everything starts from Eq. (2). Clearly from (2) there are only two options
1- Either bell did not understand how to apply probability rules or
2- He assumed that knowing the results at B should not influence the way we calculate the probability at A.
Either one is devastating to his result.

There is no other way, if you see it , let me know. The equations speak for themselves!

DrChinese

Jan7-09, 10:33 PM

You see, what is moving faster than light is not any physical influence. It is the same logical influence which caused our probability to suddenly change as soon as we knew the result of the second ball.

No one is saying that there is anything moving faster than light.

I think you are missing the bigger picture here. It is almost like you found a misspelled word and want to throw out a work for that reason. I have derived Bell's Theorem a variety of ways, and never do I bother with his "P(AB|Z) = P(A|Z)P(B|Z)" which you say is wrong. There are plenty of things in the original that could be presented differently, and something things I personally think are obscure in the extreme. But the ideas ultimately are fine. Like the EPR paper, he makes a few statements - the references to Bohmian Mechanics, for example - which leave a poor aftertaste.

What is generally accepted as the legacy of the paper is the following conclusion: No physical theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics.

The key here is the realism requirement, which is not explicitly mentioned at all! 1 >= P(A, B, C) >= 0. Einstein would have naturally agreed with this, because he insisted that particle attributes existed independently of the act of observation: "I think that a particle must have a separate reality independent of the measurements. That is: an electron has spin, location and so forth even when it is not being measured. I like to think that the moon is there even if I am not looking at it."

So I would say that the 2 main assumptions of the paper are: a) realism, which is what the inequality is based upon - and NOT separability, as is often mentioned; and b) locality, because he was trying to point out that there might be some unknown force that can communicate between entangled particles or their measuring apparati superluminally.

I would like to make it clear that no dBB-type theory I have ever seen explains any mechanism by which Bell test results occur. They always simply state that they generically reproduce QM predictions. Maybe, maybe not. But the mechanism itself is *always* missing.

On the other hand: If you assume locality, then you can still stay with QM as a non-realistic local theory and be done with it. And there is no problem at all. Look at the bigger picture, and you will see why Bell is widely praised and accepted today: Bell tests are the heart of entanglement measures, and are leading to innovative new ways to explore the quantum world. And lo and behold, there are still no local realistic theories which have survived the rigors of both theory and experiment.

mn4j

Jan7-09, 11:13 PM

JesseM

Jan7-09, 11:53 PM

This make no sense. If you had the objective facts, you will not need induction.
I didn't say anything about you knowing the objective facts. Again, the frequentist idea is to imagine a God's-eye perspective of all the facts, and knowing the causal relations between the facts, figure out what the statistics would look like for a very large number of trials. Then, if you want to know the probability that you will observe Y when you have already observed X, just look at the subset of these large number of trials where one of the facts is "experimenter observes X", and figure out what fraction of these trials would also include the fact "experimenter observed Y".

If you believe there are objective facts in each trial, even if you don't know them, then it should be possible to map any statement about subjective probabilities into a statement about what this imaginary godlike observer would see in the statistics over many trials--do you disagree? For example, suppose there is an urn with two red balls and one white ball, and the experiment on each trial is to pick two balls in succession (without replacing the first one before picking the second), and noting the color of each one. If I open my hand and see that the first one I picked was red, and then I look at the closed fist containing the other and guess if it'll be red or white, do you agree that I should conclude P(second will be white | first was red) = 1/2? If you agree, then it shouldn't be too hard to understand how this can be mapped directly to a statement about the statistics as seen by the imaginary godlike observer. On each trial, this imaginary observer already knows the color of the ball in my fist before I open it, of course. However, if this observer looks at a near-infinite number of trials of this kind, and then looks at the subset of all these trials where I saw that the first ball was red, do you agree that within this subset, on about half these trials it'll be true that the ball in my other hand was white? (and that by the law of large numbers, as the number of trials goes to infinity the ratio should approach precisely 1/2?)

If you agree with both these statements, then it shouldn't be hard to see how any statement about subjective probabilities in an objective universe should be mappable to a statement about the statistics seen by a hypothetical godlike observer in a large number of trials. If you think there could be any exceptions--objectively true statements of probability which cannot be mapped in this way--then please give an example. It would be pretty earth-shattering if you could, because the frequentist interpretation of probabilities is very mainstream, I'm sure you could find explanations of probability in terms of the statistics over many trials in virtually any introductory statistics textbook.
Take the example of the urn I gave earlier.

1. You know that there are two balls in the urn, one is red and one is white.
2. You know that the monkey picked one ball first and then another ball second.
3. You are asked to infer what the probability is of the first ball being red before seeing the result of the second ball. Then the second ball is shown to you and you are asked to again infer the probability that the first ball is red.

We both accept that "the second ball is red" has no physically causative effect on the state of the first ball, because it was picked after the second ball. At most, they have a single event in their past which caused them both.
Exactly, there was an event in their past which predetermined what the color of both the first and second ball would be (the event of the first ball being picked from the urn containing only a white and red ball). Don't you remember that this was exactly my point, that in a realist universe any statistical correlation between events must be explainable either in terms of one event causing the other or in terms of a common cause (or set of causes) in their common past? I asked if you had any counterexamples to this general statement about statistical correlations, the urn example certainly isn't one.
Yet, in calculating the probabilities in (3) above, you will not arrive at the correct result if you do not use the right equations which include logical dependence.
What exactly do you mean by "logical dependence"? The probabilities can of course be calculated in the same frequentist manner as I discussed above--if you imagine a large number of trials of this type, it's certainly true that on the subset of trials where the first ball picked was white, the second ball was always red in 100% of this subset, and likewise in the subset of trials where the first ball picked was red, the second ball was always white in 100% of this subset.
Bell's equation written for this situation is essentially,

P(AB|Z) = P(A|Z)P(B|Z) ( see Bell's equation (2) which is the same as eq 12 in Jaynes )
You are distorting Bell's claims again. He does not claim that as some sort of general rule, P(AB|Z) = P(A|Z)P(B|Z) for any arbitrary observations or facts A, B, and Z. Instead, he says that for the specific case where a and b represent the events of some experimenter's choices of what variable to measure on a given trial, we can assume that these choices are really "free" and were not predetermined by some common cause in the past which also determined the state of the hidden variables \lambda. And thus, in this specific case with a and b having that specific meaning, we can write the equality in equation (14) from the Jaynes paper you referenced (http://bayes.wustl.edu/etj/articles/cmystery.pdf):

P(AB | ab\lambda ) = P(A | a \lambda) P(B | b \lambda)

Jaynes does not disagree that this equation is correct if you make the assumption about a and b not being predetermined by factors that also determined \lambda, that's why he prefaces that equation by saying "But if we grant that knowledge of the experimenters' free choices (a,b) would give us no information about \lambda". If you want to question the assumption of "free choice" (which just means choices not determined by factors which also determined the hidden variables produced by the source on a given trial, they might be determined by other complex factors in the experimenter's brains prior to the choice), then go ahead, this is a known loophole in the proof of Bell's theorem. But don't act like Bell was making some very broad statement about probability that would be true regardless of what events/observations the symbols a and b are supposed to represent.
Remember the question is "What is the probability that both balls are red"?

Z: The premise that the urn contains two balls, one white and one red.
A: First ball is red
B: Second ball is red

Calculating based on that equation, the probability that both of those balls is red results in 0.5 * 0.5 or 0.25! Which is wrong!
And this is a strawman, since Bell never suggested such a broad equation that was supposed to work regardless of what the symbols represent. Try to think of an experiment where the symbol a represents the free choice of experimenter #1 of what measurement to perform (like which of the three boxes on the lotto card to scratch in my example), and b represents the free choice of experimenter #2 at some distant location (such that no signal moving at the speed of light can cross from the event of experimenter #1 making his choice/measurement to the event of experimenter #2 making his choice/measurement), and A represents the outcome seen by #1 while B represents the outcome #2, and \lambda represents some factors in the systems being measured that determine (or influence in a statistical way) what outcome each sees when they perform their measurement. With the symbols having this specific meaning, can you think of an experiment in a local realist universe where the equation

P(AB | ab\lambda ) = P(A | a \lambda) P(B | b \lambda)

would not work?
As you see, even though we accept that there is no physical causality from the second draw to the first draw, we still must include logical dependence to calculate the probabilities correctly.
Um, have you been ignoring my point all along that in a realist universe, statistical correlations between events are always either due to one event influencing the other or events in their common past which influenced (or predetermined) both? I don't think I was very subtle about the idea that there were two options here. You seem to have simply ignored my second option, which is a little suspicious because it's precisely the one that applies to the case of the second ball drawn from the urn (whose color is predetermined by the event of the first ball being picked from the urn, since the urn only contained two balls to begin with).
Again, when you interpret probabilities in frequentist/realist terms, all statistical correlations (which I guess is what you mean by 'logical dependence') must be explained in terms of physical causes, though the explanation may involve a common cause in the past of two events rather than either one directly influencing the other.
You are not reading what I write. I'll use a dramatic example.

"not Dead implies not Executed".
Do you agree with the above?

There is a logical dependence between "not Dead" and "not Executed". If a person is not dead, it MUST follow that the person is not Executed. However, you can not say "not Dead" physically causes "not Executed", otherwise nobody will ever be executed. Logical dependence is not the same as physical causation.
And you are not reading what I write, because I already explained that your overly narrow definition of "physical causation" is different from what I mean by the term. Read the end of post #11 again:
Why not? "Causality" just means that one physical fact determines another physical fact according to the laws of physics, and an absence of a certain physical event is still a physical fact about the universe, there's no reason it can't be said to "cause" some other fact.
Both "the prisoner is not dead" and "the prisoner was not executed" are physical facts which would be known by a hypothetical godlike being that knows every physical fact about every situation, and if this being looked at a large sample of prisoners, he'd find that for everyone to whom "not dead" currently applies, it is also true that "not executed" applies to their past history. So, according to my broad definition of "cause", it is certainly true that "not executed" is a necessary (but not sufficient) cause for the fact of being "not dead".
And Bell did not assume there'd be no statistical correlation between A and B--the whole point of including \lambda was to show there could be such a correlation
He must have. Equation (2) in his article, (12) in Jaynes, means just that.
Wow, you have really missed the most basic idea of the proof. No, of course (12) in Jaynes doesn't mean A and B are independent of \lambda, where could you possibly have gotten that idea? The equation explicitly includes the terms P(A | a \lambda) and P(B | b \lambda), that would make no sense if A and B were independent of \lambda! What equation (12) does show is that Bell was assuming A was independent of b, and B was independent of a. No other independence is implied, if you think it is you really need to work on your ability to read statistics equations.
in a local realist universe, as long as it was explained by the source creating both particles with correlated hidden variables (a common cause in the past), just like my example of sending balls to Alice and Bob and always making sure one was sent a black ball and the other was sent a white one, so their measurements results would always be opposite (here I play the role of the 'source' which determines the hidden variables of each box that determine the correlations between their observations when they open their respective boxes).
Maybe I should ask you a question. If you know the outcome at A, the settings at A and the settings at B will you be able to deduce the outcome at B? Isn't this the premise of any hidden variable theorem, that the outcome is determined by the values they left the source with and the settings at the terminals?
No, of course it isn't--you've completely left out the hidden variables here! A hidden-variables theory just says that the outcome A seen by experimenter #1 is determined by experimenter #1's choice of measurement a (like the choice of which box to scratch in my lotto card analogy in post #3 on this thread) combined with the hidden variables \lambda_1 associated with the system experimenter #1 is measuring (like the preexisting hidden fruits behind each box on the lotto card). Likewise, the outcome B seen by experimenter #2 is determined by experimenter #2's choice of measurement b along with the hidden variables \lambda_2 associated with the system experimenter #2 is measuring. If both experimenters always get the same result on trials where they both choose the same measurement, that must mean that the hidden variables associated with each system must predetermine the same outcome to any possible measurement, as long as you assume the source that's "preparing" the hidden variables on each trial has no foreknowledge of what the experimenters will choose (if it did have such foreknowledge, then it might only predetermine the same outcome to the same measurement on trials where the experimenters were, in fact, going to choose the same measurement to make).
Just like knowing the result of the first second draw should influence the way you calculate the probability of the second draw. Shouldn't it? How is it supposed to influence it if you do not have a P(A|B) term??
Because the correlation seen between results A and B is assumed to be purely a result of the hidden variables the source associated with each particle--a common cause in the past (again, in a local realist universe all correlations are understood either as direct causal relations or a result of common causes in the past, and A and B are supposed to have a spacelike separation which rules out a direct causal relation in a local realist universe). As long as you include a term for the dependence of both A and B on the hidden variables, there's no need for a separate term for the statistical correlation between A and B. Similarly, if I have an urn containing two reds and one white, and the first I pick is red, I can write the equation P(second ball seen to be white | first ball seen to be red) = 1/2; but if I explicitly include a term for the "hidden variables" associated with what's left in the urn on each pick, I can just rewrite this as:

P(second ball seen to be white | after first pick but prior to examination, first ball had 'hidden state' red and urn had 'hidden state' one red, one white) = 1

and

P(first ball seen to be red | after first pick but prior to examination, first ball had 'hidden state' red and urn had 'hidden state' one red, one white) = 1

which together with

P(after first pick but prior to examination, first ball had 'hidden state' red and urn had 'hidden state' one red, one white) = 1/2

imply the statement P(second ball seen to be be white | first ball seen to be red) = 1/2. And in general we can write the equation:

P(first ball seen to be red, second ball seen to be white) = [SUM OVER ALL POSSIBLE HIDDEN STATES X FOR URN + FIRST BALL AFTER FIRST PICK] P(first ball seen to be red | urn + first ball in hidden state X)*P(second ball seen to be white | urn + first ball in hidden state X)*P(urn + first ball in hidden state X)

...This is directly analogous to equation (12) from the Jaynes paper which you've told me is the same as (2) from Bell's paper.
Look at Bell's original article, DrChinese has it on his website. Everything starts from Eq. (2). Clearly from (2) there are only two options
1- Either bell did not understand how to apply probability rules or
2- He assumed that knowing the results at B should not influence the way we calculate the probability at A.
Either one is devastating to his result.
Nope, see above, in a local realist universe any correlation between A and B should be determined by the hidden state of each particle given to them by the source at the event of their common creation, so there is no need to include P(A | B) as a separate term, just like there's no need in my equation above for the urn as long as you include the hidden state of the urn + first ball after the first pick.

mn4j

Jan8-09, 09:18 AM

Nope, see above, in a local realist universe any correlation between A and B should be determined by the hidden state of each particle given to them by the source at the event of their common creation, so there is no need to include P(A | B) as a separate term, just like there's no need in my equation above for the urn as long as you include the hidden state of the urn + first ball after the first pick.

Read up on probability theory Jesse, what you say makes no sense. If you assume that there is a correlation between the two particles when they left the source, which you must if you are trying to model a local realist theorem, then you MUST also assume that there is logical dependence between the probability of the measurement at A and that at B. Bell did not do that. Remember Bell was trying to model a hidden variable theorem.

For example, suppose there is an urn with two red balls and one white ball, and the experiment on each trial is to pick two balls in succession (without replacing the first one before picking the second), and noting the color of each one. If I open my hand and see that the first one I picked was red, and then I look at the closed fist containing the other and guess if it'll be red or white, do you agree that I should conclude P(second will be white | first was red) = 1/2?

Write down the equations you used to arrive at your answers. Then we can talk.

mn4j

Jan8-09, 10:04 AM

If you agree with both these statements, then it shouldn't be hard to see how any statement about subjective probabilities in an objective universe should be mappable to a statement about the statistics seen by a hypothetical godlike observer in a large number of trials.

Again you apparently have a limited view of what probability means.
If you think there could be any exceptions--objectively true statements of probability which cannot be mapped in this way--then please give an example. It would be pretty earth-shattering if you could, because the frequentist interpretation of probabilities is very mainstream, I'm sure you could find explanations of probability in terms of the statistics over many trials in virtually any introductory statistics textbook.
I already did. What is the common event in the past of "not executed" and "not dead" which caused both? Yet I can assign a probability to P("not dead"|"not executed").

You are arguing against yourself. In a hidden variable local realist theorem, there was also an event in their past which predetermined the result of the measurement. So then their probabilities should be dependent and you should have a p(A|B) term.

What exactly do you mean by "logical dependence"? The probabilities can of course be calculated in the same frequentist manner as I discussed above--if you imagine a large number of trials of this type, it's certainly true that on the subset of trials where the first ball picked was white, the second ball was always red in 100% of this subset, and likewise in the subset of trials where the first ball picked was red, the second ball was always white in 100% of this subset.

Logical dependence is probably the first thing you learn in any philosophy or probability class. I have already explained in this thread what it means. Nobody said anything about not being able to calculate probabilities in a frequentist manner. The monkey did not perform the experiment 100 times. He did it once. Are you trying to say in such a case probabilities can not be assigned? Probability means much more than frequencies!!!

You are distorting Bell's claims again. He does not claim that as some sort of general rule, P(AB|Z) = P(A|Z)P(B|Z) for any arbitrary observations or facts A, B, and Z.
No I am not. If Bell's equation (2) is not a general rule, where does he get it from. What principle of mathematics or logic permits him to write down that equation the way he did??
Instead, he says that for the specific case where a and b represent the events of some experimenter's choices of what variable to measure on a given trial, we can assume that these choices are really "free" and were not predetermined by some common cause in the past which also determined the state of the hidden variables \lambda. And thus, in this specific case with a and b having that specific meaning, we can write the equality in equation (14) from the Jaynes paper you referenced (http://bayes.wustl.edu/etj/articles/cmystery.pdf):

P(AB | ab\lambda ) = P(A | a \lambda) P(B | b \lambda)

Jaynes does not disagree that this equation is correct if you make the assumption about a and b not being predetermined by factors that also determined \lambda, that's why he prefaces that equation by saying "But if we grant that knowledge of the experimenters' free choices (a,b) would give us no information about \lambda".

You are focussing on the wrong thing that is why you are not understanding Jaynes. If knowledge of the experimenters' free choices (a,b) gives us information about \lambda, then you must have a P(\lambda|ab) term in the equation. Jaynes can grant him that because he is focused on how knowledge of the RESULT at A is dependent on knownedge of the RESULT at B. That is why in the correct equation (15), you have a P(B|A) term or a P(A|B) term. Please read the article again carefully to see this.

If you want to question the assumption of "free choice" (which just means choices not determined by factors which also determined the hidden variables produced by the source on a given trial, they might be determined by other complex factors in the experimenter's brains prior to the choice), then go ahead, this is a known loophole in the proof of Bell's theorem. But don't act like Bell was making some very broad statement about probability that would be true regardless of what events/observations the symbols a and b are supposed to represent.
As pointed out above you are focusing on the wrong thing.

Both "the prisoner is not dead" and "the prisoner was not executed" are physical facts which would be known by a hypothetical godlike being that knows every physical fact about every situation, and if this being looked at a large sample of prisoners, he'd find that for everyone to whom "not dead" currently applies, it is also true that "not executed" applies to their past history. So, according to my broad definition of "cause", it is certainly true that "not executed" is a necessary (but not sufficient) cause for the fact of being "not dead".
As I suspected, your statement above tells me you do not know the deference between logical dependence and physical dependence.

What equation (12) does show is that Bell was assuming A was independent of b, and B was independent of a. No other independence is implied, if you think it is you really need to work on your ability to read statistics equations.
This is wrong. While equation (12) shows that KNOWLEDGE of A is independent of KNOWLEDGE of b, it also shows that KNOWLEDGE of A is independent of KNOWLEDGE of B. According to the rules of probability it MUST be so. Readup on "product rule", P(AB)=P(A|B)P(B).

If a source releases two electrons, one spin up and spin down, and we measure one at A and find it to be up, then we can predict with certainty that the one at B will be down. Now try to write down the equations for this exact, very simple two electron case, without any settings at the stations and a single hidden variable (spin). If A is not logically dependent on B and vice versa, how come when you do not no either outcome, the P(A) = 1/2, and p(B) = 1/2 but when you know the out come the A is "up", P(A) = 1.0 and p(B)=0. There is only one set of equations that will give you these results consistently and that is the product rule. Bell's equation (2) is not valid. Unless you are trying to say the example is not local realist.

Because the correlation seen between results A and B is assumed to be purely a result of the hidden variables the source associated with each particle--a common cause in the past (again, in a local realist universe all correlations are understood either as direct causal relations or a result of common causes in the past
So you admit that A and B should be logically dependent. Do you agree that if KNOWING A should give you more information about B?

mn4j

Jan8-09, 10:25 AM

Z: The premise that the urn contains three balls, one white and two red. Two balls are picked in succession without replacing.
A: First ball is red
B: Second ball is red

The correct equation is
p(AB|Z) = p(A|B)p(A|Z)

from Z we can calculate p(A|Z) = [number of red balls]/[total number of balls] = 2/3
p(A|B) = [number of red balls - 1 red ball]/[total number of balls - 1 red ball] = 1/2.

therefore p(AB|Z) = 2/3 * 1/2 = 1/3

Let's add another premise:

C: second ball is white.
The correct equation is p(AC|Z) = p(A|C)p(A|Z), notice that the equation is exactly the same except for the meaning of the terms. The rules of probability do not change on a case by case basis.

p(A|Z) = [number of red balls]/[total number of balls] = 2/3
p(A|C) = [number of red balls]/[total number of balls - 1 white ball] = 2/2 = 1.0. You don't subtract 1 white ball from the numerator because taking out a white ball does not change the number of red balls still in consideration.

therefore p(AC|Z) = 2/3 * 1.0 = 2/3.

Do you disagree with these equations?

DrChinese

Jan8-09, 10:31 AM

If the equations on which his conclusions are based are wrong then the conclusions are baseless. Just because you prove it differently does not mean your equations are correct. For example, where in your eqations do you account for logical dependence? If you think logical dependence is not necessary then what you are modeling is not a local-realist system. No doubt the results contradict real experiments.

You really aren't saying anything meaningful. It should be glaringly obvious, once you have seen Bell's Theorem, that you cannot model a local realistic (HV) photon to operate in a manner consistent with experiment (cos^2(theta)). You can choose to find your way to that position any of a number of ways. It's sort of like discovering there is no Easter Bunny. Once you learn, you can't go back.

And it is not logical dependence that needs to be considered; it is the independence of an observation at one point with an observation at another spacelike separated point. Really, what is so difficult about that? What we are talking about observing is the "DNA" of twin photons, which clearly act in an entangled fashion - in opposition to the local realistic model. Are you trying to be obtuse? As said before, you are completely missing the big picture by your focus on meaningless semantics. If you don't like Bell's method of representation, simply change it so the effect (and conclusion) is the same. It's not THAT hard because a jillion (perhaps a slight exaggeration) others have already passed down that road.

Don't throw the baby out with the bathwater. You can solve this "issue" yourself. Remind yourself that Bell was essentially a reply to EPR. See Einstein's quoted words above (which of course were pre-Bell). Were Einstein alive, he would be forced to concede the point. And he certainly wouldn't be mumbling something about logical vs. physical dependence.

DrChinese

Jan8-09, 10:47 AM

Unless you are trying to say the example is not local realist.

Your example is ridiculous. The perfect correlation argument was known in 1935, and has never been an issue as this must appear first in HV models. Model the Bell inequality violation in a local realist fashion and there might be something to discuss. (QM already predicts as experiment finds.) There are new models being put forth every week, in fact I believe you have referenced one such previously. Typically, they try to show that photon visibility issues lead to S>2 [CHSH inequality form] but fall apart with ever-increasing experimental precision.

JesseM, I am bowing out as it appears that our friend is stuck in a meaningless philosophical time warp. Best of luck!

JesseM

Jan8-09, 12:10 PM

Z: The premise that the urn contains three balls, one white and two red. Two balls are picked in succession without replacing.
A: First ball is red
B: Second ball is red

The correct equation is
p(AB|Z) = p(A|B)p(A|Z)
Shouldn't that be p(AB|Z) = p(B|A)p(A|Z)? In other words, you want to find the probability of A given Z, then the probability of B given A.
from Z we can calculate p(A|Z) = [number of red balls]/[total number of balls] = 2/3
p(A|B) = [number of red balls - 1 red ball]/[total number of balls - 1 red ball] = 1/2.

therefore p(AB|Z) = 2/3 * 1/2 = 1/3
Of course p(B|A) is also 1/2, so the error I noted above gives the same final answer for p(AB|Z).
Let's add another premise:

C: second ball is white.
The correct equation is p(AC|Z) = p(A|C)p(A|Z), notice that the equation is exactly the same except for the meaning of the terms. The rules of probability do not change on a case by case basis.
Again, the correct equation should actually be p(AC|Z) = p(C|A)p(A|Z)
p(A|Z) = [number of red balls]/[total number of balls] = 2/3
p(A|C) = [number of red balls]/[total number of balls - 1 white ball] = 2/2 = 1.0.
P(C|A) is the correct term to use in the equation, and it's 1/2.
You don't subtract 1 white ball from the numerator because taking out a white ball does not change the number of red balls still in consideration.

therefore p(AC|Z) = 2/3 * 1.0 = 2/3.
With the correct equation, you'd have p(AC|Z) = 2/3*1/2 = 1/3. Think about it and you'll see this is correct--if you do a lot of trials, on 2/3 of the total trials the first ball will be red (A), and in 1/2 of the subset of trials where the first ball was red the second will be white (C), so on 2/3*1/2 = 1/3 of all trials you'll find that the first was red while the second is white.
Do you disagree with these equations?
See my corrections. But note that even the corrected equations I gave, these are not the only possible correct equations according to the rules of probability. In particular, the equations above say nothing about the hidden state of the first ball and the urn after the first pick, which in causal terms explains all correlations between the observed color of the first and second balls, and which is analogous to the hidden variables in any local realist explanation for the correlations seen in entanglement experiments.

Let H = Hidden color of first ball after first pick is made + hidden colors of other balls in urn after first pick
A = first ball is red
B = second ball is white

Then we can write P(AB) = [SUM OVER ALL POSSIBLE H] P(A|H)P(B|H)*P(H)

Likewise, if we let C = second ball is red, then we have:

Then we can write P(AC) = [SUM OVER ALL POSSIBLE H] P(A|H)*P(C|H)*P(H)

Do you disagree with these equation, which are directly analogous to equation 12 in Jaynes' paper? (just substitute \lambda for H, and substitute an integral over \lambda for a sum over H, and add terms for the experimenter's choices of what to measure which don't matter in the ball/urn scenario, although you could think of another classical example where the experimenter can make a choice like my scratch lotto card example)

The "sum over all possible H" is pretty simple in a case with only three balls in the urn initially, since the only possible states for H are:

H1: first ball has hidden color red, urn contains two balls with hidden colors red and white
H2: first ball has hidden color white, urn contains two balls with hidden colors red and red

So the first equation is really just P(AB) = P(A|H1)*P(B|H1)*P(H1) + P(A|H2)*P(B|H2)*P(H2)
and the second equation is P(AC) = P(A|H1)*P(C|H1)*P(H1) + P(A|H2)*P(C|H2)*P(H2)

P(H1) = 2/3 and P(H2) = 1/3. Meanwhile P(A|H1) = 1, P(B|H1) = 1/2, and P(A|H2) = 0 and P(B|H2) = 0. Likewise P(A|H1) = 1, P(C|H1) = 1/2, and P(A|H2) = 0 and P(C|H2) = 1.

So, we have P(AB) = 1 * 1/2 * 2/3 + 0 * 0 * 1/3 = 1/3. Likewise, P(AC) = 1/3.

Any disagreement with the above?

JesseM

Jan8-09, 01:31 PM

Again you apparently have a limited view of what probability means.
I don't say that the frequentist way of interpreting probabilities is the only way to interpret them, just that it is a way. If you refuse to think in frequentist terms, it is you who has the more limited view. Do you disagree that any correct statement about subjective probabilities can be mapped onto a statement about the statistics of a large number of trials? If you do disagree, give me a single example of a correct statement about subjective probabilities that can't be mapped in this way.
If you think there could be any exceptions--objectively true statements of probability which cannot be mapped in this way--then please give an example. It would be pretty earth-shattering if you could, because the frequentist interpretation of probabilities is very mainstream, I'm sure you could find explanations of probability in terms of the statistics over many trials in virtually any introductory statistics textbook.
I already did. What is the common event in the past of "not executed" and "not dead" which caused both? Yet I can assign a probability to P("not dead"|"not executed").
First of all, you're confusing my request for a counterexample to the claim "every correct probability claim can be turned into a statement about the statistics that would be expected over many trials (whether or not many trials are actually performed)" with my request for a counterexample to the claim "every correlation can be explained by either one event causing the other or both events having a common cause". I was asking about the first one in the question above, yet you appear to be talking about causality in your response.

Second, your prisoner example is not a counterexample to either. It's easy to turn it into a statement about the statistics of a large sample of prisoners--for every prisoner, we can check if "not dead" applies to that prisoner currently, and we can also check if "not executed" applies to that prisoner's history. In frequentist terms, P(not dead|not executed) just means looking at the subset of prisoners who were not executed, and seeing what fraction are not dead. As for causality, an event A can be defined as a cause of B if A is in the past of B, if the occurrence of A has some positive effect on the probability of B over a large number of trials, and if this positive effect on the probability cannot be entirely explained in terms of some other event(s) C in the past of both A and B. In this sense, "not executed" is a direct cause of "not dead", and I said that for in a local realist universe any correlation can either be explained as one event directly causing the other or both events having a common cause.
You are arguing against yourself. In a hidden variable local realist theorem, there was also an event in their past which predetermined the result of the measurement. So then their probabilities should be dependent and you should have a p(A|B) term.
Nope, you just haven't thought things through very well. See my equation involving the urn above:

P(AB) = [SUM OVER ALL POSSIBLE H] P(A|H)*P(B|H)*P(H)

This equation gives the correct probabilities for P(AB) where A represents the observed observed color on the first pick and B represents the color on the second pick, and H represents the hidden state of the ball on the first pick before it's examined, along with the hidden state of all the other balls in the urn immediately after the first ball is removed. Note that there is no P(A|B) term.
Logical dependence is probably the first thing you learn in any philosophy or probability class. I have already explained in this thread what it means. Nobody said anything about not being able to calculate probabilities in a frequentist manner. The monkey did not perform the experiment 100 times. He did it once. Are you trying to say in such a case probabilities can not be assigned?
No, of course not, the large number of trials need not be actually realized as long as we can figure out logically what the statistics of a large number of trials would be if we (hypothetically) repeated the experiment many times.
No I am not. If Bell's equation (2) is not a general rule, where does he get it from. What principle of mathematics or logic permits him to write down that equation the way he did??
Again, Bell's equation (2) is almost identical to my equation:

P(AB) = [SUM OVER ALL POSSIBLE H] P(A|H)*P(B|H)*P(H)

And this equation does not work generally, but it does work in the specific case I was describing where any correlation between A and B is fully explainable in terms of the hidden variables H. Do you see any P(A|B) terms in there?
You are focussing on the wrong thing that is why you are not understanding Jaynes. If knowledge of the experimenters' free choices (a,b) gives us information about \lambda, then you must have a P(\lambda|ab) term in the equation.
His "fundamentally correct" equation (13) does have such a term. But I think I see your point, Jaynes is just saying that the fact that \lambda is independent of a,b allows us to substitute p(\lambda) in for p(\lambda | ab) in (13), but that still does not justify (12) without the additional substitution of (14), and it is this that Jaynes disagrees with, saying the correct substitution would instead look like (15). Is that what you're saying? If so, I think you're right, I have misunderstood what Jaynes is arguing somewhat. However, I disagree with his point. Again, in a local realist universe any statistical correlation must be explainable in terms of events either influencing one another or being mutually influenced by events in their common past. To the extend that there is a correlation between A and B--what you call "logical dependence"--then it cannot be due to any direct influence of outcome A on outcome B or vice versa, since there is a spacelike separation between them, so it must be explained by both being determined (or influenced) by events in their common past, and this influence is entirely expressed in terms of the "hidden variables" \lambda associated with both particles, imparted to them by the source that created them. It's exactly like the example where I send a ball to Alice and a ball to Bob, and I make sure that they always receive opposite-colored balls--here there is also a logical dependence between what Bob sees when he opens his box and what Alice sees when she opens hers, but this is explained entirely by the fact that I always prepare the "hidden states" of each box (i.e. the hidden color of the ball inside before the box is opened) in such a way that they will yield opposite results. If you explicitly include the hidden state in your equation, there is no need to include any additional terms of the form p(A|B). You can also see this in the urn example, where the "hidden state" of the first ball and the remaining balls in the urn after the first pick is made is what explains any correlations between the observation of the color of the first ball and the observation of the color of the second ball...there were no terms of the form p(A|B) in my equation:

P(AB) = [SUM OVER ALL POSSIBLE H] P(A|H)*P(B|H)*P(H)

...and they weren't needed, for exactly this reason.
Both "the prisoner is not dead" and "the prisoner was not executed" are physical facts which would be known by a hypothetical godlike being that knows every physical fact about every situation, and if this being looked at a large sample of prisoners, he'd find that for everyone to whom "not dead" currently applies, it is also true that "not executed" applies to their past history. So, according to my broad definition of "cause", it is certainly true that "not executed" is a necessary (but not sufficient) cause for the fact of being "not dead".
As I suspected, your statement above tells me you do not know the deference between logical dependence and physical dependence.
Or maybe I am just defining my terms a little differently from you. Do you think my way of defining things is unreasonable? Is there not a causal relation between the fact that the prisoner wasn't executed and the fact that he's not dead?

Even if you prefer to define this as "logical dependence", all that would mean is that I'd have to slightly modify my claim about correlations in a local realist universe. Leaving aside all talk of physical causality, I could say that any time two variables A and B have a correlation, then it must be true either that A lies in the past light cone of B or vice versa, or it could be true that there is some other event or events C which lie in the overlap of the past light cones of A and B, such that the correlation between A and B is fully explained by C (i.e. if you know the value of C on a given trial, and you know the logical dependence between A and C as well as the logical dependence between B and C, then this fully determines any correlations between A and B). Do you disagree that something like this must be true in a local realist universe? If you disagree, try to provide a specific counterexample, and I'll try to explain what the C is in the past light cone of the correlated events A and B that you provide which fully determines how they are correlated (assuming there is a spacelike separation between A and B, if not than no such past event is necessary according to my claim above).
This is wrong. While equation (12) shows that KNOWLEDGE of A is independent of KNOWLEDGE of b, it also shows that KNOWLEDGE of A is independent of KNOWLEDGE of B. According to the rules of probability it MUST be so. Readup on "product rule", P(AB)=P(A|B)P(B).
No, it just says that any dependence between your knowledge of A and B must be fully explainable in terms of what knowledge of A tells you about \lambda, which in turn can inform your knowledge of B. Once again, there is certainly some correlation between A and B in the urn example, but since this correlation is entirely explainable in terms of the hidden state H after the first ball was picked, the equation P(AB) = [SUM OVER ALL POSSIBLE H] P(A|H)*P(B|H)*P(H) works just fine despite lacking any P(A|B) terms.
If a source releases two electrons, one spin up and spin down, and we measure one at A and find it to be up, then we can predict with certainty that the one at B will be down. Now try to write down the equations for this exact, very simple two electron case, without any settings at the stations and a single hidden variable (spin). If A is not logically dependent on B and vice versa, how come when you do not no either outcome, the P(A) = 1/2, and p(B) = 1/2 but when you know the out come the A is "up", P(A) = 1.0 and p(B)=0. There is only one set of equations that will give you these results consistently and that is the product rule. Bell's equation (2) is not valid. Unless you are trying to say the example is not local realist.
No, this example works fine in a local realist universe. You can assume that immediately on being created, the electrons have either hidden state H1 = "electron going in direction of A is up, electron going in direction of B is down", or hidden state H2 = "electron going in direction of A is down, electron going in direction of B is up". In this case we can say that P(AB) = P(A|H1)*P(B|H1)*P(H1) + P(A|H2)*P(B|H2)*P(H2). From this you will be able to show that for A=up and B=down, P(AB)=1/2, and likewise for A=down and B=up, whereas for A=up and B=up you have P(AB)=0 and likewise for A=down and B=down. The correlation between A and B is fully explained by their dependence on the hidden state, there's no need to include any P(A|B) terms in the equation.
So you admit that A and B should be logically dependent. Do you agree that if KNOWING A should give you more information about B?
Yes, of course. But as I say, that doesn't mean you actually need a P(B|A) term in an equation for P(AB).

mn4j

Jan8-09, 02:15 PM

Shouldn't that be p(AB|Z) = p(B|A)p(A|Z)? In other words, you want to find the probability of A given Z, then the probability of B given A.

That is correct. My bad. Still you have a P(B|A) term and a P(C|A) term. It doesn't change what I am saying.

Let H = Hidden color of first ball after first pick is made + hidden colors of other balls in urn after first pick
A = first ball is red
B = second ball is white

Why did you change the formulation. Your's has problems. You are trying to eliminate the prior information in Z. The H you have introduced is an outcome because it changes every time a pick is made.

Then we can write P(AB) = [SUM OVER ALL POSSIBLE H] P(A|H)P(B|H)*P(H)

Conditional probabilities do not work like that. What is the condition that AB is premised on? In other words, what is your hypothesis space? You must include it in the equation. What you call H is not a hypothesis space.
The consequence is that P(H) is a meaningless term in your equation -- "The probability that 'hidden color of first ball after first pick is made plus hidden colors of other balls after first pick is made'" makes no sense.

mn4j

Jan8-09, 02:26 PM

P(AB) = [SUM OVER ALL POSSIBLE H] P(A|H)*P(B|H)*P(H)

Again I am waiting for you to show where you got this equation. What rule or theory of mathematics or logic permits you to write this equation the way you did. Bell did not give a citation and neither have you. Jaynes equations are the result of the well known and established product rule of probability theory. What is the equation you or Bell is using?

JesseM

Jan8-09, 02:38 PM

Why did you change the formulation. Your's has problems. You are trying to eliminate the prior information in Z. The H you have introduced is an outcome because it changes every time a pick is made.
If you wish you can include Z as a separate condition, but every valid H will already include the total number of red and white balls.
Conditional probabilities do not work like that. What is the condition that AB is premised on? In other words, what is your hypothesis space? You must include it in the equation. What you call H is not a hypothesis space.
As I said, Z can be added if you want, it would just convert the equation into this:

Then we can write P(AB|Z) = [SUM OVER ALL POSSIBLE H] P(A|HZ)P(B|HZ)*P(H|Z)

However, it is not generally necessary to include the hypothesis space explicitly as a variable, as long as it is understood what it is from the definition of the problem. For example, if I am talking about flipping a coin, I am free to just write P(heads) rather than P(heads|a coin that can land on either of two sides, heads or tails).
The consequence is that P(H) is a meaningless term in your equation -- "hidden color of first ball after first pick is made plus hidden colors of other balls after first pick is made" makes no sense.
That's because "hidden color of first ball after first pick is made plus hidden colors of other balls after first pick is made" is not a phrase representing a particular description of the hidden colors, any more than \lambda is supposed to represent a particular description of the value of the hidden variables. Both H and \lambda are variables which can take multiple values. For example, H can take the value H1, where H1 means "first ball has hidden color red, urn contains two balls with hidden colors red and white". It can also take the value H2, "first ball has hidden color white, urn contains two balls with hidden colors red and red". Hopefully you would agree that P(H1) and P(H2) are both meaningful probabilities.

What did you think "sum over all possible H" was supposed to mean, anyway? Did you totally miss the part where I explicitly broke down the equation "P(AC) = [SUM OVER ALL POSSIBLE H] P(A|H)*P(C|H)*P(H)" into a sum including H1 and H2? Read this part again:
The "sum over all possible H" is pretty simple in a case with only three balls in the urn initially, since the only possible states for H are:

H1: first ball has hidden color red, urn contains two balls with hidden colors red and white
H2: first ball has hidden color white, urn contains two balls with hidden colors red and red

So the first equation is really just P(AB) = P(A|H1)*P(B|H1)*P(H1) + P(A|H2)*P(B|H2)*P(H2)

and the second equation is P(AC) = P(A|H1)*P(C|H1)*P(H1) + P(A|H2)*P(C|H2)*P(H2)

mn4j

Jan8-09, 04:06 PM

P(AC) = [SUM OVER ALL POSSIBLE H] P(A|H)*P(C|H)*P(H)"
It appears you are looking at multiple experiments that is why you are summing over different H. But the result you get by summing over multiple experiments can not be correct if you are not treating the individual experiment properly. So focus on just one experiment for now and once that is settled, you can then sum over all the experiments.

Let us take what you call the H1 case (first ball has hidden color red, urn contains two balls with hidden colors red and white).
Compare this with my initial premises:

Z: The premise that the urn contains three balls, one white and two red. Two balls are picked in succession without replacing.
A: First ball is red
B: Second ball is red
As you see, H is not necessary because it is just A.

In any case, even if we assume that H is not the same as A, you say the correct equation is:
P(AB|HZ) = P(A|HZ)P(B|HZ)*P(H|Z)

I say it should be according to the product rule of probability theory.
P(AB|HZ) = P(A|BHZ)*P(B|HZ)

If you want to sum over all H, this is the equation you should use. Where did you get your equation from??? Again we are looking at a the single experiment probability which is everything right of the intergral sign in bell's equation(2), eq. 12 in Jaynes.

In other words, can you prove that equation, or point me to a textbook or article where it is proven??? If you want a formal proof of the product rule, look at Chapter 1 of Jaynes book. "Probability Theory: The Logic of Science" which is available for free here: http://bayes.wustl.edu/etj/prob/book.pdf

JesseM

Jan8-09, 04:28 PM

Again I am waiting for you to show where you got this equation. What rule or theory of mathematics or logic permits you to write this equation the way you did. Bell did not give a citation and neither have you. Jaynes equations are the result of the well known and established product rule of probability theory. What is the equation you or Bell is using?
The equation just comes from thinking about the statistics of this particular problem, but I'll spell it out a bit. Do you agree that if X and Y are statistically independent, then P(XY)=P(X)*P(Y)? If so, would you also agree that even if X and Y are not independent in the sample space as a whole, if they are independent in the subset of trials where Z occurred, then we can write P(XY|Z)=P(X|Z)*P(Y|Z)? Clearly this sort of thing applies if you consider statements about the hidden state of the first ball and urn after the first drawing, like H1 and H2. If you already know H1, "first ball has hidden color red, urn contains two balls with hidden colors red and white", then knowing A (that the first ball was observed to have the color red when it was revealed) gives you no further knowledge about the likelihood that the second ball will be red (B), and likewise knowing B gives you no further knowledge about the likelihood that the first ball will be observed to be red (in fact knowing H1 leads you to predict A with probability 1).

So, it's reasonable that P(AB|H1) = P(A|H1)*P(B|H1) in this problem (not as a general statement about probabilities in arbitrary problems), and likewise for any other hidden state that might exist immediately after the first ball is picked. It's also reasonable that if every point in the sample space has some H_i associated with it (i.e. on every trial there is some fact of the matter of what the hidden color of the first ball was after it was picked, and what hidden colors remained in the urn), and all the different H_i's are mutually exclusive, then we can write P(AB) = \sum_i P(AB|H_i)*P(H_i). Do you disagree with that? If not, it's easy to combine these equations to get P(AB) = \sum_i P(A|H_i)P(B|H_i)P(H_i).

So if there is some point of disagreement here, please be specific about where it is. Also, if you disagree, please play around with the above equation a little and see if you can find any urn examples where the equation would give the wrong numbers for the probabilities.

mn4j

Jan8-09, 08:26 PM

Do you agree that if X and Y are statistically independent, then P(XY)=P(X)*P(Y)?

Of course. Working from the product rule
P(XY|Z) = P(X|Z)*P(Y|Z)
the reason P(XY|Z) = P(X|Z)*P(Y|Z) in the case of logical independence is because P(Y|XZ) = P(Y|Z). This means that knowledge of X gives us no additional information about Y, so the equation reduces to P(XY|Z) = P(X|Z)*P(Y|Z).

If so, would you also agree that even if X and Y are not independent in the sample space as a whole, if they are independent in the subset of trials where Z occurred
This makes no sense. If you write P(XY|Z) = P(X|Z)*P(Y|Z), you are saying that based on all the information Z, X is logically independent of Y. The reason this is false in this particular case is precisely because there is logical dependence.

To see this you only need to ask the question, does knowing the spin at A tell me anything about the spin at B? If it does, then they are logically dependent and you can NOT reduce P(Y|XZ) to P(Y|Z). Now remember we are still talking about a specific experiment NOT repeated events. The fact that the experiment can be repeated is inconsequential.

The definitive premise in any hidden variable theorem is that the particles left the source with enough information to determine the outcome at the stations (together with the settings at the station). This means for two entangled particles, any information you gain about one, MUST tell you something about the other. This was the basis of the EPR paradox which Bell was trying to refute. So logical dependence is a MUST in the specific case Bell is treating. Therefore there is no justification for why he reduced P(Y|XZ) to P(Y|Z). Do you understand this?

JesseM

Jan8-09, 09:44 PM

This makes no sense. If you write P(XY|Z) = P(X|Z)*P(Y|Z), you are saying that based on all the information Z, X is logically independent of Y.
I'm not sure what you mean by the phrase "based on all the information Z, X is logically independent of Y". The equation P(XY|Z)=P(X|Z)*P(Y|Z) certainly doesn't mean X is logically independent of Y in the series of trials as a whole, it just means that in the subset of trials where Z occurred, the frequency of X is independent of Y and vice versa. Another way of putting this is that the statistical dependence of X and Y is wholly accounted for by Z, meaning that if you know Z, then knowing X gives you no additional information about Y, and likewise if you know Z, knowing Y gives you no additional information about X.

Consider the urn example again. If I've already learned that H1 is true (that the first ball selected had the hidden color red prior to being observed) and I want to bet on whether or not B is true (that the second ball selected will be observed to be red), do you agree that knowing A (that the first ball selected was observed to be red) tells me absolutely nothing new that should cause me to modify my bet when all I knew was H1?

Of course this example is somewhat trivial since H1 implies A with probability 1. But you could consider a more complicated example too. Suppose that instead of colored balls, the urn contains three plastic balls with dimmed lights inside, and that pressing a button on top of each ball causes either the internal red light to turn on, or the internal green light. The balls are externally indistinguishable but their hidden internal mechanisms are somewhat different, so to distinguish them we'll call the three balls D, E and F. Each ball has a random element that decides which light will go on when the button is pressed, but because of the different hidden mechanisms the probabilities are different in each case--D has a 70% chance of lighting up red, E has a 90% chance of lighting up red, and F has a 25% chance of lighting up red. Two balls are taken from the urn, one is given to Alice and the second is given to Bob, then they go into separate rooms and press the buttons and note the results.

Now, since the balls are taken from the same urn without replacement, there is some statistical dependence between the results seen by Alice and the results seen by Bob--for example, if Alice sees her ball light up green, that makes it more likely that the she got ball F, which in turn makes it less likely that Bob got F and thus more likely that Bob will see his light up red. Now consider the "hidden variables" condition H1: "ball D was given to Alice, ball E was given to Bob". Do you agree that if we look at the subset of trials where H1 is true, then in this subset, knowing that Alice's ball lit up red gives us no additional information about the likelihood that Bob's ball will light up red? In other words, P(Bob's ball lights up red | ball D given to Alice, ball E given to Bob) = P(Bob's ball lights up red | ball D given to Alice, ball E given to Bob AND Alice's lit up red) = P(Bob's ball lights up red | ball D given to Alice, ball E given to Bob AND Alice's lit up green) = 90%. The fact that Bob got ball E is enough to tell us there's a 90% chance his will light up red, knowing the color Alice saw doesn't further affect our probability calculation at all. In the same way, knowing that Alice got ball D tells us there's a 70% chance hers will light up red, knowing the result Bob got doesn't change the probability further either. So, P(Alice saw red, Bob saw red|H1) = P(Alice saw red|H1)*P(Bob saw red|H1) = 0.9*0.7=0.63. It's only if we don't know the hidden truth about which ball each one got that knowing what color one of them saw can influence our estimate of the probability the other one saw red. Do you disagree with any of this? If you do, please tell me what you would give for the probability P(Alice saw red, Bob saw red|H1) along with the probabilities P(Alice saw red|H1) and P(Bob saw red|H1).
The definitive premise in any hidden variable theorem is that the particles left the source with enough information to determine the outcome at the stations (together with the settings at the station). This means for two entangled particles, any information you gain about one, MUST tell you something about the other.
Of course. But if God came and told you the hidden state \lambda associated with each particle, then in that case knowing the information gained by one experimenter tells you nothing additional about the result the other experimenter will see, just like if God told Alice that she had gotten ball D, then knowing Bob had seen his ball light up red would tell her nothing additional about the probability her own would light up red. That's why for every possible hidden state H (H1, H2, etc.), it is true that P(Alice sees red, Bob sees red|H) = P(Alice sees red|H)*P(Bob sees red|H). Similarly, in a hidden variables theory of QM that's why if you include the complete set of hidden variables \lambda in the conditional, in that case you can write P(AB|\lambda) = P(A|\lambda)*P(B|\lambda)--you're only looking at the subset of trials where \lambda took on some particular value. But I know you'll probably find reason to dispute this, which is why I want to first deal with the simple example of the light-up balls, and see if in that case you dispute that P(Alice sees red, Bob sees red|H) = P(Alice sees red|H)*P(Bob sees red|H).

mn4j

Jan8-09, 10:29 PM

I'm not sure what you mean by the phrase "based on all the information Z, X is logically independent of Y". The equation P(XY|Z)=P(X|Z)*P(Y|Z) certainly doesn't mean X is logically independent of Y in the series of trials as a whole

You keep mixing up consideration of an individual experiment with consideration of multiple experiments that is why you are confused. We are not yet talking about multiple experiments, that is where the intergral comes in. We are talking about A SINGLE experiment!

Secondly, if P(XY|Z)=P(X|Z)*P(Y|Z) does not mean X is logically independent of Y, then where did you get the equation from. I have already shown you how you can derive P(XY|Z)=P(X|Z)*P(Y|Z) from the product rule P(XY|Z)=P(X|YZ)*P(Y|Z) WHEN AND ONLY WHEN X is logically independent of Y. You have not shown how you can arrive at that equation without assuming logical independence.

This is very basic probability theory. Unless you understand this, we can not even begin to discuss about hidden variables and multiple experiments.

JesseM

Jan8-09, 11:04 PM

You keep mixing up consideration of an individual experiment with consideration of multiple experiments that is why you are confused. We are not yet talking about multiple experiments, that is where the intergral comes in. We are talking about A SINGLE experiment!
As I already explained, the probability of X on a single experiment is always guaranteed to be equal to the fraction of trials on which X occurred in a large number (approaching infinity) of repeats of the experiment with the same conditions--it doesn't matter if a large number of trials actually happens, all that matters is that this would be true hypothetically the experiment were repeated in this way. Do you disagree with this? Please give me a simple yes or no answer.
Secondly, if P(XY|Z)=P(X|Z)*P(Y|Z) does not mean X is logically independent of Y, then where did you get the equation from.
It comes from the fact that if you only look at the trials where Z occurred and throw out all the trials where it did not, then in that set of trials X is independent of Y. It would help if you addressed my simple numerical example:

Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance.

H1 is the condition that "ball D was given to Alice, ball E was given to Bob".

Do you agree that in this case, P(Alice saw red, Bob saw red|H1) = 0.63, exactly the same as P(Alice saw red|H1)*P(Bob saw red|H1)? If not, what probability would you assign to P(Alice saw red, Bob saw red|H1)? Please give me a specific answer to this question.
I have already shown you how you can derive P(XY|Z)=P(X|Z)*P(Y|Z) from the product rule P(XY|Z)=P(X|YZ)*P(Y|Z) WHEN AND ONLY WHEN X is logically independent of Y. You have not shown how you can arrive at that equation without assuming logical independence.
Of course I have. As I already explained, I am simply assuming that X is logically independent of Y in the subset of trials where Z occurs, even though it they are not independent in the set of all trials. Another way of putting this is that if we know Z occurred, then knowing X tells us nothing further about the probability of Y, even though in the absence of knowledge of Z knowledge of X would have given us some new information about the probability of Y. You apparently are having problems understanding this explanation, but I can't help you unless you are willing to actually think about some specific examples like the one I ask above. Can you please just do me the courtesy of answering my simple questions about these examples?

Here's another example--we have a 6-sided die and a 12-sided die, Alice rolls one and Bob rolls the other. If there's a 50/50 chance the 12-sided die will be given to either one of them, and we aren't told who got which die, then the results they see aren't logically independent--for example, if Alice gets a 10 that allows us to infer Bob will get some number between 1 and 6, whereas before we knew what Alice rolled we would have considered it possible Bob could get any number from 1-12. However, in the subset of cases where Alice gets the 12-sided die and Bob gets the 6-sided die (or if you prefer, in a single case where we know Alice got the 12-sided die and Bob got the 6-sided die), Bob has a probability 1/6 of getting any number 1-6, and a probability 0 of getting a number 7-12, and knowing what Alice rolled has absolutely no further effect on these probabilities. Do you disagree? Again, please think about this specific example and answer yes or no.

mn4j

Jan9-09, 01:05 PM

mn4j

Jan9-09, 01:52 PM

I decided to to post the question as a separate post and changed it a bit so that the result is clearer.

Now answer this one.

Z: Two red LEDs D, E were made on a circuit so that when D was observed to be lit, the probability of E lighting up was 0.2 and when E was lit the probability of D lighting up was 0.2. Also, the circuit was designed such that at least one LED was lit on each button press with no other bias imposed on the LED other than the correlation above. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

What probability will you assign to P(AB|Z).

JesseM

Jan9-09, 02:40 PM

As I already explained, the probability of X on a single experiment is always guaranteed to be equal to the fraction of trials on which X occurred in a large number (approaching infinity) of repeats of the experiment with the same conditions
Yes. This makes sense provided your sampling is unbiased. However, this point is irrelevant here. We are not arguing about how Bell calculated P(X|Z), we are arguing about how he calculated P(XY|Z) to be P(X|Z)*P(Y|Z) for a single experiment.
Yes, but P(XY|Z) can easily be made into a statement about a large number of trials too--just do the experiment many times with the same conditions, then look at the subset of trials where Z did in fact occur, and look at the fraction of these in which X and Y occurred as well. Do you disagree that this fraction should always be the same as P(XY|Z)?

If you don't disagree, then again, my point is just that you can have a situation where X is independent of Y in this subset, even though it is not independent of Y in the complete set of all trials.
Again, when you write P(XY|Z)=P(X|Z)*P(Y|Z), Z is not considered as an outcome, but as a premise.
That's not exactly true, we're still dealing with the same larger sample space that includes events in which Z did not occur. It's just that when we write P(XY|Z), we want to look only at the subset of points in the sample space where Z did occur, and then look at how frequently X and Y occurred in this subset. Of course this is logically equivalent to just defining a new sample space that includes only events where Z occurred, and looking at how frequently X and Y occur in the complete sample space (i.e. P(XY) in this smaller sample space which includes only events where Z occurs is equal to P(XY|Z) in the larger sample space which includes events where Z does not occur).
P(AB|Z) means: Given that Z is true, what is the probability that both A and B are True.
Let's look at the Bell's specific case.
Z: Two spin 1/2 particles denoted by 1 and 2 were jointly in a pure singlet state in the past, with 1 moving toward station A, and 2 moving toward station B , but they remain jointly in a pure singlet state, in which their spins are perfectly anti-correlated.

A: spin of particle 1 is found in the up direction at station A
B: spin of particle 2 is found in the up direction at station B

Therefore P(AB|Z) means: What is the probability that both particle 1 and particle 2 are found in the up direction given Z.

The second term in P(AB|Z)=P(A|BZ)*P(B|Z), ie P(A|BZ), means that if we know for sure that B is true, ie that that spin at station 2 is found to be 'up', then that INFORMATION should influence the probability we assign to A. Do you disagree with this?
I agree that this is true in your equation with the meaning you've assigned to the terms, but I disagree that it'd necessarily still be true if we change the meaning of Z so that it now represents some complete state of the hidden variables. With Z given this meaning, P(A|BZ)=P(A|Z), and in those situations P(AB|Z)=P(A|Z)*P(B|Z), meaning that if you know Z and use that to calculate the probability of A, then knowing B gives you no further information that would cause you to alter your calculation of the probability of B. Note that this was true in my thought-experiment with the light-up balls, if A and B represented statements about the colors seen by Alice and Bob when they pushed their buttons, and Z represented a statement about which of the three balls D, E, F were given to Alice and Bob.
Now to the answer of your question:

Z: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = P(A|Z) since knowing that Bob saw red, does not tell me anything about what Alice might see.
Yes, that's the key. So you agree that in this example, P(AB|Z)=P(A|Z)P(B|Z).
In other words,
A and B are logically independent.
Yes, But A and B are not logically independent in the sample space for this problem as a whole, which includes events where Z is not true (for example, it also includes events where F is given to Alice and D is given to Bob). In other words, P(AB) is not equal to P(A)P(B) in this sample space, despite the fact that P(AB|Z)=P(A|Z)P(B|Z). Do you agree or disagree?
Now you should answer this one:
Z: Two red LEDs D, E were made to be anti-correllated so that when D was observed to lit, the probability of E lighting up was 0.25 and E was lit the probability of D lighting up was 0.6. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

What probability will you assign to P(AB|Z).
This problem seems to be not very well-defined, I don't think enough information is given to calculate P(A|Z) and P(B|Z) as a whole. All we can say is that P(A|BZ)=0.6, and P(B|AZ)=0.25. From these facts alone how do you calculate P(AB|Z)?

Also, this example is not really in the spirit of a "local hidden variables" explanation because it's not clear from how it's described whether the correlation involves actual communication between the two LEDs ('spooky action at a distance'), or whether the correlation is ensured by internal mechanisms (hidden variables) in each LED which on each trial were given to them by the "source" that is sending them out to Alice and Bob, such that if you know the mechanism associated with a particular LED on a given trial that allows you to calculate the probability it'll light up, and knowing about what happened to the second LED gives you no further information about the probability the first one will light up, just like with my example of the light-up balls above where you apparently agreed this was true (you agreed that P(A|BZ)=P(A|Z), where Z gave you the information about the internal mechanism about the ball given to Alice). The latter is how it's supposed to work in a local hidden variables theory.

We could assume a very simple local hidden-variables explanation in your example--suppose the "source" assigning LEDs to Alice and Bob has only two types, a type L (for light) that always lights up when the button is pushed, and a type D (for dark) that always fails to light up. Then on x trials, the source sends an L to Alice and a D to Bob; on y trials, the source sends an L to both; and on z trials, the source sends a D to Alice and an L to Bob (assume the source never sends a D to both, otherwise the problem won't be well-defined). Assume x, y, and z are all fractions, so that x + y + z = 1. Then we can write P(Bob's lights up|Alice's lit up) = P(Bob's lights up AND Alice's lit up)/P(Alice's lit up) = y/(x + y) = 0.25, which gives us x=3y. Likewise, P(Alice's lights up|Bob's lit up)=P(Bob's lights up AND Alice's lit up)/P(Bob's lit up)=y/(z + y) = 0.6, which gives z=(2/3)y. So, we can substitute x=3y and z=(2/3)y into x + y + z = 1 to get (14/3)y = 1 which implies y=3/14. So, x=9/14 and z=2/14. So, now we can define a new condition Z' which implies your Z but contains more detail:

Z': The source has a collection of L LEDs that always light up, and D LEDs that never do, and there is a probability of 9/14 it gives Alice an L and Bob a D, and probability of 3/14 it gives both Alice and Bob an L, and a probability of 2/14 it gives Alice a D and Bob an L.

A: Alice's lights up
B: Bob's lights up

In this case P(A|BZ')=0.6 and P(B|AZ')=0.25 as before. But here we can also show that P(AB|Z')=3/14, and that P(A|Z')=12/14 while P(B|Z')=5/14. Here, note that it is not true that P(A|BZ')=P(A|Z'), so it isn't true that P(AB|Z')=P(A|Z')P(B|Z'). However, this is just because Z' does not give us the complete information about the hidden variables on a given trial. Consider Z1, Z2 and Z3 that do give us this information:

Z1: The source has a collection of L LEDs that always light up, and D LEDs that never do, and it gives Alice an L and Bob a D
Z2: The source has a collection of L LEDs that always light up, and D LEDs that never do, and it gives both Alice and Bob an L
Z3: The source has a collection of L LEDs that always light up, and D LEDs that never do, and it gives Alice a D and Bob an L.

In this case we can say that in our sample space, P(Z1)=9/14, P(Z2)=3/14, and P(Z3)=2/14. In this case it is true that P(AB) = \sum_i P(AB|Z_i)P(Z_i), and it's also true that P(AB|Z1)=P(A|Z1)P(B|Z1) (because P(A|BZ1) = P(A|Z1), and likewise that P(AB|Z2)=P(A|Z2)P(B|Z2) and that P(AB|Z3)=P(A|Z3)P(B|Z3). So, we can rewrite the above summation as P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i). Do you disagree with any of this?

mn4j

Jan9-09, 02:45 PM

mn4j

Jan9-09, 02:56 PM

If you don't disagree, then again, my point is just that you can have a situation where X is independent of Y in this subset, even though it is not independent of Y in the complete set of all trials.

That's not exactly true, we're still dealing with the same larger sample space that includes events in which Z did not occur. It's just that when we write P(XY|Z), we want to look only at the subset of points in the sample space where Z did occur, and then look at how frequently X and Y occurred in this subset. Of course this is logically equivalent to just defining a new sample space that includes only events where Z occurred, and looking at how frequently X and Y occur in the complete sample space (i.e. P(XY) in this smaller sample space which includes only events where Z occurs is equal to P(XY|Z) in the larger sample space which includes events where Z does not occur).

It all boils down to the point that you do not understand the meaning of logical independence. Look at this

http://math-cs.cns.uni.edu/~campbell/stat/prob4.html#prod

This problem seems to be not very well-defined, I don't think enough information is given to calculate P(A|Z) and P(B|Z) as a whole. All we can say is that P(A|BZ)=0.6, and P(B|AZ)=0.25. From these facts alone how do you calculate P(AB|Z)?
You apparently did not see the correct question in the following post. Could you answer that one instead? Sorry about the mixup.

Let me answer it also.

P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = 0.2
P(B|Z) = P(A|Z) = 0.5 since there is no bias between A and B, they are both equally likely. I suspected you will have a problem with this one because it appears, you do not understand probabilities as meaning more than frequencies.

therefore P(AB|Z) = 0.2 * 0.5 = 0.1

JesseM

Jan9-09, 03:17 PM

Here's another example--we have a 6-sided die and a 12-sided die, Alice rolls one and Bob rolls the other. If there's a 50/50 chance the 12-sided die will be given to either one of them, and we aren't told who got which die, then the results they see aren't logically independent--for example, if Alice gets a 10 that allows us to infer Bob will get some number between 1 and 6, whereas before we knew what Alice rolled we would have considered it possible Bob could get any number from 1-12. However, in the subset of cases where Alice gets the 12-sided die and Bob gets the 6-sided die (or if you prefer, in a single case where we know Alice got the 12-sided die and Bob got the 6-sided die), Bob has a probability 1/6 of getting any number 1-6, and a probability 0 of getting a number 7-12, and knowing what Alice rolled has absolutely no further effect on these probabilities. Do you disagree? Again, please think about this specific example and answer yes or no.
The examples you are giving are deliberately cases in which there is no logical dependence.
But of course there is a logical dependence in the sample space as a whole! Remember I said there was a 50/50 chance the 12-sided die would be given to either one, so the sample space includes both events where Alice got the 12-sided die and events where Bob got it. If we don't know who got it (i.e. we don't know the hidden variables which determine the probability distribution of each one's rolls), and then we find out Alice got a 10, this will cause us to revise the probabilities that Bob will get a given number--do you disagree?

It seems that part of the problem here is that you are fairly confused about the difference between the sample space for a problem and the conditions in a statement about conditional probability in a problem. They are not the same thing! For a simple example, consider the following:

Sample space--pairs of flips of a fair coin, possible outcomes HH, HT, TH, and HH, each with probability 1/4.

Event A: first flip was H.
Event B: Second flip was H.
Event C: Both flips gave the same result.

For this sample space, all of the following are true:
P(A) = P(B) = P(C) = 1/2
P(B|A) = 1/2
P(B|C) = 1/2
P(B|AC) = 1

Do you disagree?

Now, consider the more complex example of the balls with the hidden internal mechanisms. Here, we have:

Sample space: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. The source randomly chooses which balls to give to Alice and Bob, so P(Alice gets D, Bob gets E)=P(Alice gets D, Bob gets F)=P(Alice gets E, Bob gets D)=P(Alice gets E, Bob gets F)=P(Alice gets F, Bob gets D)=P(Alice gets F, Bob gets E)=1/6.

H1 is the event "Alice gets D, Bob gets E".
H2 is the event "Alice gets D, Bob gets F".
H3 is the event "Alice gets E, Bob gets D".
H4 is the event "Alice gets E, Bob gets F".
H5 is the event "Alice gets F, Bob gets D".
H6 is the event "Alice gets F, Bob gets E".
A is the event that Alice saw red when she pushed the button on her ball.
B is the event that Bob saw red when he pushed the button on his ball.

Do you agree that in this sample space as a whole, A is not logically independent of B, so that if we don't know which hidden-variable condition H1-H6 obtained on a given trial, then knowing that B occurred should cause us to revise our estimate of the probability of A, and P(AB) is not equal to P(A)*P(B) in this sample space? On the other hand, do you also agree that if we pick a particular hidden-variable condition H1-H6, then it will be true that P(AB|H1)=P(A|H1)*P(B|H1), and so on for conditions H2, H3, H4, H5 and H6? Do you agree that in this sample space, P(AB) = \sum_i P(AB|H_i )P(H_i ), so if we combine this with the fact that P(AB|Hi)=P(A|Hi)*P(B|Hi), we get the equation P(AB) = \sum_i P(A|H_i )P(B|H_i )P(H_i ) which is directly analogous to the hidden-variable integral in Bell's paper, or equation (12) in the Jaynes paper?

JesseM

Jan9-09, 03:34 PM

It all boils down to the point that you do not understand the meaning of logical independence. Look at this

http://math-cs.cns.uni.edu/~campbell/stat/prob4.html#prod
Of course I understand it. Do you agree that it may be possible to find a sample space where it is true that P(AB) is not equal to P(A)P(B), meaning they are not logically independent, yet it's also true that P(AB|Z)=P(A|Z)P(B|Z) in this sample space? This would mean that if we consider a different sample space where Z is included as part of the conditions of the sample space (i.e. the subset of trials in the original sample space where Z occurred), then in this new sample space A is logically independent of B.
You apparently did not see the correct question in the following post. Could you answer that one instead? Sorry about the mixup.
Sure, your new problem was this:
Z: Two red LEDs D, E were made on a circuit so that when D was observed to be lit, the probability of E lighting up was 0.2 and when E was lit the probability of D lighting up was 0.2. Also, the circuit was designed such that at least one LED was lit on each button press with no other bias imposed on the LED other than the correlation above. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red
Your own answer contains a leap that can't really be justified in any rigorous terms:
Let me answer it also.

P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = 0.2
P(B|Z) = P(A|Z) = 0.5 since there is no bias between A and B, they are both equally likely.
Your original statement of the condition Z did not say "there is no bias between A and B", it simply didn't comment on the overall likelihood that A and B occurred. If you want to include it as an additional part of Z that A and B are equally likely overall, then that's fine, but the problem as stated didn't contain enough information to give a well-defined answer.
I suspected you will have a problem with this one because it appears, you do not understand probabilities as meaning more than frequencies.
I recognize that in Bayesian reasoning observers can have "prior" probability distributions which simply represent their beliefs about the likelihood different events will occur; however, in problems that don't contain enough information to be interpreted in frequentist terms, this is highly subjective, different people can have different prior probability distributions and there are no well-defined rules for deciding what the "correct" prior is. So I should say that in any situation where enough information is given so that everyone will be forced to assign events the same probabilities, in those situations you should always be able to re-interpret probabilities in frequentist terms. All the examples we have been dealing with up until now, like the light-up balls or the urn examples, are sufficiently well-defined so that there should be no subjective elements in assigning probabilities. And Bell's proof assumes that there is a complete set of local physical facts which determine probabilities, such that if an omniscient observer had access to all possible physical facts, he could assign probabilities to events like P(AB) without any subjective element either. The variable \lambda is supposed to represent the complete set of hidden facts on any given trial--we don't know the value of \lambda on a given trial, but that's similar to the fact that we don't know which of the six hidden conditions H1-H6 obtained on a given trial in the example with the light-up balls in my previous post, we nevertheless can agree that from the perspective of an omniscient observer who does know, it must be true that P(AB) = \sum_i P(A|H_i )P(B|H_i )P(H_i ) (and of course, we do know whether A and B obtained on each trial). This equation is obviously very similar to (12) in Jaynes' paper, and the reasoning behind both is analogous, if you don't understand my summation equation above you won't be able to understand the reasoning behind (12).

mn4j

Jan9-09, 03:37 PM

On the other hand, do you also agree that if we pick a particular hidden-variable condition H1-H6, then it will be true that P(AB|H1)=P(A|H1)*P(B|H1), and so on for conditions H2, H3, H4, H5 and H6?
No I do not. Simply knowing the values of the hidden variables is not enough to reduce P(A|BH1) to P(A|H1). You MUST also know that A is logically independent of B in the specific event. That is why you should focus on only one event and show me how it is that knowledge of the hidden variable reduces P(A|BH1) to P(A|H1).

JesseM

Jan9-09, 03:52 PM

No I do not. Simply knowing the values of the hidden variables is not enough to reduce P(A|BH1) to P(A|H1).
But in post #36 you already agreed with something equivalent:
Now to the answer of your question:

Z: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = P(A|Z) since knowing that Bob saw red, does not tell me anything about what Alice might see.
The only difference is that in my statement of the problem I have more adequately distinguished what was the sample space and what was a specific hidden-variables condition which only obtained in some events in the sample space (see my questions about the coinflip in post #41--I'd like to know whether you agree or disagree there, and whether in general you understand the difference between the sample space and conditions in a conditional probability equation). Specifically I wrote:
Sample space: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. The source randomly chooses which balls to give to Alice and Bob, so P(Alice gets D, Bob gets E)=P(Alice gets D, Bob gets F)=P(Alice gets E, Bob gets D)=P(Alice gets E, Bob gets F)=P(Alice gets F, Bob gets D)=P(Alice gets F, Bob gets E)=1/6.

H1 is the event "Alice gets D, Bob gets E".
Do you agree that in any event (i.e. single trial) with this sample space where H1 occurred, the proposition you labeled Z will also be true?
You MUST also know that A is logically independent of B in the specific event. That is why you should focus on only one event and show me how it is that knowledge of the hidden variable reduces P(A|BH1) to P(A|H1).
"Logical independence" refers to the sample space, asking whether A is logically independent of B in a specific event doesn't make much sense. Do you agree that any statement about probabilities must involve a well-defined sample space?

mn4j

Jan9-09, 04:13 PM

But in post #36 you already agreed with something equivalent:

Not correct! In your statement of the problem, there was no dependence between D E and F. That is why I could reduce P(A|BZ) to P(A|Z), not because I knew values of the probabilities P(E), P(D), P(F). Do not confuse the value of a specific hidden variable with the probability of having that particular value in an event.

asking whether A is logically independent of B in a specific event doesn't make much sense.
False. My very first post on this issue gave you an example. The case of the monkey picking two balls.
1 event in that case is the the picking of the two balls and yet in that event, I showed you that the probability of picking the first ball, is logically dependent on the outcome of the second and vice versa. That example was not taken from thin air. It was chosen precisely because it mirrors the EPR experiment.

1 event in the EPR case is a measurement at 2 stations A and B for two electrons with a common past.

JesseM

Jan9-09, 04:33 PM

Not correct! In your statement of the problem, there was no dependence between D E and F.
What does "dependence between D E and F" mean? D E and F aren't events, they are just names of balls. Also, when you say "in my statement of the problem", do you mean the original statement you were responding to when you agreed P(A|BZ) = P(A|Z), or do you mean the more recent statement where I distinguished between the sample space and the events H1-H6?
That is why I could reduce P(A|BZ) to P(A|Z), not because I knew values of the probabilities P(E), P(D), P(F).
Like I said, not events, so these don't make sense as stated, but I would guess that by P(E) you probably mean the probability that ball E will light up when pressed, and similarly for D and F?
In other words, it is impossible to tell just from the probabilities P(E), P(D), P(F) whether E, D, F are dependent on each other or not.
I see, you're worried about whether there's a logical dependence between the event of ball E lighting up and the event of ball D lighting up, and so on for other pairs? Well, this is supposed to be a local hidden variables explanation where the probability a given ball lights up is wholly determined by its internal mechanism (internal hidden variables), so you can assume there is no such logical independence. I'll restate the problem as follows:
Sample space: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. The probability a given ball will light up is wholly determined by its internal mechanism, so there is no logical dependence between the event of ball D lighting up when the button is pushed and any other outside events, and likewise for balls E and F. The source randomly chooses which balls to give to Alice and Bob, so P(Alice gets D, Bob gets E)=P(Alice gets D, Bob gets F)=P(Alice gets E, Bob gets D)=P(Alice gets E, Bob gets F)=P(Alice gets F, Bob gets D)=P(Alice gets F, Bob gets E)=1/6.

H1 is the event "Alice gets D, Bob gets E".
H2 is the event "Alice gets D, Bob gets F".
H3 is the event "Alice gets E, Bob gets D".
H4 is the event "Alice gets E, Bob gets F".
H5 is the event "Alice gets F, Bob gets D".
H6 is the event "Alice gets F, Bob gets E".
A is the event that Alice saw red when she pushed the button on her ball.
B is the event that Bob saw red when he pushed the button on his ball.
With this revised description of the sample space and various possible events, do you now agree that P(AB|H1) = P(A|H1)P(B|H1)? If so, do you also agree that A and B are not logically independent in the sample space as a whole, i.e. P(AB) is not equal to P(A)P(B)?
asking whether A is logically independent of B in a specific event doesn't make much sense.
False. My very first post on this issue gave you an example. The case of the monkey picking two balls.
1 event in that case is the the picking of the two balls and yet in that event, I showed you that the probability of picking the first ball, is logically dependent on the outcome of the second and vice versa. That example was not taken from thin air.
Yes, but the logical dependence here is based on your knowledge of the sample space--the reason one event is logically dependent on the other is because if we consider the sample space of all possible outcomes for both the first and second pick, we can assign probabilities to each point based on the way the balls are picked from the urn, and we find that if we know we picked a point in that sample space where the first ball was red, that tells us something more about the probability that we picked a point in the sample space where the second ball was also red. If you didn't know anything about the sample space or the probability of different points in the sample space--for example, if you had no idea whether on the second pick the monkey would get a ball, an egg, or a squirrel, or if you did know the urn contained only balls but didn't know what color balls were in the urn originally, or didn't know whether new balls were being placed in the urn between picks to replace the one chosen, or didn't know whether the monkey picks balls randomly from the urn or whether he has a particular taste for picking balls of a particular color--in that case, knowing the result of the first pick wouldn't tell you anything definite about the probability of getting a given color on the second pick.
It was chosen precisely because it mirrors the EPR experiment.
My light-up ball experiment mirrors the EPR experiment in this way too--P(AB) is not equal to P(A)P(B), so if you find out B and you don't already know anything about the hidden variables, this gives you new information about P(A).

mn4j

Jan9-09, 05:25 PM

What does "dependence between D E and F" mean? D E and F aren't events, they are just names of balls.

P(D) is the probability for observing the event that "D lights up when the button is pressed" etc.

Also, when you say "in my statement of the problem", do you mean the original statement you were responding to when you agreed P(A|BZ) = P(A|Z), or do you mean the more recent statement where I distinguished between the sample space and the events H1-H6?

I mean (see midway on post #35):
Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. ball D is given to Alice, ball E is given to Bob and they both press the button. What is the probability P(Alice saw red, Bob saw red).

Like I said, not events, so these don't make sense as stated, but I would guess that by P(E) you probably mean the probability that ball E will light up when pressed, and similarly for D and F?

I thought that was obvious.

My light-up ball experiment mirrors the EPR experiment in this way too

No! It doesn't because in the EPR case the two electrons are from the same source and are anti-correlated and you do not know prior to measurement whether spin up will go left or right. If you put the two balls in a box and at each event Alice and Bob each pull a ball at random without replacing and press on the button, then it will be closer.

Do you agree that any statement about probabilities must involve a well-defined sample space?
Yes. But this does not mean the sample space should not be modified in calculating the answer. That is precisely the meaning of the product rule.

When you write P(AB|Z) = P(A|BZ)P(B|Z),
Everything right of the "|" is the hypothesis space in which you are calculating the particular term. The hypothesis space for calculating P(AB) is Z. P(A|BZ) means that knowing B changes the hypothesis space for calculating P(A) by adding information to it. This is just another way of saying "we now assume that, not only is Z true but B is also True".

In the EPR case, since you do not know which electron came to which station, if you now know that station B measured the spin up electron, that information changes the hypothesis space such that station A can only measure the spin down electron therefore P(A) = P(A|BZ).

mn4j

Jan9-09, 05:36 PM

My light-up ball experiment mirrors the EPR experiment in this way too--P(AB) is not equal to P(A)P(B), so if you find out B and you don't already know anything about the hidden variables, this gives you new information about P(A).
What do you mean by find out B. What are the hidden variables in this case that could give me any information about A?

Sample space: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. The probability a given ball will light up is wholly determined by its internal mechanism, so there is no logical dependence between the event of ball D lighting up when the button is pushed and any other outside events, and likewise for balls E and F. The source randomly chooses which balls to give to Alice and Bob, so P(Alice gets D, Bob gets E)=P(Alice gets D, Bob gets F)=P(Alice gets E, Bob gets D)=P(Alice gets E, Bob gets F)=P(Alice gets F, Bob gets D)=P(Alice gets F, Bob gets E)=1/6.

H1 is the event "Alice gets D, Bob gets E".
H2 is the event "Alice gets D, Bob gets F".
H3 is the event "Alice gets E, Bob gets D".
H4 is the event "Alice gets E, Bob gets F".
H5 is the event "Alice gets F, Bob gets D".
H6 is the event "Alice gets F, Bob gets E".
A is the event that Alice saw red when she pushed the button on her ball.
B is the event that Bob saw red when he pushed the button on his ball.
With this revised description of the sample space and various possible events, do you now agree that P(AB|H1) = P(A|H1)P(B|H1)?

I do not agree. The statement I highlighted in bold above shows you clearly what I have been saying from the beginning. Logical dependence is different from physical dependence. In that sentence you are implying that lack of physical dependence implies lack of logical dependence. Although physical dependence implies logical dependence, lack of physical dependence does not imply lack of logical dependence. Do you disagree?

You can not impose a logical independence condition at will in your hypothesis space. It is part of the mechanism by which you reason out the problem. What you have done is to break the calculator before asking me to use it to calculate a problem. The problem therefore becomes ill-formed because if I know that Bob got F, it DOES tell me something about the probability that Alice got D. You can't force me to not consider that information. By imposing independence in the premise. It doesn't work like that.

This whole argument boils down to the meaning of logical dependence. And I conclude that you either do not understand it, or you understand but refuse to accept it.

JesseM

Jan9-09, 06:13 PM

I thought that was obvious.
OK, I guessed that was probably what you meant, which is why I presented you with the revised description of the sample space and asked you some questions about it--can you please look over the revised description and answer those questions?
Sample space: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. The probability a given ball will light up is wholly determined by its internal mechanism, so there is no logical dependence between the event of ball D lighting up when the button is pushed and any other outside events, and likewise for balls E and F. The source randomly chooses which balls to give to Alice and Bob, so P(Alice gets D, Bob gets E)=P(Alice gets D, Bob gets F)=P(Alice gets E, Bob gets D)=P(Alice gets E, Bob gets F)=P(Alice gets F, Bob gets D)=P(Alice gets F, Bob gets E)=1/6.

H1 is the event "Alice gets D, Bob gets E".
H2 is the event "Alice gets D, Bob gets F".
H3 is the event "Alice gets E, Bob gets D".
H4 is the event "Alice gets E, Bob gets F".
H5 is the event "Alice gets F, Bob gets D".
H6 is the event "Alice gets F, Bob gets E".
A is the event that Alice saw red when she pushed the button on her ball.
B is the event that Bob saw red when he pushed the button on his ball.
With this revised description of the sample space and various possible events, do you now agree that P(AB|H1) = P(A|H1)P(B|H1)? If so, do you also agree that A and B are not logically independent in the sample space as a whole, i.e. P(AB) is not equal to P(A)P(B)?
My light-up ball experiment mirrors the EPR experiment in this way too
No! It doesn't because in the EPR case the two electrons are from the same source and are anti-correlated and you do not know prior to measurement whether spin up will go left or right. If you put the two balls in a box and at each event Alice and Bob each pull a ball at random without replacing and press on the button, then it will be closer.
I didn't mean to imply the light-up ball experiment was analogous to the EPR experiment in every respect, but it's analogous to an attempt at a local hidden-variables explanation for the EPR experiment in the specific respect that there is a statistical correlation between the results seen by Alice and the results seen by Bob (i.e. P(AB) is not equal to P(A)P(B)), but this correlation is wholly explained by the hidden internal mechanism associated with each ball being measured. If you want an example that's even more closely analogous to a local hidden-variable explanation for the EPR experiment, look at my lotto card example in post #3 in this thread.
Yes. But this does not mean the sample space should not be modified in calculating the answer. That is precisely the meaning of the product rule.
No, this shows your confusion between the sample space of a problem and a statement of conditional probability. When you write P(AB|Z) that does not imply a modification to the original sample space, it just means you are looking at the subset of points in the sample space where Z occurred, then looking at how frequently AB occurred in this subset. Read page 81 here (http://books.google.com/books?id=zAp2wBpSvioC&pg=PA81&lpg=PA81&dq=%22conditional+probability%22+subset+%22sample+ space%22&source=web&ots=BHyXj2NEj5&sig=hH-4k2hBoK50UdY763oqNMKHAEc&hl=en&sa=X&oi=book_result&resnum=3&ct=result), for example.
In the EPR case, since you do not know which electron came to which station, if you now know that station A measured the spin up electron, that information changes the hypothesis space such that station B can only measure the spin down electron.
See above for disagreement about your use of "hypothesis space", although I agree that knowing station A measured spin-up changes your estimate of the probability that B will measure spin-up or spin-down, assuming you don't know what hidden variables (if any) are associated with each electron (if you did know the full hidden variables associated with B, then in a local hidden variables theory that would tell you whether it would be spin-up or spin-down, so you'd gain no additional information from knowing what station A measured). Similarly, in the light-up ball case where you don't know which case H1-H6 applied (i.e. you don't know which of balls D, E, and F were given to Alice and Bob), if you know whether Bob's ball lit up, that will change your estimate of the probability that Alice's ball will light up: P(A|B) is not equal to P(A).

mn4j

Jan9-09, 06:14 PM

I recognize that in Bayesian reasoning observers can have "prior" probability distributions which simply represent their beliefs about the likelihood different events will occur; however, in problems that don't contain enough information to be interpreted in frequentist terms, this is highly subjective, different people can have different prior probability distributions and there are no well-defined rules for deciding what the "correct" prior is.
This is not correct. There are well-defined rules. Read up on the "principle of indifference" or "maximum entropy".

When physicists say "any two configurations of atoms with the same total energy are equally probable at equilibrium" you don't expect that someone has actually counted the atoms to calculate their probability do you? They are using the principle of indifference.

Still I would have liked to see how you calculate on this question:
Z: Two red LEDs D, E were made on a circuit so that when D was observed to be lit, the probability of E lighting up was 0.2 and when E was lit the probability of D lighting up was 0.2. Also, the circuit was designed such that at least one LED was lit on each button press with no other bias imposed on the LED other than the correlation above. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

What probability will you assign to P(AB|Z).

And please point out the error you claim exists in my answer to it:
P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = 0.2
P(B|Z) = P(A|Z) = 0.5 since there is no bias between A and B, they are both equally likely. I suspected you will have a problem with this one because it appears, you do not understand probabilities as meaning more than frequencies.

therefore P(AB|Z) = 0.2 * 0.5 = 0.1

JesseM

Jan9-09, 06:45 PM

P(AB) is not equal to P(A)P(B), so if you find out B and you don't already know anything about the hidden variables, this gives you new information about P(A).
What do you mean by find out B. What are the hidden variables in this case that could give me any information about A?
Again, the assumption is that the two balls given to Alice and Bob were taken from the set D, E, and F, with the assumptions already given about the probabilities associated with each ball. But I'm talking about a situation where you know the balls were selected this way, but you don't know which specific ball was given to Alice and which specific ball was given to Bob. In this case, if you find out that Bob's ball lit up red, this will change your estimate of the probability that Alice's ball will light up red: P(A) is different from P(A|B).
I do not agree. The statement I highlighted in bold above shows you clearly what I have been saying from the beginning. Logical dependence is different from physical dependence. In that sentence you are implying that lack of physical dependence implies lack of logical dependence. Although physical dependence implies logical dependence, lack of physical dependence does not imply lack of logical dependence. Do you disagree?
No, I don't disagree with that statement in general. But we are talking about the specific case where the outcome of a given ball's button being pressed is assumed to be fully determined physically by the internal mechanisms in that ball, which are not in communication with the outside world. Are you saying that even when we already know which physical mechanisms are inside the balls given to Alice and Bob, you still think that knowing whether Bob's ball lit up or not would cause us revise our estimate of the probability that Alice's ball lit up? Suppose instead Alice and Bob were flipping coins, and we know that each coin's probability of coming up heads or tails is physically determined by its physical structure, and that both coins are fair coins that have a 50/50 chance of landing heads or tails. Would knowing that Bob's coin landed heads somehow cause you to revise your estimate of the probability that Alice's came up heads?

In a universe with local realist laws, the results of a physical experiment on any system are assumed to be determined by some set of variables specific to the region of spacetime where the experiment was performed. There can be a statistical correlation (logical dependence) between outcomes A and B of experiments performed at different locations in spacetime with a spacelike separation, but the only possible explanation for this correlation is that the variables associated with each system being measured were already correlated before the experiment was done, that the systems had "inherited" correlated internal variables from some event or events in the overlap of their past light cones. Do you disagree? If so, try to think of a counterexample that we can be sure is possible in a local realist universe (no explicitly quantum examples).

If you don't disagree, then the point is that if the only reason for the correlation between A and B is that the local variables \lambda associated with system #1 are correlated with the local variables associated with system #2, then if you could somehow know the full set of variables \lambda associated with system #1, knowing the outcome B when system #2 is measured would tell you nothing additional about the likelihood of getting A when system #1 is measured. In other words, while P(A|B) may be different than P(B), P(A|B\lambda ) = P(A | \lambda ). If you disagree with this, then I think you just haven't thought through carefully enough what "local realist" means.
You can not impose a logical independence condition at will in your hypothesis space.
No, but you can if it makes sense given the physical assumptions of the problem. For example, while I can't impose the condition P(AB)=P(A)P(B) in general, if I know I'm dealing with a situation where Bob and Alice are both given different fair coins and asked to flip them, and A=Alice got heads and B=Bob got heads, then of course in this case it makes sense to say P(AB)=P(A)P(B). Similarly, if I know I'm dealing with a situation where Alice got a ball that has a 90% chance of lighting up when the button was pressed due to some internal mechanisms that aren't in communication with anything outside the ball, and where Bob got a ball that has a 70% chance of lighting up due to similar internal mechanisms, then if A=Alice's ball lit up and B=Bob's ball lit up, I can of course say that P(AB)=P(A)P(B) here.
It is part of the mechanism by which you reason out the problem. What you have done is to break the calculator before asking me to use it to calculate a problem. The problem therefore becomes ill-formed because if I know that Bob got F, it DOES tell me something about the probability that Alice got D.
Huh? I never asked you to calculate the probability that Alice got D given that Bob got F. (of course I'd agree that if you don't know what Alice got, knowing Bob got F increases the probability Alice got D!) The only two problems I gave were: #1, the one where you don't know which ball either of them got, and you have to calculate the probability that both balls lit up; and #2, the one where you already know which balls both of them got, and you have to calculate the probability both balls lit up given that knowledge.

JesseM

Jan9-09, 07:58 PM

This is not correct. There are well-defined rules. Read up on the "principle of indifference" or "maximum entropy".
They are well-defined, but you have been misled by Jaynes if you think they are universally agreed upon by all people who study statistics--see this page (http://www.statisticalengineering.com/frequentists_and_bayesians.htm) on the internal division between frequentists and bayesians among statisticians. There is no objective reason why adopting the principle of indifference is more correct than adopting some other rule--indeed, it is possible to think of well-defined physical situations where, if you don't give enough information about the situation for the frequentist probabilities to be clear, then if one person fills in the blanks using the principle of indifference while second person uses a different prior distribution, the second person may be closer to the "correct" frequentist probabilities. For example, if I tell two people "where I live, on some days it rains and on others it doesn't, what's the probability it rains on a given day", then if person 1 uses the principle of indifference here he'll say the answer is 0.5, while if the other person blindly guesses the answer is 0.2, it may in fact be correct that the ratio of rainy days to sunny days over a long period in my area is closer to 0.2 than 0.5.

In any case, I hope you can agree that in some cases enough information about a problem is given so that there is no need to make use of the principle of indifference--these are precisely the problems where enough information is given so we can see exactly what the frequencies of different events would be over a large number of trials. And in any discussion of the proof of Bell's theorem, it should be understood that the proof takes the point of view of some omniscient being who knows the value of all physical variables that are relevant to determining the outcome of experiments, even if some of these variables would be "hidden" to human experimenters--based on this hypothetical point of view, it shows that it's possible to prove that in a local realist universe there'd be certain probability relations between the types of events which can be seen by human experimenters, and that these probability relations are in fact violated by quantum mechanics.
When physicists say "any two configurations of atoms with the same total energy are equally probable at equilibrium" you don't expect that someone has actually counted the atoms to calculate their probability do you? They are using the principle of indifference.
It's a physical fact of our universe that the principle of indifference happens to work well in situations where we want to calculate the future evolution of a system in a certain observed macrostate. On the other hand, since the laws of physics are time-symmetric, if you know the complete physical state of a system at a given time you can calculate backwards to see what its state would have been at earlier times--if you can only see the current macrostate, assuming the principle of indifference with regard to microstates would lead you to predict the system was in a higher entropy state in the past, a prediction would be wrong in most cases. See Loschmidt's paradox (http://en.wikipedia.org/wiki/Loschmidt%27s_paradox).

Anyway, as I said, discussions of the principle indifference are irrelevant in the context of Bell's theorem, because the theorem is explicitly based on imagining that we (or some imaginary omniscient observer) could know the full state of every system, with no information lacking.
Still I would have liked to see how you calculate on this question:
Z: Two red LEDs D, E were made on a circuit so that when D was observed to be lit, the probability of E lighting up was 0.2 and when E was lit the probability of D lighting up was 0.2. Also, the circuit was designed such that at least one LED was lit on each button press with no other bias imposed on the LED other than the correlation above. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

What probability will you assign to P(AB|Z).
And please point out the error you claim exists in my answer to it:
P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = 0.2
P(B|Z) = P(A|Z) = 0.5 since there is no bias between A and B, they are both equally likely. I suspected you will have a problem with this one because it appears, you do not understand probabilities as meaning more than frequencies.

therefore P(AB|Z) = 0.2 * 0.5 = 0.1
Actually I realize I was mistaken when I said the problem didn't give enough information, I was thrown off by your "since there is no bias between A and B" comment. In fact it is possible to deduce the likelihood of A and B in a frequentist picture here. Just let x be the fraction of trials where only Alice's LED lit up, y be the fraction where they both lit up, and z be the fraction where only Bob's lit up. Then the fraction of trials where Bob's lit up is (y + z), in which case the fact that Alice's LED has only an 0.2 probability of lighting up as well when Bob's is lit up means y/(y + z) = 0.2. Likewise, the fraction of trials where Alice's lit up is (x + y), in which case y/(x + y) = 0.2. From these equations we can conclude that z=4y and x=4y, so since we know that x + y + z = 1, that gives us y=1/9, which means x=z=4/9 (note that this means P(A|Z) = P(B|Z) = 5/9, not 0.5...you apparently forgot that these possibilities are not mutually exclusive so they don't have to add up to 1!) So, the correct probability for both lighting up is just y, i.e. 1/9. Obviously this is different from the answer you got of 0.1. If you think your answer is right and mine is wrong, please give me your own answers for P(Alice saw red, Bob didn't see red | Z), and P(Alice didn't see red, Bob saw red | Z). Hopefully you'd agree that since at least one LED always lights up red, if we take these probabilities and add them to P(Alice saw red, Bob saw red | Z), i.e. P(AB|Z) which you claim is 0.1, the sum should be equal to 1?

mn4j

Jan10-09, 07:46 AM

You are correct that my answer of 0.1 was wrong. And yours (1/9) is correct. And the reason why mine is wrong is because in calculating one of the probabilities, I did not consider all the information provided in Z. Even though I knew the information. In other words, Z did not mean the same thing in my calculation of every term.

Since we have come full circle, please answer the following simply with yes or no.

1. When you reduce
P(AB) = \sum_i P(AB|Z_i)P(Z_i)
to
P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i)
because of independence in the specific Z_i subset. Do you agree that Z_i MUST mean the same thing in all the terms P(A|Z_i) , P(B|Z_i) and P(Z_i)?

2. Do you agree that if Z_i can be split up into individual pieces of information a_i, b_i, \lambda then
P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i)
is equivalent to
P(AB) = \sum_i P(A|a_ib_i \lambda)P(B|a_ib_i \lambda)P(a_ib_i \lambda)
but is NOT equivalent to
P(AB) = \sum_i P(A|a_i)P(B|b_i)P(\lambda)

3. Do you believe that if knowledge of a_i gives us no addition information about B and if knowledge of b_i gives us no addition information about A, and knowledge of a_i and b_i give us no additional information about \lambda, then
P(AB) = \sum_i P(A|a_ib_i\lambda)P(B|a_ib_i\lambda)P(\lambda)
should give us the same result as
P(AB) = \sum_i P(A|a_i\lambda)P(B|b_i\lambda)P(\lambda)

4. Do you agree that in Bell's proof, calculating with P(AB) = \sum_i P(A|a_ib_i\lambda)P(B|a_ib_i\lambda)P(\lambda) gave a result which was in agreement with Quantum mechanics but calculating with P(AB) = \sum_i P(A|a_i\lambda)P(B|b_i\lambda)P(\lambda) gave a result which was not.

5. If you agree with 3 and 4, then can you explain to me how the two statements can both be true.

JesseM

Jan10-09, 11:21 AM

1. When you reduce
P(AB) = \sum_i P(AB|Z_i)P(Z_i)
to
P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i)
because of independence in the specific Z_i subset. Do you agree that Z_i MUST mean the same thing in all the terms P(A|Z_i) , P(B|Z_i) and P(Z_i)?
Yes, for any value of i, Z_i should mean the same thing everywhere. Of course Z_1 means something different than Z_2 and so forth...one might involve the hidden-variable condition that Alice got ball D and Bob got ball E, another might involve the hidden-variable condition that Alice got ball D and Bob got ball F. You have to sum over all possible hidden-variable conditions in the above equation to get the total probability of P(AB). Agreed?
2. Do you agree that if Z_i can be split up into individual pieces of information a_i, b_i, \lambda then
P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i)
is equivalent to
P(AB) = \sum_i P(A|a_ib_i \lambda)P(B|a_ib_i \lambda)P(a_ib_i \lambda)
but is NOT equivalent to
P(AB) = \sum_i P(A|a_i)P(B|b_i)P(\lambda)
Yes, I agree that if Z_i can be split up in the way you suggest, the sum is equivalent to the first option but not in general equivalent to the second option. But your equation actually has little relevance to Bell's proof, because you haven't put any subscript on the \lambda, implying you think the hidden variables should be exactly the same on every trial! Of course this was not what Bell assumed, he imagined the hidden variables associated with the particles could be different on different trials, and that these hidden variables would explain why an experimenter measuring on a particular axis sometimes gets spin-up and sometimes gets spin-down. So, you really need to give \lambda a subscript and sum over all possible values of this subscript. Also, of course on every trial where the first experimenter makes measurement a1 we do not assume the other experimenter makes b1, so you need multiple subscripts on those letters too. So if you want to write an equation more in keeping with Bell's proof, it should be something like:
P(AB) = \sum_i \sum_j \sum_k P(A|a_i b_j \lambda_k)P(B |a_i b_j \lambda_k)P(a_i b_j \lambda_k)
Or if you assume we are looking at the subset of trials where experimenter #1 made measurement a and experimenter #2 made measurement b (where a and b now stand for two specific measurements rather than variables) we can write:
P(AB|ab) = \sum_k P(A|ab \lambda_k)P(B |ab \lambda_k)P(ab \lambda_k)
3. Do you believe that if knowledge of a_i gives us no addition information about B and if knowledge of b_i gives us no addition information about A, and knowledge of a_i and b_i give us no additional information about \lambda, then
P(AB) = \sum_i P(A|a_ib_i\lambda)P(B|a_ib_i\lambda)P(\lambda)
should give us the same result as
P(AB) = \sum_i P(A|a_i\lambda)P(B|b_i\lambda)P(\lambda)
Yes, although here the equation more in keeping with Bell's proof would be P(AB) = \sum_i \sum_j \sum_k P(A|a_i \lambda_k)P(B |b_j \lambda_k)P(\lambda_k)
4. Do you agree that in Bell's proof, calculating with P(AB) = \sum_i P(A|a_ib_i\lambda)P(B|a_ib_i\lambda)P(\lambda) gave a result which was in agreement with Quantum mechanics but calculating with P(AB) = \sum_i P(A|a_i\lambda)P(B|b_i\lambda)P(\lambda) gave a result which was not.
As before, the idea of Bell's proof is to do a sum over possible values of \lambda (he did an integral because he imagined it taking a continuous range of possible values), but it'd be easy to modify your equations above in this way. But I don't know what you mean by "calculating with"--of course we have no idea of the actual possible values of \lambda_k and thus no idea of the exact value of terms like P(A|a_i\lambda_k) or P(\lambda_k)! The idea is just that if we imagine there is a "conspiracy" in the initial conditions that determines both \lambda_k on a given trial and determines what choices of measurements a_i and b_j the experimenters will make on the same trial, then it's possible to imagine that they could be correlated in a way that would be consistent with quantum predictions even in a universe with local realist physics (I can give you a specific numerical example if you like). It's only if you specifically assume no correlation between the experimenter's choice of measurement on a given trial and the source's "choice" of hidden variables to assign the particles on a given trial that you can derive the Bell inequalities which are inconsistent with QM.

mn4j

Jan10-09, 09:23 PM

JesseM

Jan12-09, 03:08 AM

Let's see here:

(Q1 & Q2): So you agree that in
P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i) ,
Z_i MUST mean exactly the same thing in all the terms (see your answer to Q1.), and you also believe that

P(AB) = \sum_i P(A|a_ib_i \lambda_i)P(B|a_ib_i \lambda_i)P(a_ib_i \lambda_i)
can be equivalent to
P(AB) = \sum_i P(A|a_i)P(B|b_i)P(\lambda_i) in the case being considered by Bell (your answer to Q2).
I said in my answer that this equation is not really correct because you use the same subscript for both the measurement choices a_i and b_i and the hidden variable states \lambda_i, implying that each hidden variable state is associated with a unique measurement. In reality, if there's no correlation between the hidden variable states and measurements, then it should be possible to have trials where you have measurement a_1 and hidden variable state \lambda_1, trials where you have measurement a_1 and hidden variable state \lambda_4, trials where you have measurement a_3 and hidden variable state \lambda_1, etc. This is why you have to write it with a multiple sum like I did:

P(AB) = \sum_i \sum_j \sum_k P(A|a_i \lambda_k ) P(B|b_j \lambda_k ) P(\lambda_k)

Also, you made an error when you left the \lambda's out of the P(A|ai) and P(B|bi), they should be included as I did above. I also noticed I made a small error of my own when writing the above equation--it's assumed we set things up so the experimenters make each combination of measurements equally frequently, so P(a1b1)=P(a2b1)=P(a3b1) etc., all have the same probability 1/N where N is the number of possible combinations. So really we should have P(AB) = \sum_i \sum_j \sum_k P(A|a_i \lambda_k ) P(B|b_j \lambda_k ) P(a_i b_j \lambda_k) which implies P(AB) = \sum_i \sum_j \sum_k P(A|a_i \lambda_k ) P(B|b_j \lambda_k ) P(a_i b_j ) P(\lambda_k) based on the assumption of the independence of measurements and hidden variables, which means the proper equation would have the extra constant factor 1/N:

P(AB) = \sum_i \sum_j \sum_k (1/N) P(A|a_i \lambda_k ) P(B|b_j \lambda_k ) P(\lambda_k)

Which implies that in Bell's thinking (and yours) there MUST be no dependence between a_i, b_i and \lambda_i, AND there MUST be no dependence between any pair of variables in the complete set (a_i,b_i and \lambda_i), irrespective of which what i is. For example, there MUST be no dependence between a_1 and a_2 or between b_5 and \lambda_{11} . Do you agree with this?
Yes. Of course with the independence of a-measurements from b-measurements, this is a situation that we arrange just by telling the experimenters to choose randomly on each trial (perhaps each uses a separate random-number generator, or each rolls a separate die or something), it isn't an assumption about the way the laws of physics work.
Now do you agree that time t_i can also be a piece of the information contained in Z_i. If you do, can you also envision that the settings at both stations, a_i, b_i could have time-like correlated components in which case integration over time can not be factorized and the settings a_i will be correlated with b_i without any need for spooky action at a distance? In other words, will this new scenario, not considered by Bell, be a local hidden variable model?
The a's and b's represent choices made by the experimenters, unless you think it would be impossible to set things up so that their choices are uncorrelated with one another, I don't see the relevance here. Remember, Bell's theorem is about picking the optimum experimental situation for ruling out local hidden variables, we don't have to consider arbitrary variations in the experimental settings that we control. We do have to consider variations in the nature of the hidden variables associated with the particles, since we don't control those--so, it would be appropriate to imagine if the hidden variables associated with each particle might vary over time. But remember that according to QM, if both experimenters measure along the same axis they'll always get opposite spins (or the same spins, depending on what particles are used and how they are entangled), even if they measure at different times. And in a local hidden variables theory we are assuming that there is no physical "communication" between the particles once they are separated, any correlation in their measured behavior must be due to local physical variables--which could be time-varying functions--that were "assigned" to each one in a correlated way when the particles were at a common location.

So, if we make the assumption that there's no correlation between the hidden variable functions assigned to each particle when they were in the same location and the experimenters' later choices about how/when to measure them, the only way to explain this perfect correlation when they are measured on the same axis (regardless of when the measurements are made) is if the hidden variables predetermine a single answer each particle will give to being measured on any given axis, and there's no time variation in this answer (though there could be time variation in other aspects of the hidden variables as long as they don't change what answer a given particle would give when measured on a particular axis at different times). Do you agree?
In other words, Bell's proof is only valid for the set of hidden variable theories consistent with his assumptions about independence outlined above. Do you agree? (I know you believe that every possible hidden variable theory is covered by it)
Yes, I agree--the proof covers every possible hidden variable theory that meets the stated conditions--i.e. locality (there are nonlocal hidden variables theories which Bell's theorem doesn't apply to), realism, and the assumption about a lack of "conspiracy" in the

Now let us look at this equation which is just the product rule:
P(AB) = \sum_i P(A|a_ib_i \lambda_i)P(B|a_ib_i \lambda_i)P(a_ib_i\lambda_i)
Consider a hidden variable theory in which time is considered a variable as well so that a_i, b_i and \lambda_i are time dependent variables. Note that time dependence of the settings at the stations does not take away the experimenters free will to change a_i or b_i. For example, the measuring device could be a pendulum and the experimenter has the free will to choose the length of the string. It is also not difficult to imagine that the Stern-Gerlach magnet could be made up of electrons exhibiting harmonic motion, even though the experimenter can freely choose the angle of the magnet. At the same time, it is not difficult to imagine that the electrons leaving the source would have the same harmonic motion, for instance due to the fact that they are governed by the same physical law, without any spooky action at a distance.
I don't understand how it would be relevant if the electrons in the magnet are exhibiting harmonic motion. Even if we assume there are some hidden variables in the electrons making up the magnet that are correlated with the hidden variables being measured, this does not in any way imply a correlation between the hidden variables of the particles being measured and the experimenter's choice of which setting to use. The different "settings" like a1 or a2 don't contain information about all the physical details of the measuring device, they only refer to the single visible aspect of the measurement that's being varied, in this case the detection angle. It may well be that hidden variables associated with the magnet are different on one trial with setting a2 than they are on a different trial with setting a2, it doesn't mean we treat them as different settings. You can include any hidden variables associated with the measuring device in \lambda if you like, it doesn't only have to refer to hidden variables associated with the particles being measured. All that matters is that in local realism, any correlation between physical variables (hidden or otherwise) in the local neighborhood of measurement #1 and physical variables in the local neighborhood of measurement #2 must be explained by common events in the overlap of the past light cones in these two regions, the idea I was talking about in post #51 when I said:
In a universe with local realist laws, the results of a physical experiment on any system are assumed to be determined by some set of variables specific to the region of spacetime where the experiment was performed. There can be a statistical correlation (logical dependence) between outcomes A and B of experiments performed at different locations in spacetime with a spacelike separation, but the only possible explanation for this correlation is that the variables associated with each system being measured were already correlated before the experiment was done, that the systems had "inherited" correlated internal variables from some event or events in the overlap of their past light cones. Do you disagree? If so, try to think of a counterexample that we can be sure is possible in a local realist universe (no explicitly quantum examples).

If you don't disagree, then the point is that if the only reason for the correlation between A and B is that the local variables \lambda associated with system #1 are correlated with the local variables associated with system #2, then if you could somehow know the full set of variables \lambda associated with system #1, knowing the outcome B when system #2 is measured would tell you nothing additional about the likelihood of getting A when system #1 is measured. In other words, while P(A|B) may be different than P(B), P(A|B\lambda ) = P(A | \lambda ). If you disagree with this, then I think you just haven't thought through carefully enough what "local realist" means.
In this case, one can refactor the above equation easily to
P(AB) = \sum_i P(A|a_ib_i t_i\lambda_i)P(B|a_ib_it_i \lambda_i)P(a_ib_it_i\lambda_i)

Do you agree that in this case, since all the variables after | are dependent on time and thus on each other, it is not valid to reduce this equation to
P(AB) = \sum_i P(A|a_i)P(B|b_i )P(\lambda_i)[/tex]
As I said before, "Bell's theorem is about picking the optimum experimental situation for ruling out local hidden variables, we don't have to consider arbitrary variations in the experimental settings that we control." The time of the two measurements is one of those settings that we control. If we want to arrange things so that each experimenter makes their measurement at the same prearranged time on every trial, we're free to do so, in this case when summing over many trials we don't have to sum over variations in time--if we can rule out local hidden variables theories in this experiment, then that means local hidden variables theories can't account for all the physics of our universe, period. And even if the time is varied randomly (each experimenter has a randomized timer that tells them when to choose what axis to measure, for example), we should still be able to arrange things so there's no correlation between the time an experimenter makes the choice and what angle they choose. So in this case it would be valid to reduce your equation above to one where only \lambda is a function of time:

P(AB) = \sum_i \sum_j \sum_k \sum_l (Constant) P(A|a_i \lambda_k (t_l) ) P(B|b_j \lambda_k (t_l) ) P(\lambda_k (t_l) )

And in this case, remember my comments earlier about a time-varying \lambda. As I said, in local realism any correlation between physical variables in the region of the two spacelike-separated measurements--whether the physical variables are hidden variables associated with the particles, hidden variables associated with the measuring devices, or the actual observed choice of measurement settings--must be explained by a common inheritance from the overlap of the past light cone of the measurements. As long as we assume no correlation between the experimenter's choices about what measurement settings to use and the physical variables inherited from this overlap region that explain correlations in hidden variables and outcomes at the two measurement-events, then it must be true that on every trial, the answers for each possible measurement choice were already predetermined in this overlap region, in order to explain how they always get opposite answers when they happen to choose the same measurement setting.
In which case such a hidden variable theory was not considered by Bell. If you do, how can you say Bell's theorem disproves all hidden variable theorems. If you think this is not a hidden variable theory, tell me why.
All we need is a single type of experiment that gives results that contradict local hidden variables theories, and we've shown that such theories can't explain the physics of our universe. If we want to design the experiment so that both measurements are always performed after the same time interval, or so that there is no correlation between the time the measurements are made and the choice of detector angles, we are free to do so. You can consider time variation in the hidden variables themselves since that's out of our control, but I gave an argument above as to why this doesn't make a difference.
Consider a different hidden variable theory in which the material of which the magnet is a deterministic learner. By this I mean, on interacting with an electron from the i-th measurement event, the material updates it's state based on the value of hidden variable of the electron and it's own hidden variable value. In other words, there is some memory effect left over from the interaction. Then when the electron from the (i+1)th even arrives, the same process repeats over and over.

Do you now see that in such a case, a_i is not independent from a_{i+1} and therefore Bell's factorisation P(AB) = \sum_i P(A|a_i)P(B|b_i )P(\lambda_i) is no longer valid? Do you agree with this? If you think this is not a hidden variable theorem, explain how?
Again, a_i only refers to the single visible aspect of the device which we vary, not to other hidden aspects of the device which can be included in \lambda.

To make this more concrete, I think it would really help if you'd address the example of the scratch-off lotto cards I gave in post #3. You are free to imagine that instead of a static fruit printed underneath the scratch-off material on each box, behind the scratch off material is a screen connected to a computer in the card which can vary what fruit will be revealed depending on when an experimenter scratches a box. You can also imagine that the experimenter is using a coin to scratch one of the boxes and reveal the fruit, and that the coin contains all sorts of complicated internal hidden variables that can be in communication with the card it's scratching (including some kind of 'learning material' which remembers which boxes have been scratched in the past and communicates this to the computer in the card). None of this would change the basic fact that if Alice and Bob always get opposite fruits on trials where they pick the same box to scratch, then under a local hidden variables theory where the source creating the cards has no advanced knowledge of what choices they'll make, it should be absolutely impossible for them to get opposite fruits less than 1/3 of the time when they pick different boxes to scratch. Do you disagree?
As you hopefully can see now, Bell's inequalities is just a mathematical theorem, whose result is a consequence of the specific assumptions imposed, outside of which it will NOT be valid. Bell's theorem has no experimental basis and has NEVER been proven experimentally! All known experiments violate Bell's theorem! Quantum Mechanics violates Bell's theorem! Think about that for a moment.
You're getting the terminology confused here--QM violates the Bell inequalities, but Bell's theorem is essentially the statement that "in any universe where the laws of physics obey local realism (along with the no-conspiracy assumption), no experiment should violate the Bell inequalities". So, if Bell's theorem is valid, then experimental violations of Bell inequalities just shows that we do not live in a universe where the laws of physics obey local realism (along with the no-conspiracy assumption).

mn4j

Jan12-09, 01:25 PM

it's assumed we set things up so the experimenters make each combination of measurements equally frequently, so P(a1b1)=P(a2b1)=P(a3b1) etc., all have the same probability 1/N where N is the number of possible combinations.
Do you agree that to be consistent, you MUST include the possibility that magnets are also governed by local hidden variables so that a_i and b_i represents not only the subset of settings that the experimenter freely chose, but the COMPLETE state of the magnet at the time of the measurement?

I already gave you the example of the measuring device being like a pendulum hidden in a black box where the experimenter freely changes the length of the string but has no other control over the inner working of the box. I also showed you how in fact this is a possible scenario for a local-hidden variable governed Stern-Gerlach magnet where, even though the experimenter can freely choose the angle, they have no control over the harmonic motion of the individual particles making up the magnet. I need a simple yes or no from you whether you think this is possible local-hidden variable description of the behaviour of the Magnet.

Yes. Of course with the independence of a-measurements from b-measurements, this is a situation that we arrange just by telling the experimenters to choose randomly on each trial (perhaps each uses a separate random-number generator, or each rolls a separate die or something), it isn't an assumption about the way the laws of physics work.

Are you aware that any two objects, exhibiting harmonic motion are correlated, by virtue of circular symmetry, irrespective of differences of frequency or phase and such correlation is not necessarily due to spooky action at a distance? If you disagree, consider two harmonic oscilators which obey the following wave equation,

y(t) = A sin(\omega t + \theta)
Pick any two combinations (1,2) of (A, \omega and \theta) and plot y1 vs y2 for the same t for a given time range and see if you change your mind.

The a's and b's represent choices made by the experimenters, unless you think it would be impossible to set things up so that their choices are uncorrelated with one another, I don't see the relevance here. Remember, Bell's theorem is about picking the optimum experimental situation for ruling out local hidden variables, we don't have to consider arbitrary variations in the experimental settings that we control.

Do you believe that the experimenters can control the harmonic behaviour of the atoms and subatomic particles within their magnets? If you don't then you must agree as I said above that a_i and b_i MUST represent not only the subset of settings that the experimenter freely chose, but the COMPLETE state of the magnet at the time of the measurement, including all local-hidden variables of the magnets. For the two oscillations which you plotted above and saw that they correlated, can you explain how it is possible to design an experiment in which such correlation will not be observed, without using any information about the HIDDEN behaviour?

We do have to consider variations in the nature of the hidden variables associated with the particles, since we don't control those--so, it would be appropriate to imagine if the hidden variables associated with each particle might vary over time. But remember that according to QM, if both experimenters measure along the same axis they'll always get opposite spins (or the same spins, depending on what particles are used and how they are entangled), even if they measure at different times.
This is circular reasoning. Bell did not use QM to derive his inequalities. So what QM predicts should happen, is irrelevant to the derivation of Bell's inequalities.

And in a local hidden variables theory we are assuming that there is no physical "communication" between the particles once they are separated, any correlation in their measured behavior must be due to local physical variables--which could be time-varying functions--that were "assigned" to each one in a correlated way when the particles were at a common location.

Bell believed (and apparently you do too), that the only possible way to have any correlation between
a_i and b_i is by psychokinesis (spooky action at a distance). I have just give you above a situation in which there can be correlation between any two harmonic oscillators without psychokinesis and if you are consistent in not only assigning local-hidden variables to the particles but also to the measuring devices, and the local variables can exhibit harmonic time dependent motion, there will be a correlation without any psychokinesis.

So, if we make the assumption that there's no correlation between the hidden variable functions assigned to each particle when they were in the same location and the experimenters' later choices about how/when to measure them, the only way to explain this perfect correlation when they are measured on the same axis (regardless of when the measurements are made) is if the hidden variables predetermine a single answer each particle will give to being measured on any given axis, and there's no time variation in this answer (though there could be time variation in other aspects of the hidden variables as long as they don't change what answer a given particle would give when measured on a particular axis at different times). Do you agree?

No! I disagree, because the assumption of no correlation, excludes other valid local-hidden variable theories explained above, and if indeed this was the assumption Bell made, his theorem is only valid within the confines of the assumption.

Yes, I agree--the proof covers every possible hidden variable theory that meets the stated conditions--i.e. locality (there are nonlocal hidden variables theories which Bell's theorem doesn't apply to), realism, and the assumption about a lack of "conspiracy" in the

I have already shown you above that there are hidden variables which meet the criteria of locality, realism and lack of conspiracy which Bell did not consider. In other words, those are not the only condidions Bell imposed. He also implicity left out time-varying hidden variables of the type I've mentioned.

I don't understand how it would be relevant if the electrons in the magnet are exhibiting harmonic motion.

It is relevant because any two harmonic oscillators are correlated as demonstrated above. Therefore a_i and b_i understood as complete representations of the local state of the measuring stations are not logically independent.

The different "settings" like a1 or a2 don't contain information about all the physical details of the measuring device, they only refer to the single visible aspect of the measurement that's being varied
Why should it matter, if some of these settings are part of the natural dynamics of the measuring device? Why is it inappropriate to also describe the electrons in the devices with local hidden variables in addition to the 'settings'?

You can include any hidden variables associated with the measuring device in \lambda if you like
No. It has to be associated with a_i and b_i not \lambda because \lambda represents the hidden variable shared between the particles and to avoid consipiracy, those variables have to be separate from those of the measuring devices.

it doesn't only have to refer to hidden variables associated with the particles being measured. All that matters is that in local realism, any correlation between physical variables (hidden or otherwise) in the local neighborhood
Wrong. Then it would be a global variable not a local one. Read Bell's article. Global variables don't come in at all. It is very easy to explain spooky action at a distance using global variables!

So in this case it would be valid to reduce your equation above to one where only \lambda is a function of time.

P(AB) = \sum_i \sum_j \sum_k \sum_l (Constant) P(A|a_i \lambda_k (t_l) ) P(B|b_j \lambda_k (t_l) ) P(\lambda_k (t_l) )

No! Give me a good reason why each entity should not get it's own local variables, with the only variables in common being the ones shared by the particles from their source?

..in order to explain how they always get opposite answers when they happen to choose the same measurement setting.
Again, Bell did not use QM to derive the inequalities so this statement is completely out of place. The result in one orientation, says nothing about the mechanism by which the results are obtained!

Again, a_i only refers to the single visible aspect of the device which we vary, not to other hidden aspects of the device which can be included in \lambda.
Give me a good reason why it should not describe the complete state of the measuring device, just like in any real experiment which will ever be performed?

You're getting the terminology confused here--QM violates the Bell inequalities, but Bell's theorem is essentially the statement that "in any universe where the laws of physics obey local realism (along with the no-conspiracy assumption), no experiment should violate the Bell inequalities". So, if Bell's theorem is valid, then experimental violations of Bell inequalities just shows that we do not live in a universe where the laws of physics obey local realism (along with the no-conspiracy assumption).
No. I'm not. Both Bell's theorem and Bell's inequality are only valid within the narrow set of conditions he imposed while deriving Bell's inequalities. Can you point me to a single experiment that confirms Bell's inequalities. If you can't then how can you claim that it has be validated. If Bell's inequalities have never been validated experimentally, how can you claim that Bell's theorem, which is based on the inequalities has been validated.

The argument is like saying:
All real spiders must have 6 legs. Any spiders with more than 6 legs are not real. And then when somebody finds a spider with 8 legs, instead of evaluating the first premise, you instead conclude that the 8 legged spider is not real.

JesseM

Jan14-09, 02:34 AM

Do you agree that to be consistent, you MUST include the possibility that magnets are also governed by local hidden variables so that a_i and b_i represents not only the subset of settings that the experimenter freely chose, but the COMPLETE state of the magnet at the time of the measurement?
No, as I said the different a's and b's are defined to simply represent the distinct orientations of the spin-measuring device, if you think there are other properties of the measuring devices which vary on different trials and are relevant to determining the measurement outcome, these properties should be included in the \lambda's.
I already gave you the example of the measuring device being like a pendulum hidden in a black box where the experimenter freely changes the length of the string but has no other control over the inner working of the box. I also showed you how in fact this is a possible scenario for a local-hidden variable governed Stern-Gerlach magnet where, even though the experimenter can freely choose the angle, they have no control over the harmonic motion of the individual particles making up the magnet. I need a simple yes or no from you whether you think this is possible local-hidden variable description of the behaviour of the Magnet.
Yes, I already said it was possible, and I already said it should be included in \lambda, the a's and b's are defined to refer just to the single property of the measuring device that the experimenters vary.
Are you aware that any two objects, exhibiting harmonic motion are correlated, by virtue of circular symmetry, irrespective of differences of frequency or phase and such correlation is not necessarily due to spooky action at a distance? If you disagree, consider two harmonic oscilators which obey the following wave equation,

y(t) = A sin(\omega t + \theta)
Pick any two combinations (1,2) of (A, \omega and \theta) and plot y1 vs y2 for the same t for a given time range and see if you change your mind.
Is this equation derived from Newtonian equations where it's assumed that forces are transmitted instantaneously? If so it's not relevant to the question of how things work in a local realist universe with a speed-of-light limit on physical effects. Maybe an equation like that could also apply to something like charged particles being bobbed along by an electromagnetic plane wave, I dunno (though in this case the charged particles would not be influencing one another, they'd both just be passively influenced by electromagnetic waves which must have been generated by other charges in the overlap of their past light cones). It should be obvious that in a relativistic universe, any correlation between events with a spacelike separation must be explainable in terms of other events in the overlap of their past light cones. If you disagree, please give a detailed physical model of a situation in electromagnetism (the only non-quantum relativistic theory of forces I know of) where this would not be true. Or just give a simpler situation compatible with relativity, like two balls being drawn from an urn and shipped off in boxes at sublight speeds to Alice and Bob, where it wouldn't be true.
Do you believe that the experimenters can control the harmonic behaviour of the atoms and subatomic particles within their magnets? If you don't then you must agree as I said above that a_i and b_i MUST represent not only the subset of settings that the experimenter freely chose, but the COMPLETE state of the magnet at the time of the measurement, including all local-hidden variables of the magnets.
Why "must" it? Again, the a's and b's are defined to mean just the settings that the experimenters control. If there are other physical variables associated with the measuring devices, and we choose to define \lambda to include these variables as well as variables associated with the particles being measured, what problem do you see with this? Can't we define symbols to mean whatever we want them to, and isn't it still true that in this case the combination of the a-setting and the \lambda value will determine the probability of the physical outcome A?
For the two oscillations which you plotted above and saw that they correlated, can you explain how it is possible to design an experiment in which such correlation will not be observed, without using any information about the HIDDEN behaviour?
As always, the "information about the hidden behavior" is assumed to be included in the value of \lambda. \lambda can be understood to give the value of all local physical variables in the immediate spacetime region of one measurement which are relevant to determining the outcome of that measurement.
We do have to consider variations in the nature of the hidden variables associated with the particles, since we don't control those--so, it would be appropriate to imagine if the hidden variables associated with each particle might vary over time. But remember that according to QM, if both experimenters measure along the same axis they'll always get opposite spins (or the same spins, depending on what particles are used and how they are entangled), even if they measure at different times.
This is circular reasoning. Bell did not use QM to derive his inequalities. So what QM predicts should happen, is irrelevant to the derivation of Bell's inequalities.
No, but the fact that we always see opposite results on trials where the settings are the same is an observed experimental fact, and a variant of Bell's theorem can be used to show that if we observe this experimental fact and if the experiment is set up in the way Bell describes (with each experimenter making a random choice among three distinct detector angles) and if the universe is a local realist one (with the no-conspiracy assumption), then we should expect to see opposite results at least 1/3 of the time on the subset of trials where the experimenters chose different measurement settings. Since this Bell inequality is violated in real life, that means at least one of the "if" statements must fail to be true as well, and since we can verify directly that the first true were true, it must be the third one about the universe being local realist that's false (see my next post for an elaboration of this logic).
Bell believed (and apparently you do too), that the only possible way to have any correlation between
a_i and b_i is by psychokinesis (spooky action at a distance).
I assume you are still incorrectly defining the a's and b's to refer to all physical aspects of the measuring devices, and that if you used the correct definitions, what you really mean here is that Bell believed any correlation in the values of variables in \lambda associated with one spacetime region and the values of variables in \lambdaassociated with another spacetime region at a spacelike separation from the first would by spooky action at a distance. But of course this isn't true either, the whole point of a hidden variables explanation for correlations in measurement outcomes is that there can be correlations in the values of local hidden variables in different regions with a spacelike separation, as long as these correlations were determined by events in the overlap of the past light cones of the two regions. I've repeated this over and over so there's really no excuse for your continued mischaracterization of the argument.
I have just give you above a situation in which there can be correlation between any two harmonic oscillators without psychokinesis and if you are consistent in not only assigning local-hidden variables to the particles but also to the measuring devices, and the local variables can exhibit harmonic time dependent motion, there will be a correlation without any psychokinesis.
And as I said, in any relativistic model of a harmonic oscillator (which I don't think your equation is, though as I said it might be possible to find a situation in electromagnetism where the equation applies), correlations in the values of physical variables in different regions with a spacelike separation would be explained by physical causes in the overlap of the past light cones of these two regions.
So, if we make the assumption that there's no correlation between the hidden variable functions assigned to each particle when they were in the same location and the experimenters' later choices about how/when to measure them, the only way to explain this perfect correlation when they are measured on the same axis (regardless of when the measurements are made) is if the hidden variables predetermine a single answer each particle will give to being measured on any given axis, and there's no time variation in this answer (though there could be time variation in other aspects of the hidden variables as long as they don't change what answer a given particle would give when measured on a particular axis at different times). Do you agree?
No! I disagree, because the assumption of no correlation, excludes other valid local-hidden variable theories explained above, and if indeed this was the assumption Bell made, his theorem is only valid within the confines of the assumption.
Well, you're simply confused about the physical meaning of a "local realist" universe then. The statement I give above is a general truth about perfect correlations in regions with a spacelike separation in any universe with local realist laws--the only way to explain perfect correlations between events with a spacelike separation is to assume that the events were totally predetermined by other events in the overlap of the past light cones of the two regions. Again, if you disagree, please think up a situation compatible with relativistic physics (no instantaneous Newtonian forces) where this wouldn't be true.
The different "settings" like a1 or a2 don't contain information about all the physical details of the measuring device, they only refer to the single visible aspect of the measurement that's being varied
Why should it matter, if some of these settings are part of the natural dynamics of the measuring device? Why is it inappropriate to also describe the electrons in the devices with local hidden variables in addition to the 'settings'?
Where did you get the idea I said it was inappropriate? I explicitly said you could include these local hidden variables, but they should be included in \lambda, not in the a's and b's.
You can include any hidden variables associated with the measuring device in if you like
No. It has to be associated with a_i and b_i not \lambda because \lambda represents the hidden variable shared between the particles and to avoid consipiracy, those variables have to be separate from those of the measuring devices.
No, there is no rule that \lambda cannot include hidden variables not directly associated with the particles, it can include any physical variables that are local to the spacetime regions of the two measurements. You misunderstand the "no-conspiracy" condition if you think there can't be a correlation between the value of hidden variables associated with the particle and hidden variables associated with the measuring device--it's only a correlation between such hidden variables and the experimenter's free choice of how to set the angle that would be called a "conspiracy".
it doesn't only have to refer to hidden variables associated with the particles being measured. All that matters is that in local realism, any correlation between physical variables (hidden or otherwise) in the local neighborhood
Wrong. Then it would be a global variable not a local one. Read Bell's article. Global variables don't come in at all. It is very easy to explain spooky action at a distance using global variables!
Of course it wouldn't be global, I just said I was talking about variables in the local neighborhood of each measurement. If it makes it more clear, you can use the symbol \lambda to refer to the value of local physical variables in the spacetime region of one experimenter's measurement, and some other symbol like \phi to refer to local physical variables in the spacetime region of the other experimenter's measurement. In this case we can say that if the experimenters always get opposite outcomes when they both pick identical detector angles (call these identical settings a1 and b1), then it must be true that the result for experimenter #1 is fully determined by the combination of a1 and \lambda, while the result for experimenter #2 is fully determined by the combination of b1 and \phi, and that events in the overlap of the past light cones of these two regions cause \lambda and \phi to be correlated in such a way that the predetermined outcome given a1 + \lambda is guaranteed to be the opposite of the predetermined outcome given b1 + \phi.
No! Give me a good reason why each entity should not get it's own local variables, with the only variables in common being the ones shared by the particles from their source?
Each spacetime region can get its own separate local variables as above if you want to write it that way. But once the particle is in the same region as the measuring-device, there's no reason it couldn't have a physical influence on the hidden variables associated with that measuring-device. Of course it would still be true that any correlations in the hidden variables associated with the two measuring-devices in different regions would still be explained by causal influences from the overlap of the past light cones of the regions (in this case, the causal influences would be the hidden variables carried by the two particles which influenced the hidden variables of their respective measuring devices, with the value of each particle's hidden variables having been determined when they both came from the source, an event which was indeed in the overlap of the two past light cones).
Again, Bell did not use QM to derive the inequalities so this statement is completely out of place. The result in one orientation, says nothing about the mechanism by which the results are obtained!
See above, the fact that it's an experimental observation that you get opposite results on the same setting is part of the derivation of the conclusion that in a local realist universe, you should get opposite results at least 1/3 of the time when the experimenters choose different settings. Of course this particular inequality was not actually the one Bell derived in his original paper, though it is a valid Bell inequality--for the inequality he derived in the original paper, see post #8 of this thread (http://www.physicsforums.com/showthread.php?t=210793) which I linked to back in post #3 here. However, this inequality also includes in the derivation the fact that a perfect correlation (or anticorrelation) is seen when the experimenters choose the same detector setting.
Give me a good reason why it should not describe the complete state of the measuring device, just like in any real experiment which will ever be performed?
There is obviously no cosmic force that compels you to assign particular variables a particular physical meaning. Symbols can mean whatever we define them to mean. But by the same token, there is obviously nothing stopping us from using the a's and b's to refer only to the settings chosen by the experimenters, and to include any other physical variables associated with the detectors in the variable representing all the physical hidden variables \lambda (and as I said you could make a minor tweak to the proof to have two different variables for the two distinct spacetime regions if you prefer). Bell's proof definitely depends on the assumption that the a's and b's refer only to the choices made by the experimenters, so if you want to follow Bell's proof you should adopt this convention, which is as good as any other convention.

(continued in next post)

JesseM

Jan14-09, 02:35 AM

(continued from previous post)
No. I'm not. Both Bell's theorem and Bell's inequality are only valid within the narrow set of conditions he imposed while deriving Bell's inequalities. Can you point me to a single experiment that confirms Bell's inequalities. If you can't then how can you claim that it has be validated. If Bell's inequalities have never been validated experimentally, how can you claim that Bell's theorem, which is based on the inequalities has been validated.

The argument is like saying:
All real spiders must have 6 legs. Any spiders with more than 6 legs are not real. And then when somebody finds a spider with 8 legs, instead of evaluating the first premise, you instead conclude that the 8 legged spider is not real.
You really have the logic totally confused--your analogy has nothing to do with deriving certain conclusions from theoretical assumptions about the laws of physics, saying "in a universe where the laws of physics take form X, under experimental conditions Y we should be guaranteed to see results Z" is a theoretical deduction, nothing like the arbitrary definition "all real spiders must have 6 legs".

Your comment "Can you point me to a single experiment that confirms Bell's inequalities" also shows confusion about the logic of what Bell was trying to do--the whole point is that Bell's inequalities are violated, thus demonstrating that the assumption of local realism (along with the no-conspiracy assumption) must be false! Are you familiar with the idea of "the contrapositive" or "contraposition" in logic? (see here (http://whyslopes.com/etc/ThreeSkillsForAlgebra/ch29.html) and here (http://en.wikipedia.org/wiki/Contraposition)). The idea here is that if you can prove that A logically implies B, that is logically equivalent to the statement that if B is false, then logically A must be false as well. As it says at the bottom of the second article from wikipedia, this can be a good way of doing proofs by contradiction:
Because the contrapositive of a statement always has the same truth value (truth or falsity) as the statement itself, it can be a powerful tool for proving mathematical theorems via proof by contradiction, as in the proof of the irrationality of the square root of 2. By the definition of a rational number, the statement can be made that "If \sqrt{2} is rational, then it can be expressed as an irreducible fraction". This statement is true because it is a restatement of a true definition. The contrapositive of this statement is "If \sqrt{2} cannot be expressed as an irreducible fraction, then it is not rational". This contrapositive, like the original statement, is also true. Therefore, if it can be proven that \sqrt{2} cannot be expressed as an irreducible fraction, then it must be the case that \sqrt{2} is not a rational number.
The logic of Bell's theorem, and how when combined with the observed confirmation of quantum predictions it can be used to show that QM is incompatible with local hidden variables, is essentially the same. Here, the A in "A implies B" can be divided into three conditions:

A1: A universe where the laws of physics respect local realism with the no-conspiracy assumption.
A2: An experimental setup where two experimenters make measurements at a spacelike separation, and each is choosing from three possible detector settings which can be labeled a1-a3 for the first experimenter and b1-b3 for the second. The experimenters are making free random choices of which setting to use on each trial.
A3: It is observed to be the case that whenever the experimenters both choose the same setting, they always get opposite results.

Now for B, one version of Bell's theorem proves that these conditions lead logically to the following conclusion:

B: On the subset of trials where experimenters choose different settings, the probability that they get opposite results should be greater than or equal to 1/3.

Now, this experiment can be done, and we can verify that conditions A2 and A3 both apply, yet B is false. So using contraposition, we know some part of A must be false, and since A2 and A3 can be directly verified to be true, the false part must be the theoretical assumption A1.

mn4j

Jan14-09, 08:14 PM

Yes, I already said it was possible, and I already said it should be included in \lambda, the a's and b's are defined to refer just to the single property of the measuring device that the experimenters vary.

You still have not said why it should not be separate. It would seem that if Bell's proof was robust, it should be able to accommodate hidden variables at the sources in addition to source parameters. It should tell you a lot that the hidden variables must be defined a specific way in order for the proof to work. Since you are the one claiming that Bell's proof eliminates all possible local-hidden variable theorems, the onus is on you to explain why the stations should not be able to get separate local hidden variables. Probably because you already know that Bell's theorem can not be formulated in that context. Therefore, you can not claim with a straight face that all real local hidden variable theorems are out.

Is this equation derived from Newtonian equations where it's assumed that forces are transmitted instantaneously?
You don't recognize a simple wave equation? This has nothing to do with physics, it is mathematics. If you did the plot as I explained you will see that any two sinusoidal waves are correlated irrespective of phase, amplitude and frequency. In other words, there is no way you can design an experiment that will eliminate the correlation if you do not know the specific parameters for each wave.

If so it's not relevant to the question of how things work in a local realist universe with a speed-of-light limit on physical effects.
It is relevant. Especially since we know about wave-particle duality. It should tell you that we do not need psychokinesis to explain correlations between distant objects.

It should be obvious that in a relativistic universe, any correlation between events with a spacelike separation must be explainable in terms of other events in the overlap of their past light cones. If you disagree, please give a detailed physical model of a situation in electromagnetism (the only non-quantum relativistic theory of forces I know of) where this would not be true.
You obviously have not thought it through well enough. Two objects can be correlated because they are governed by the same physical laws, whether or not they share a common past or not. This is obvious. A pendulum clock on opposite sides of the globe made by different local manufacturers are correlated by virtue of the fact that they exhibit harmonic motion. What do you claim is the common event in their past that is the source of their correlations?

Why "must" it? Again, the a's and b's are defined to mean just the settings that the experimenters control. Can't we define symbols to mean whatever we want them to, and isn't it still true that in this case the combination of the a-setting and the \lambda value will determine the probability of the physical outcome A?

It must, because you claim that Bell's theorem eliminates ALL hidden variable theorems. It is telling that the terms were so narrowly defined that other possible local hidden variable theorems do not fit.
No you can't define the terms to mean whatever you want them to. You have to define them so that they include all possible hidden variable theorems. Therefore the conclusion of Bell's theorem is handicapped.

No, but the fact that we always see opposite results on trials where the settings are the same is an observed experimental fact, and a variant of Bell's theorem can be used to show that if we observe this experimental fact and if the experiment is set up in the way Bell describes (with each experimenter making a random choice among three distinct detector angles) and if the universe is a local realist one (with the no-conspiracy assumption), then we should expect to see opposite results at least 1/3 of the time on the subset of trials where the experimenters chose different measurement settings. Since this Bell inequality is violated in real life, that means at least one of the "if" statements must fail to be true as well, and since we can verify directly that the first true were true, it must be the third one about the universe being local realist that's false (see my next post for an elaboration of this logic).

You forgot a very important "if" that is the very topic of this thread, ie

"if we observe this experimental fact"
"if a local realist universe behaves only as described by Bell's assumptions"
"if the universe is a local realist one (with the no-conspiracy assumption)"
"if the experiment is set up in the way Bell describes"
then we should expect to see opposite results at least 1/3 of the time on the subset of trials where the experimenters chose different measurement settings.

As you can see, violation of 5 can imply that either (2), (3) or (4) or combinations of them are wrong. For some probably religious reason, proponents of Bell's theorem, jump right to (3) and claim that it must be (3) that is wrong.

I have given you already two examples of hidden variable theorems that point to the falsity of (2). In fact, (2) is the proverbial "a spider must have 6 legs". Do you deny that the validity of Bell's theorem rests as much on (2) as on (3) or (4). It remains to be seen whether any experiment has ever been performed which exactly reproduced Bell's assumptions. But that is a different topic.

For other more rigorous proofs why (2) is wrong, see:

Brans, CH (1988). Bell's theorem does not eliminate fully causal hidden variables. 27, 2 , International Journal of Theoretical Physics, 1988, pp 219-226

Joy Christian, "Can Bell's Prescription for Physical Reality Be Considered Complete?"
[http://arxiv.org/pdf/0806.3078v1

See, Hess K, and Philipp W (2000). PNAS ͉ December 4, 2000 ͉ vol. 98 ͉ no. 25 pp 14228-14233 for a proof that Bell's theorem can not be derived for time-like correlated parameters, and that such variables produce the QM result.

See also, Hess K, and Philipp W (2003), "Breakdown of Bell's theorem for certain objective local parameter spaces"
PNAS February 17, 2004 vol. 101 no. 7 1799-1805

Well, you're simply confused about the physical meaning of a "local realist" universe then. The statement I give above is a general truth about perfect correlations in regions with a spacelike separation in any universe with local realist laws--the only way to explain perfect correlations between events with a spacelike separation is to assume that the events were totally predetermined by other events in the overlap of the past light cones of the two regions. Again, if you disagree, please think up a situation compatible with relativistic physics (no instantaneous Newtonian forces) where this wouldn't be true.
I suppose all those people cited above are also confused, as is Jaynes. Yet you have not shown me a single reason why my descriptions of the two scenarios in pos #55 are not valid realist local hidden variable theorems. For some reason you ignored the second scenario completely and did not even bother to say whether a "deterministic learning machine" is local or not.

You can see the following articles, for proof that a local deterministic learning hidden variable model reproduces the quantum result:

Raedt, KD, et. al.
A local realist model for correlations of the singlet state
The European Physical Journal B - Condensed Matter and Complex Systems, Volume 53, Number 2 / September, 2006, pp 139-142

Raedt, HD, et. al.
Event-Based Computer Simulation Model of Aspect-Type Experiments Strictly Satisfying Einstein's Locality Conditions
J. Phys. Soc. Jpn. 76 (2007) 104005

Peter Morgan,
Violation of Bell inequalities through the coincidence-time loophole
http://arxiv.org/pdf/0801.1776

More about the coincidence time loophole here:
Larson, JA, Gill, RD, Europhys. Lett. 67, 707 (204)

JesseM

Feb13-09, 04:28 PM

Hi mn4j, apologies for not replying to your last post before now, I started it a while ago but realized it would require a somewhat involved response, so I kept putting off writing it for weeks. Anyway, I've finally finished it up:
You still have not said why it should not be separate. It would seem that if Bell's proof was robust, it should be able to accommodate hidden variables at the sources in addition to source parameters.
It is able to do so. I already said "the hidden variables can be included in \lambda"--did you miss that, or are you not understanding it somehow?
It should tell you a lot that the hidden variables must be defined a specific way in order for the proof to work.
Physically the hidden variables can be absolutely anything, but for the proof to work you do need to assign them separate variables from the experimental choices. This is like just about any proof where you can't redefine terms willy-nilly and expect it to still make sense. If the proof is mathematically and logically valid, then you have to accept the conclusions follow from the premises, you can't somehow object to it on the basis that you wish the symbols meant different things than what they are defined to mean.
Since you are the one claiming that Bell's proof eliminates all possible local-hidden variable theorems, the onus is on you to explain why the stations should not be able to get separate local hidden variables.
Do you understand the difference between "the stations should not be able to get separate local hidden variables" and "there can be hidden variables associated with the stations, but the symbols used to refer to them should be separate from the symbols used to refer to the experimenters' choice of measurements angles"? Remember, each value of \lambda is supposed to stand for an array of values for all the hidden variables--we are supposed to have some function that maps values of \lambda to a (possibly very long) list of values for all the different physical variables which may be in play, like "\lambda=3.8 corresponds to hidden variable #1 having value x=7.2 nanometers, hidden variable #2 having value 0.03 meters/second, hidden variable #3 having value 34 cycles/second, ... , hidden variable #17,062,948,811 having value 17 m/s^2", something along those lines. There's no reason at all why the long list of values included in a given value of \lambda can't be values of hidden variables associated with the measuring-device.
You don't recognize a simple wave equation?
Of course I recognize a wave equation--you weren't tipped off by the fact that I immediately suggested the idea of particles being bobbed along by an electromagnetic plane wave? My point was that I wanted to see a well-defined physical scenario, compatible with local realism, in which the equation would actually apply to physical elements with a spacelike separation, but no common cause in their mutual past light cone to explain why they were both obeying this equation (for example, in the example of two particles at different locations being bobbed up and down by an electromagnetic plane wave, the oscillations of the charges which generated this wave would lie in the overlap of the past light cones). However, I've since realized that they might both be synchronized just because of coincidental similarity in their initial conditions, so I've modified my comments about the relevance to Bell's theorem accordingly--see below.
It is relevant. Especially since we know about wave-particle duality. It should tell you that we do not need psychokinesis to explain correlations between distant objects.
Of course wave-particle duality is part of QM, and you can't treat it as a foregone conclusion that QM itself is compatible with local realism.
It should be obvious that in a relativistic universe, any correlation between events with a spacelike separation must be explainable in terms of other events in the overlap of their past light cones. If you disagree, please give a detailed physical model of a situation in electromagnetism (the only non-quantum relativistic theory of forces I know of) where this would not be true.
You obviously have not thought it through well enough. Two objects can be correlated because they are governed by the same physical laws, whether or not they share a common past or not. This is obvious.
If two experimenters at a spacelike separation happen to choose to do the same experiment, then since the same laws of physics govern them they'll get correlated results--but this is a correlation due to the coincidence of their happening to independently replicate the same experiment, not the type of correlation where seeing the results of both experimenters' measurements tells us something more about the system being measured than we'd learn from just looking at the results that either experimenter gets on their own. This does show that my statement above is too vague though, and needs modification. One way to sharpen things a little would be to specify we're talking about experiments where even with the same settings the experimenters can get different results on different trials, with the results being seemingly random and unpredictable; if we find that the results of the two experimenters are nevertheless consistently correlated, with a spacelike separation between pairs of measurements, this is at least strongly suggestive of the idea that each result was conditioned by events in the past light cones of the two measurements. But this is still not really satisfactory, because in principle there might actually be some hidden deterministic pattern behind the seemingly random results, and it might be that the two systems they were studying coincidentally had identical and synchronized deterministic patterns (for example, they might both be looking out a series of numbers generated by a pseudorandom deterministic computer program, with the programmers at different locations coincidentally having written exactly the same program without having been influenced to do so by a common cause in their mutual past light cone). So, back to the drawing board!

Let me try a different tack. Consider the claim I was making about correlations in a local realist universe earlier, which you were disputing for a while but then stopped after my post #51, so I'm not really sure if I managed to convince you with that post...here's the statement from post #51:
In a universe with local realist laws, the results of a physical experiment on any system are assumed to be determined by some set of variables specific to the region of spacetime where the experiment was performed. There can be a statistical correlation (logical dependence) between outcomes A and B of experiments performed at different locations in spacetime with a spacelike separation, but the only possible explanation for this correlation is that the variables associated with each system being measured were already correlated before the experiment was done ... Do you disagree? If so, try to think of a counterexample that we can be sure is possible in a local realist universe (no explicitly quantum examples).

If you don't disagree, then the point is that if the only reason for the correlation between A and B is that the local variables \lambda associated with system #1 are correlated with the local variables associated with system #2, then if you could somehow know the full set of variables \lambda associated with system #1, knowing the outcome B when system #2 is measured would tell you nothing additional about the likelihood of getting A when system #1 is measured. In other words, while P(A|B) may be different than P(B), P(A|B\lambda ) = P(A | \lambda ). If you disagree with this, then I think you just haven't thought through carefully enough what "local realist" means.
Note that I put some ellipses in the quote above, the statement I removed was "that the systems had 'inherited' correlated internal variables from some event or events in the overlap of their past light cones". I want to retract that part of the post since it does have some problems as you've pointed out, but I stand by the rest. The statement about "variables specific to the region of spacetime where the experiment was performed" could stand to be made a little more clear, though. To that end, I'd like to define the term "past light cone cross-section" (PLCCS for short), which stands for the idea of taking a spacelike cross-section through the past light cone of some point in spacetime M where a measurement is made; in SR this spacelike cross-section could just be the intersection of the past light cone with a surface of constant t in some inertial reference frame (which would be a 3D sphere containing all the events at that instant which can have a causal influence on M at a later time). Now, let \lambda stand for the complete set of values of all local physical variables, hidden or non-hidden, which lie within some particular PLCCS of M. Would you agree that in a local realist universe, if we want to know whether the measurement M yielded result A, and B represents some event at a spacelike separation from M, then although knowing B occurred may change our evaluation of the probability A occurred so that P(A|B) is not equal to P(A), if we know the full set of physical facts \lambda about a PLCCS of M, then knowing B can tell us nothing additional about the probability A occurred at M, so that P(A|\lambda) = P(A|\lambda B)?

If so, consider two measurements of entangled particles which occur at spacelike-separated points M1 and M2 in spacetime. For each of these points, pick a PLCCS from a time which is prior to the measurements, and which is also prior to the moment that the experimenter chose (randomly) which of the three detector settings under his control to use (as before, this does not imply the experimenter has complete control over all physical variables associated with the detector). Assume also that we have picked the two PLCCS's in such a way that every event in the PLCCS of M1 lies at a spacelike separation from every event in the PLCCS of M2. Use the symbol \lambda_1 to label the complete set of physical variables in the PLCCS of M1, and the symbol \lambda_2 to label the complete set of physical variables in the PLCCS of M2. In this case, if we find that whenever the experimenters chose the same setting they always got the same results at M1 and M2, I'd assert that in a local realist universe this must mean the results each of them got on any such trial were already predetermined by \lambda_1 and \lambda_2; would you agree? The reasoning here is just that if there were any random factors between the PLCCS and the time of the measurement which were capable of affecting the outcome, then it could no longer be true that the two measurements would be guaranteed to give identical results on every trial.

Now, keep in mind that each PLCCS was chosen to be prior to the moment each experimenter chose what detector setting to use. So, if we assume that the experimenters' choices were uncorrelated with the values of physical variables \lambda_1 and \lambda_2, either because the choice involved genuine randomness (using the decay of a radioactive isotope and assuming this is a truly random process, for example), or because the choice involved "free will" (whatever that means), then if it's true that \lambda_1 and \lambda_2 predetermine the result on every trial where they happen to make the choice, in a local realist universe we must assume that on each trial \lambda_1 and \lambda_2 predetermine what the results would be for any of the three choices each experimenter can make, not just the result for the choice they do actually make on that trial (since the values of physical variables in the PLCCS cannot 'anticipate' which choice will be made at a later time), the assumption known as counterfactual definiteness (http://en.wikipedia.org/wiki/Counterfactual_definiteness). And if at the time of the PLCCS there was already a predetermined answer for the result of either of the three choices the experimenter could make, then if they always get the same results when they make the same choice, we must assume that on every trial the two PLCCSs had the same predetermined answers for all three results, which is sufficient to show that the Bell inequalities should be respected (see my post #3). It would be simplest to assume that the reason for this perfect matchup between the PLCCSs on every trial was that they had "inherited" the same predetermined answers from some events in the overlap of the past light cones of the two measurements, but this assumption is not strictly necessary.

The deterministic case

If the experimenters' choices are not assumed to be truly random or a product of free will, but instead are pseudorandom events that do follow in some deterministic (but probably chaotic) way from the complete set of physical variables in the PLCCS, then showing that the results for each possible measurement must be predetermined by the PLCCS is trickier. I think we can probably come up with some variant of the "no-conspiracy" assumption discussed earlier that applies in this case, though. To see why it would seem to require a strange "conspiracy" to explain the perfect correlations in a local realist universe without the assumption that there was a predetermined answer for each possible choice (i.e. without assuming counterfactual definiteness), let's imagine we are trying to perform a computer simulation to replicate the results of these experiments. Suppose we have two computers A and B which will simulate the results of each measurement, and a middle computer M which can send signals to A and B for a while but then is disconnected, leaving A and B isolated and unable to communicate at some time t, after which they simulate both an experimenter making a choice and the results of the measurement with the chosen detector setting. Here the state of the information in each computer at time t represents the complete set of physical variables in the PLCCS of the measurement, while the fact that M was able to send each computer signals prior to t represents the fact that the state of each PLCCS may be influenced by events in the overlap of the past light cone of the measurement events.

Also, assume that in order to simulate the seemingly random choices of the experimenters on each trial, the computer uses some complicated pseudorandom algorithm to determine their choice, using the complete set of information in the computer at time t as a seed (http://www.lycos.com/info/pseudorandom-number-generator--seeds.html) for this algorithm, to simulate the idea that an experimenter's brain exhibits sensitive dependence on initial conditions (http://en.wikipedia.org/wiki/Butterfly_effect) so that even in a deterministic universe, everything in the past light cone of the choice has the potential to influence the choice. Finally, assume the initial conditions at A and B are not identical, so the two experimenters are not just perfect duplicates of one another. Then the question becomes: is there any way to design the programs so that the simulated experimenters always get the same outcome when they make the same choice about detector settings, but counterfactual definiteness does not apply, meaning that each computer didn't just have a preset answer for each detector setting at time t, but only a preset answer for the setting the simulated experimenter would, in fact, choose on that trial? Well, if the computer simulations are deterministic over multiple trials so we just have to load some initial conditions at the beginning and then let them run over as many trials as we want, rather than having to load new initial conditions for each trial, then in principle we could imagine some godlike intelligence looking through all possible initial conditions (probably a mind-bogglingly vast number, if N bits were required to describe the state of the simulation at any given moment there'd be 2^N possible initial conditions), and simply picking the very rare initial conditions where it happened to be true that whenever the two experimenters made the same choice, they always get the same results. Then if we run the simulation forward from those initial conditions, it will indeed be guaranteed with probability 1 that they'll get the same results whenever they make the same choice, without the simulation needing to have had predetermined answers for what they would have gotten on these trials if they had made a different choice. But this preselecting of the complete initial conditions, including all the elements of the initial conditions that might influence the experimenters' choices, is exactly the sort of "conspiracy" that the no-conspiracy assumption is supposed to rule out.

So, let's make some slightly different assumptions about the degree to which we can control the initial conditions. Let's say we do have complete control over the data that M sends to A and B on each trial, corresponding to the notion that we want to allow the source to attach hidden variables to the particles it sends to the experimenters in any fiendishly complicated way we can imagine. If you like we are also free to assume we have complete control over any variables, hidden or otherwise, associated with the measuring-devices being simulated in the A and B computers initially at time t (after M has already sent its information to A and B but before the simulated experimenters have made their choice), to fit with your idea that hidden variables associated with the measuring device may be important too. But assume there are other aspects of the initial conditions at A and B that we don't control--perhaps we can only decide what the "macrostate" of the neighborhood of the two experimenters looks like, but the detailed "microstate" is chosen randomly, or perhaps we can decide the values of all non-hidden variables in their neighborhood but not the hidden ones (aside from the ones associated with the particles sent by the source and the measuring devices, as noted above). Since the pseudorandom algorithm that determines each experimenter's choice takes the entire initial state as a seed, this means that without knowing every single precise detail of the initial state, we can't predict what choices the experimenters will make on each trial. So, for all practical purposes this is just like the situation I discussed earlier where the experimenters' choices were truly random and unpredictable, which means that if we only control some of the initial data at time t (the variables sent from M and the variables associated with the measuring-device) but after that must let the simulation run without any further ability to intervene, the only way to guarantee that the experimenters always get the same result when they make opposite choices is to make sure that the data we control at time t guarantees with 100% certainty what results the experimenters would get for any of the three possible choices, in such a way that the predetermined answers match up for computer A and computer B.

JesseM

Feb13-09, 04:29 PM

Part 2 of response

Simulations as a test of proposed hidden-variables theories

That was a somewhat long discussion of the case where the experimenters' brains make their choice in a deterministic way, and given that most people discussing Bell's theorem are willing to grant for the sake of the argument that the choice can be treated as random, perhaps unnecessary. But I think the idea I introduced of trying to simulate EPR type experiments on computers is a very useful one regardless. If anyone proposes that a local hidden variables theory can explain the results of these experiments, there's no reason that such a theory could not be simulated in the setup I described, where a middle computer M can send signals to two different computers A and B until some time t when the computers are disconnected, and some time after t the experimenters (real or simulated) make choices about which orientation to use for the simulated detector (if the experimenters are real people interacting with the simulation they could make this choice by deciding whether to type 1, 2, or 3 on the keyboard, for example), and each computer A and B must return a measurement result. On p. 15 of the Jaynes paper (http://bayes.wustl.edu/etj/articles/cmystery.pdf) you linked to, Jaynes seemed to acknowledge that if there was a local realist theory which could replicate the violations of Bell inequalities, then it should be possible to simulate on independent computers:
The Aspect experiment may show that such theories are untenable, but without further analysis it leaves open the status of other local causal theories more to Einstein's liking.

That future analysis is, in fact, already underway. An important part of it has been provided by Steve Gull's "You can't program two independently running computers to emulate the EPR experiment" theorem, which we learned about at this meeting. It seems, at first glance, to be just what we have needed because it could lead to more cogent tests of these issues than did the Bell argument. The suggestion is that some of the QM predictions can be duplicated by local causal theories only by invoking teleological elements as in the Wheeler-Feynman electrodynamics. If so, then a crucial experiment would be to verify the QM predictions in such cases. It is not obvious whether the Aspect experiment serves this purpose.

The implication seems to be that, if the QM predictions continue to be confirmed, we exorcise Bell's superluminal spook only to face Gull's teleological spook. However, we shall not rush to premature judgments. Recalling that it required some 30 years to locate von Neumann's hidden assumptions, and then over 20 years to locate Bell's, it seems reasonable to ask for a little time to search for Gull's, before drawing conclusions and possibly suggesting new experiments.
So, do you agree with the idea that this is a good way to test claims that someone has thought up a way to reproduce the EPR results with a local realist theory? Earlier you seemed to suggest that they could be reproduced by a theory in which the hidden variables associated with the particle interacted with hidden variables associated with the measuring apparatus in some way--can you explain in a schematic way how this could be simulated? Do you disagree with my statement earlier that in order to explain how experimenters always get the same result when they make the same choice about how to set the simulated detector orientation (which is not to imply there couldn't be other variables associated with the simulated detector that are out of their control), we must assume that at the time t the two computers are disconnected, the state of each computer at that time already predetermines what final result the simulation will give for each possible choice made by the experimenter?
Why "must" it? Again, the a's and b's are defined to mean just the settings that the experimenters control. Can't we define symbols to mean whatever we want them to, and isn't it still true that in this case the combination of the a-setting and the \lambda value will determine the probability of the physical outcome A?
It must, because you claim that Bell's theorem eliminates ALL hidden variable theorems. It is telling that the terms were so narrowly defined that other possible local hidden variable theorems do not fit.
No you can't define the terms to mean whatever you want them to. You have to define them so that they include all possible hidden variable theorems. Therefore the conclusion of Bell's theorem is handicapped.
See the first part of my response--you are simply confused here, adopting a particular labeling convention for what physical facts are labeled with what symbols has no physical implications whatsoever, I cannot possibly be ruling out any local hidden variables theories by choosing to let the letter "a" stand for the choice made by the experimenter. Nothing about this convention rules out the idea that there could be other physical variables associated with the measuring device that the experimenter does not control, it's just that they must be denoted by some symbol other than "a" (I suggested that these other variables could be folded into \lambda, although if you wished you could define a separate symbol for physical variables associated with the measuring-device).
No, but the fact that we always see opposite results on trials where the settings are the same is an observed experimental fact, and a variant of Bell's theorem can be used to show that if we observe this experimental fact and if the experiment is set up in the way Bell describes (with each experimenter making a random choice among three distinct detector angles) and if the universe is a local realist one (with the no-conspiracy assumption), then we should expect to see opposite results at least 1/3 of the time on the subset of trials where the experimenters chose different measurement settings. Since this Bell inequality is violated in real life, that means at least one of the "if" statements must fail to be true as well, and since we can verify directly that the first true were true, it must be the third one about the universe being local realist that's false (see my next post for an elaboration of this logic).
You forgot a very important "if" that is the very topic of this thread, ie

"if we observe this experimental fact"
"if a local realist universe behaves only as described by Bell's assumptions"
"if the universe is a local realist one (with the no-conspiracy assumption)"
"if the experiment is set up in the way Bell describes"
then we should expect to see opposite results at least 1/3 of the time on the subset of trials where the experimenters chose different measurement settings.

As you can see, violation of 5 can imply that either (2), (3) or (4) or combinations of them are wrong. For some probably religious reason, proponents of Bell's theorem, jump right to (3) and claim that it must be (3) that is wrong.

I have given you already two examples of hidden variable theorems that point to the falsity of (2). In fact, (2) is the proverbial "a spider must have 6 legs". Do you deny that the validity of Bell's theorem rests as much on (2) as on (3) or (4). It remains to be seen whether any experiment has ever been performed which exactly reproduced Bell's assumptions. But that is a different topic.
I disagree that #2 is necessary there, no assumptions about the type of hidden-variable theory are needed aside from the fact that it is a local realist one. I confused the issue a bit by making a statement about statistical correlations between spacelike separated events in a local hidden variables theory that you correctly pointed out could be violated in certain cases, but see my revised statements above. Do you agree that in a local realist universe, if \lambda is taken to mean the complete set of local variables in a PLCCS of some point is spacetime S, and we want to know the probability that an event A will take place at S given the knowledge of some other event B at a spacelike separation from S, then P(A|\lambdaB) = P(A|\lambda), i.e. knowing that B occurred gives us no additional information about the likelihood of A if we already know the complete set of information about \lambda?
For other more rigorous proofs why (2) is wrong, see:

Brans, CH (1988). Bell's theorem does not eliminate fully causal hidden variables. 27, 2 , International Journal of Theoretical Physics, 1988, pp 219-226

Joy Christian, "Can Bell's Prescription for Physical Reality Be Considered Complete?"
[http://arxiv.org/pdf/0806.3078v1

See, Hess K, and Philipp W (2000). PNAS ͉ December 4, 2000 ͉ vol. 98 ͉ no. 25 pp 14228-14233 for a proof that Bell's theorem can not be derived for time-like correlated parameters, and that such variables produce the QM result.

See also, Hess K, and Philipp W (2003), "Breakdown of Bell's theorem for certain objective local parameter spaces"
PNAS February 17, 2004 vol. 101 no. 7 1799-1805

I suppose all those people cited above are also confused, as is Jaynes.
Most likely there are some confusions in any papers that claim to show a local realist theory with the no-conspiracy assumption can reproduce QM results, yes (I don't know if this is what all the papers above are claiming since I don't have access to any but Joy Christian's paper)--if any such demonstration was valid it would have won widespread acceptance in the physics community and this would be very big news, but that hasn't happened. On the subject of Joy Christian's paper, I remember it being discussed earlier on this forum and it being mentioned that other physicists had claimed to find flaws in the argument, see for example ZapperZ's post #18 here (http://www.physicsforums.com/showthread.php?t=244806&page=2) which links to responses here (http://arxiv.org/PS_cache/arxiv/pdf/0704/0704.2038v1.pdf) and here (http://arxiv.org/abs/quant-ph/0703218v1). Wikipedia refers to Christian's work as "controversial" here (http://en.wikipedia.org/wiki/Bell's_Theorem#Theoretical_challenges), and says "The controversy around his work concerns his noncommutative averaging procedure, in which the averages of products of variables at distant sites depend on the order in which they appear in an averaging integral. To many, this looks like nonlocal correlations, although Christian defines locality so that this type of thing is allowed". Once again, I think the best way to cut through the fog is just to ask if Christian's proposal, whatever the details, could allow us to create computer programs which would correctly simulate QM statistics on pairs of computers which have been separated from connections to any other computers prior to the time the experimenters make random choices as to how to orient their simulated detectors on each trial. If you've read and understood Christian's proposal (I was not able to follow it myself because I'm not familiar with Clifford algebra), do you think this could be done?
Yet you have not shown me a single reason why my descriptions of the two scenarios in pos #55 are not valid realist local hidden variable theorems. For some reason you ignored the second scenario completely and did not even bother to say whether a "deterministic learning machine" is local or not.
You didn't give enough details there for me to be able to tell what you're proposing, or how it would reproduce violations of Bell inequalities. Any "deterministic learning machine" is certainly local if you could simulate it with a program running on a computer, but there's no way that loading this program on the two computers A and B in the setup I described would allow you to reproduce both the fact that the experimenters always get the same result when they choose the same setting on a given trial and the fact that on trials where they choose different settings they get the same result less than 1/3 of the time. Again, the basic point is that if the computers have been disconnected from communication with other computers at time t prior to the moment each experimenter makes their choice, then the only way you can guarantee a 100% chance that they'll return identical results if the experimenters make the same choice is to have the state of each computer at time t predetermine what answer they'll give for each of the three choices the experimenters can make (with both computers having the same predetermined answers), and this predetermination is enough to guarantee that if the experimenters make different choices they'll get the same answer at least 1/3 of the time.
You can see the following articles, for proof that a local deterministic learning hidden variable model reproduces the quantum result:

Raedt, KD, et. al.
A local realist model for correlations of the singlet state
The European Physical Journal B - Condensed Matter and Complex Systems, Volume 53, Number 2 / September, 2006, pp 139-142

Raedt, HD, et. al.
Event-Based Computer Simulation Model of Aspect-Type Experiments Strictly Satisfying Einstein's Locality Conditions
J. Phys. Soc. Jpn. 76 (2007) 104005

Peter Morgan,
Violation of Bell inequalities through the coincidence-time loophole
http://arxiv.org/pdf/0801.1776

More about the coincidence time loophole here:
Larson, JA, Gill, RD, Europhys. Lett. 67, 707 (204)

Are any of these other than the Morgan paper available online? Also, it's important to distinguish between two fundamentally different types of claims of "loopholes" in discussions of Bell's theorem. The first category says that there might be types of local hidden variables theories that fully reproduce the predictions of orthodox QM--for example, a theory involving a conspiracy in the initial conditions of the universe would fall in this category. This is the category I've been discussing so far on this thread. But there's a second category which doesn't actually dispute the basic idea of Bell's theorem that orthodox QM is incompatible with local realism, but instead suggests that existing tests of orthodox QM's predictions about EPR-type experiments have not adequately reproduced the conditions assumed by Bell, so that there might be a local realist theory which makes the correct predictions about experiments that have actually been performed but which would not actually violate Bell inequalities if better tests were performed that sealed off certain experimental loopholes seen in tests that have been done so far (meaning in these cases the theory would disagree with the predictions of orthodox QM). For example, one experimental loophole in some previous tests is that there may not have actually be a spacelike separation between the events of the two detector settings being chosen and the events of the two particles' spins being measured, so in principle the choice of detector settings could have had a causal influence on hidden variables associated with the particle before the particle was detected. This is known as the "communication loophole", and as discussed here (http://plato.stanford.edu/entries/bell-theorem/#5) the latest experiments have managed to seal it off. Another is the detection loophole (http://plato.stanford.edu/entries/bell-theorem/#4) which apparently has not yet been fully dealt with by existing experiments.

I haven't really read over the Morgan paper you link to in detail, but it sounds to me like he's talking about an experimental loophole rather than a theoretical loophole--on p. 1 he specifically compares it to the detection loophole, saying that the computer model under discussion "is a local model that can be said to exploit the 'coincidence-time' loophole, which was identified by Larsson and Gill as 'significantly more damaging than the well-studied detection problem'". If you have followed the details of Morgan's discussion, can you tell me if he's talking about an experimental loophole akin to the communication loophole and the detection loophole, or if he's proposing a genuine theoretical loophole involving a local hidden variables model that he thinks can precisely reproduce the predictions of QM in every possible experiment?

Glenns

Feb13-09, 07:42 PM

This is plain wrong, and on a lot of levels. Besides, you are basically hijacking the OP's thread to push a minority personal opinion which has been previously discussed ad nauseum here. Start your own thread on "Where Bell Went Wrong" (and here's a reference (http://arxiv.org/abs/0812.3058) as a freebee) and see how far your argument lasts. These kind of arguments are a dime a dozen.

For the OP: You should try my example with the 3 coins. Simply try your manipulations, but then randomly compare 2 of the 3. You will see that the correlated result is never less than 1/3. The quantum prediction is 1/4, which matches experiments which are done on pretty much a daily basis.
Lol, Dr.Chinese says that these "experiments" are done routinely; as if on a "daily basis".
Please, Dr. Chinese, tell us these experiments you are talking about that are carried out on a daily basis, using no special crystals; not specific radiation wavelengths, and no unorthodox equipage! [If you are unable to do so then you fail.]

Lol. Direct the author to the thread all you want, but he, like you, will never explain the basis of it. Certainly not under local environments! Einstein was once fond of saying that it should be simple. That it should always be kept simple.

DrChinese

Feb14-09, 11:29 AM

Please, Dr. Chinese, tell us these experiments you are talking about that are carried out on a daily basis, using no special crystals; not specific radiation wavelengths, and no unorthodox equipage! [If you are unable to do so then you fail.]

I really have no idea what you are saying. Bell tests are done in undergrad classrooms these days. They do require special PDC crystals and the appropriate laser source to create entangled photon pairs.

JesseM: Nice detailed response to mn4j. Raedt's work does involve the so-called "coincidence time loophole" also referenced by Morgan. See here (http://arxiv.org/abs/0706.2957) for a related article. (There are 2 authors named Raedt and I assume they are related as they sometimes write together.)

These types of attacks on Bell tests attempt to explain the results as being a form of a biased sample, and as such always comes back to the fair sampling assumption. Of course, as technology improves these attacks always get weaker and weaker and the results NEVER get any closer to the local realistic requirements. And note that IF THEY DID, then the QM prediction would be wrong. And now we are back to Bell's result anyway, that no local realistic theory can reproduce the predictions of QM. So ultimately, the local realist must state: QM is wrong, or they are wrong. Can't both be right!

ZapperZ

Feb14-09, 11:39 AM

I really have no idea what you are saying. Bell tests are done in undergrad classrooms these days. They do require special PDC crystals and the appropriate laser source to create entangled photon pairs.

To back up DrChinese claim that these experiments are now routinely done in undergraduate curriculum, please see this link:

http://people.whitman.edu/~beckmk/QM/

I too am puzzled by the requirement of not using any PDC crystal, etc. What's wrong with using those to get the entangled photons?

Zz.

mn4j

Feb15-09, 12:33 PM

JesseM: Nice detailed response to mn4j. Raedt's work does involve the so-called "coincidence time loophole" also referenced by Morgan. See here (http://arxiv.org/abs/0706.2957) for a related article. (There are 2 authors named Raedt and I assume they are related as they sometimes write together.)

These types of attacks on Bell tests attempt to explain the results as being a form of a biased sample, and as such always comes back to the fair sampling assumption. Of course, as technology improves these attacks always get weaker and weaker and the results NEVER get any closer to the local realistic requirements. And note that IF THEY DID, then the QM prediction would be wrong. And now we are back to Bell's result anyway, that no local realistic theory can reproduce the predictions of QM. So ultimately, the local realist must state: QM is wrong, or they are wrong. Can't both be right!

This seems rather dismissive. Raedt's work is not an attack on QM. They have developed a local realistic hidden variable model which gives the same result as QM in EPR type experiments and explains double-slit diffraction among other phenomena.
The matter is very simple, do you claim their model is not local realistic? If it is, then you must be alarmed that it reproduces the Quantum result, contrary to the claims of Bell. If it is not, then you must explain why it is not.

The model is described in the following articles:

http://arxiv.org/abs/0712.3781
http://arxiv.org/abs/0809.0616
http://arxiv.org/abs/0712.3693

The essence of the model is that quantum particles are Deterministic Learning Machines. Using this model, they are able to simulate EPR experiments, delayed-choice experiments, and double-slit experiments event-by-event in a local realist manner. You can't just brush this off.

DrChinese

Feb15-09, 01:32 PM

This seems rather dismissive. Raedt's work is not an attack on QM. They have developed a local realistic hidden variable model which gives the same result as QM in EPR type experiments and explains double-slit diffraction among other phenomena.
The matter is very simple, do you claim their model is not local realistic? If it is, then you must be alarmed that it reproduces the Quantum result, contrary to the claims of Bell. If it is not, then you must explain why it is not.

Well, actually they say that the sample is not representative due to choice of the time window for coincidence counting. Their conclusion (quote): "In general, these results support the idea that the idealized EPRB gedanken experiment that agrees with quantum theory cannot be performed". In other words, they claim: a) The experimental results of their purported local realistic theory will be biased to agree with the predictions of QM; b) On the other hand, no suitable Bell test that supports QM can be performed - ever; And finally c) QM is wrong and their local realistic theory is correct.

Why do these attacks get dismissed? Because it is not actually a proof of anything. Can you imagine saying experimental proof supporting X is actually proof of not-X? That is what is being asserted.

Let me put it a different way: there is NO alternative theory presented in these papers. Period. They try to say they have a simulation. OK, fine. Show me the THEORY that matches the scope of QM. Then we can get to the meat and potatoes. The evidence from Bell tests supports the predictions of QM. When we see their theory (which we never will of course) - let's call it LR - then we can say:

Experimental Evidence=> QM
True Theory=>LR (different predictions)
QM-LR=Delta (the difference they purport to explain)

Now we have the problem of why - regardless of approach - every Bell test has a growing Delta and not a shrinking Delta. Delta should decrease to zero as test sampling improves. Instead, Delta is now at about 150+ standard deviations. That is way up from about 10 SD a few decades ago.

So please, get serious.

mn4j

Feb15-09, 03:36 PM

Let me put it a different way: there is NO alternative theory presented in these papers. Period.

So what?. You still did not answer the following:
1. Do you deny that they presented event-by-event simulation of EPRB?
2. Do you claim that the model of their simulations is not local realistic?

These are the only two important questions. If you agree that they have indeed presented an event-by-event simulation of EPRB, then you end up with only two options

a) Their model is not local realistic or
b) Their model is local realistic contrary to the claims of Bell.

They don't need to have a complete theory which matches QM. All they need to demonstrate is that a local realistic model can reproduce the QM result, to refute Bell.

You probably have seen the following as well, although I can guess your response will be to ask them to get serious:

http://arxiv.org/abs/0901.2546

Maybe what you need is to spell out what evidence it will take for you to see the problem with Bell's theorem. Surely if your belief in it is rational, it must be falsifiable. What will it take to falsify it? Seriously, have you ever considered this question even?

Pio2001

Feb15-09, 07:26 PM

Hello,
Sorry, I didn't read all the thread. I just studied the http://arxiv.org/abs/0712.3781 article that you linked.

It seems to me that they do have a point.

They simulate measurments and associate to each of them a time t. Then, they count coincidences within a given time window only.

Their model violates Bell's inequality in the following way : they make the time t depend on the spin of the particle and of the orientation of the detector (locally). The delay between both detections associated to a pair thus depends on the spins and orientations of both particles and detectors. The coincidence count, that violates Bell's inequality is then a subset of the total coincidence count, that respects Bell's inequality. The selection of this subset depends on the delay between the events, thus on the spin and orientations of the detectors. This is a non-local hidden variable, and this is why it can violate Bell's inequality.

The most interesting point in their simulation, in my eyes, is that a real electric coincidence counter, in a real laboritory, can do exactly the same thing ! It can count a subset of results that violates Bell's inequality, from a total set of physical results that respects it, as long as a physical dependance exists between the extra correlations and the delay between the signals from the twin particules.

Pio2001

Feb15-09, 07:40 PM

Actually, this loophole is testable experimentally : we just have to emit the pairs of particles one by one, so that the time window for coincidences can be extended far beyond the maximum processing time for the detection.
This way, we can count all detections, whatever the delay between them.

If the idea in the paper is right, Bell's inequality should become respected.
If the idea in the paper is wrong, Bell's inequality should still be violated.

Maybe this have been already done.

mn4j

Feb15-09, 08:17 PM

Pio2001

Feb16-09, 05:38 AM

I mean decreasing the emission rate until pairs of photons are emitted slower (both photons being still emitted at the same time, of course).
we can then set the window of the coincidence counter very large, so that it counts the detection of two photons whatever the small time delay introduced by the authors in order to violate Bell's inequality.

If I have understood correctly their simulation, De Raedt et al.show that Bell's inequality violation in Aspect-like experiments is not necessarily caused by quantum non-local effects, but may come from an artefact caused by the coincidence counter setup.

Quantum theory predicts that as long as the two photons are from the same entangled pair, Bell's inequality will be violated.
In my understanding, De Raedt et al. simulation violates Bell's inequality introducing a delay between the photons AND setting the coincidence window narrower than this delay. So it predicts that if the coincidence window is widened enough for counting all coincidence, whatever the delay between the recording of the events, Bell's inequality will become respected.

In practice, that's exactly what happens in the real data set that they took as an example, BUT, it seems logical to assume that this is because widening the time window, we count more and more false coincidences, thus decreasing the correlations.
Decreasing the physical emission rate at the source, we should be able to widen the coincidence window without increasing the false coincidence rate at all.

This way, if Bell's inequality is still violated, the hypothesis of an artifact in the coincidence counter setup will be rejected, and the quantum non local correlations will remain the only explanation.

zonde

Feb16-09, 08:53 AM

we just have to emit the pairs of particles one by one, so that the time window for coincidences can be extended far beyond the maximum processing time for the detection.
There is something that can be done without decreasing emission rate.
You can use two coincidence windows of different width. Say that coincidences that are in one widow but are outside other will be poorly synchronized coincidences and coincidences that are inside smaller widow are decently synchronized coincidences. Now if you calculate rate "poorly synchronized coincidences"/"decently synchronized coincidences" for different relative polarization angles you should not see any correlation between relative angle and this rate for fair sampling assumption to hold.
And if there is no such correlation there will be match less possible models for coincidence loophole if any.
Good thing is that analysis like that can be done without performing any new experiments based only on existing data from experiment where all detections are recorded with timestamps (and coincidences are found later from recorded data).

DrChinese

Feb16-09, 08:58 AM

Maybe what you need is to spell out what evidence it will take for you to see the problem with Bell's theorem. Surely if your belief in it is rational, it must be falsifiable. What will it take to falsify it? Seriously, have you ever considered this question even?

Let's see if I get this right. The experimental evidence is X, and we are supposed to use that evidence to conclude not-X.

The thing about Bell is that it is more or less independent of whether local reality or QM is correct. It says they both cannot be correct, which was not obvious at the time. So let's get specific.

QM says the coincidence rate for entangled photon pairs at 60 degrees is 25%. Local realistic theories say the true coincidence rate is at least 33%. Raedt is saying that the true rate is [insert your guess here since he skips this step]% but that experiments will always support QM.

Now, once again, how are we supposed to conclude there is anything wrong with Bell? Clearly, the entire issue here is Raedt trying to explain why a LR theory, which makes predictions incompatible with QM, actually provides an experimental result compatible with QM. So clearly, this is not about Bell at all. You may as well say that all experiments supporting General Relativity are actually evidence of Newtonian gravity.

Now, get serious. Even Raedt ought to be able to see why the argument falls flat. It is going to take experimental evidence IN FAVOR of a local realistic theory to convince anyone of their result. If they really had a bead on anything, they would be proposing an experiment to test their ideas. Rather than writing a paper saying they are correct in the face of evidence to the contrary.

kobak

Feb16-09, 10:31 AM

I spent some time reading de Raedt's articles and talking with him about them. Let me answer some of the mn4j's questions in the last postings ("the only two important questions", as you write).

So what?. You still did not answer the following:
1. Do you deny that they presented event-by-event simulation of EPRB?
2. Do you claim that the model of their simulations is not local realistic?

These are the only two important questions. If you agree that they have indeed presented an event-by-event simulation of EPRB, then you end up with only two options

a) Their model is not local realistic or
b) Their model is local realistic contrary to the claims of Bell.

1. Yes, they did present event-by-event simulations of the certain experiments (like Aspect's and Weihs' et al.), that often are thought to be conclusive evidences that Bell's equality is violated.

2. Yes, their models are local realistic.

But you are totally wrong about two options that I'm supposedly left with. The important thing to realize is that no conclusive test of EPRB has ever been done. Every experiment that has been conducted has certain loopholes (there's even a special wikipedia article about those). This means that all those experiments are not ideal, and it's possible to explain their results with a local realist theory. This is well-known to everybody who is interested in QM foundations, and has been known already for ages (Philip Pearle showed in the late 1970s how this can be done using one of the loopholes).

de Raedt presents yet another model of how these loopholes can be used to still "give some chance" to local realism. This is certainly not a big deal, and has no consequences to Bell's theorem. The usual hope is that in some years the conclusive experiment will be performed (I've heard people hoping that it will occur in 10-20 years).

What I find particularly confusing in de Raedt's articles is that they are totally out of context: he never mentions a word "loophole", leave alone all existing bulk of knowledge about them. This is misleading to say the least.

See http://arxiv.org/abs/quant-ph/0703120 for this critique (yes, I do know that there's a reply by de Raedt; I think that his reply misses the point).

This seems rather dismissive. Raedt's work is not an attack on QM. They have developed a local realistic hidden variable model which gives the same result as QM in EPR type experiments and explains double-slit diffraction among other phenomena.
The matter is very simple, do you claim their model is not local realistic?

This paper about double-slit (http://arxiv.org/abs/0809.0616) is a different story, but also very telling. Have you read it? Did you realize that this model works only because many photons "get lost"? If we imagine a perfect emitter that emits 1 photon per second and we let it emit 1000 photons, and then we count how many photons hit the screen and how many photons hit the double-slit screen, and then we add those two numbers together, then the result according to this model will be a lot less then 1000.

This is clearly a prediction different from that of QM. This model can be tested and falsified. I'm absolutely sure that it's just wrong.

Of course such an experiment is tremendously difficult to perform, but there's an easier test. This model works because the detectors have memory and are "learning". Now if we start to jiggle the screen back and forth (parallel to itself) sufficiently fast, then de Raedt's model predicts that the interference image will get smeared (I think it's stated in the paper). Now, here's the question for you: what is the prediction of QM?

I think that QM predicts that the interference picture will stay the same. I asked de Raedt this question, and he replied that in his opinion QM predicts nothing, because it requires the experimental apparatus to be completely fixed during the experiment and not "jiggled". Well, then I asked him what would he say if such an experiment is performed and the interference picture does not change.

He said that in that case he would (I quote here) retire.

The bottomline is that what de Raedt proposes are some local realistic explanations of the certain experiments. All his models are in principle distinguishable from QM (as Bell always told us). And personally I'm quite sure that when the tests are done, these models will be proven false.

Pio2001

Feb16-09, 12:00 PM

QM says the coincidence rate for entangled photon pairs at 60 degrees is 25%. Local realistic theories say the true coincidence rate is at least 33%. Raedt is saying that the true rate is [insert your guess here since he skips this step]% but that experiments will always support QM.

Yes, it follows from figure 6 in the first paper (http://arxiv.org/PS_cache/arxiv/pdf/0712/0712.3781v2.pdf) that according to their simulation, the true rate, in your example, may actually be 25 %, while being measured at 33 % because of the least correlated photons being registered by the two detectors at a time interval bigger than the time window of the counter. Which leads to discarding the least correlated pairs.

DrChinese

Feb16-09, 01:20 PM

I spent some time reading de Raedt's articles and talking with him about them. Let me answer some of the mn4j's questions in the last postings ("the only two important questions", as you write).

1. Yes, they did present event-by-event simulations of the certain experiments (like Aspect's and Weihs' et al.), that often are thought to be conclusive evidences that Bell's equality is violated.

2. Yes, their models are local realistic.

But you are totally wrong about two options that I'm supposedly left with. The important thing to realize is that no conclusive test of EPRB has ever been done. Every experiment that has been conducted has certain loopholes (there's even a special wikipedia article about those). This means that all those experiments are not ideal, and it's possible to explain their results with a local realist theory. This is well-known to everybody who is interested in QM foundations, and has been known already for ages (Philip Pearle showed in the late 1970s how this can be done using one of the loopholes).

de Raedt presents yet another model of how these loopholes can be used to still "give some chance" to local realism. This is certainly not a big deal, and has no consequences to Bell's theorem.

...

The bottomline is that what de Raedt proposes are some local realistic explanations of the certain experiments. All his models are in principle distinguishable from QM (as Bell always told us). And personally I'm quite sure that when the tests are done, these models will be proven false.

Welcome to PhysicForums, kobak! And thank you very much for this insight on de Raedt.

I was just looking at the papers in a bit more detail. I have been disappointed by the approach, as it obscures what is being asserted in favor of trying to prove Bell wrong (which I think is overreaching). I do not personally consider these to be counter-examples to Bell, and I seriously doubt they will sway others either.

1. Their model (at least in one paper (http://arxiv.org/PS_cache/arxiv/pdf/0712/0712.3693v1.pdf)) does not provide fair sampling (assuming I read it correctly) to deliver an explicitly biased sample. As such, it exploits the loopholes you mention and doesn't really provide anything new (as you also mention). Quote:

"The mathematical structure of Eq. (18) is the same as the one that is used in the derivation of Bell’s results and if we would go ahead in the same way, our model also cannot produce the correlation of the singlet state. However, the real factual situation in the experiment [8] is different: The events are selected using a time window W that the experimenters try to make as small as possible. ...

"In our simulation model, the time delays ti are distributed uniformly over the interval [0, Ti] where T1 = [not random]."

In other words, there is tinkering with the time window and by their choice of how the time window is chosen, combined with time delay parameter choice, they bias the sample. They have to, because otherwise the raw source data would run afoul of Bell.

2. The other paper (also Dec 2007/Feb 2008 (http://arxiv.org/PS_cache/arxiv/pdf/0712/0712.3781v2.pdf)) relies on so-called DLMs (Deterministic Learning Machines). These purport to satisfy local causality and involve a form of memory from trial to trial:

"A DLM learns by processing successive events but does not store the data contained in the individual events. Connecting the input of a DLM to the output of another DLM yields a locally connected network of DLMs. A DLM within the network locally processes the data contained in an event and responds by sending a message that may be used as input for another DLM. Networks of DLMs process messages in a sequential manner and only communicate with each other by message passing: They satisfy Einstein’s criterion of local causality. For the present purpose, we only need the simplest version of the DLM [11]. The DLM that we use to simulate the operation of the Stern-Gerlach magnet is defined as follows. The internal state of the ith DLM, after the nth event, is described by one real variable un,i. Although irrelevant for what follows, this variable may be thought of as describing the fluctuations of the applied field due to the passage of an uncharged particle that carries a magnetic moment."

and

"A key ingredient of these models, not present in the textbook treatments of the EPRB gedanken experiment, is the time window W that is used to detect coincidences. We have demonstrated (see Section IIG) the importance of the choice of the time window by analyzing a data set of a real EPRB experiment with photons [32].

3. With both of these, the critique is really the same: why not point to the specific difference? They do everything humanly possible to obscure what should be a simple point: what is the difference between QM and their LR? Clearly, they could show how their data points satisfy the Inequality if all trials are considered and are fully independent, while the sub-sample within the time window is biased to yield a result consistent with QM but violating the Inequality.

Specifically: the QM prediction of entangled photon coincidences is .250 at 60 degrees. So we know their adjusted result must therefore also be .250. The LR value must be .333 or greater, so the delta is .0833. Which data items were excluded to get this result? Or why would the results be biased specifically towards that of a wrong theory (QM)? These are the lines in the sand, and they really are not addressed. I can see the hand waving in the equations, but without this simple explanation I don't see where they have anything. Quoting again:

"Extensive tests (data not shown) lead to the conclusion that for d = 3 and to first order in W, our simulation model reproduces the results of quantum theory of two S = 1/2 objects, for both Case I and Case II."

Clearly, for the algorithm to work, the delta must be .0833 at 60 degrees; delta=0 at 0 and 45 degrees; and so on. That delta function, in my opinion, should jump off the page. In reality, I don't think they have identified such a function. They should be the ones to point out the source of the delta. I have tried, but can't really follow their algorithm far enough to generate values.

My point is basically: why not make a testable prediction showing how using the algorithm, the experimental results vary in good agreement with the model but NOT according to any quantum mechanical prediction? I.e. if I change the time window and delay parameters in an actual experiment, the results match the LR model but are not explained by QM. After all, according to the LR model, it is strictly an accident of chance that QM happens to be correct in its predictions regarding how entangled photons behave (since there are no such things as entangled photons in LR, by definition).

Pio2001

Feb16-09, 02:10 PM

I have tried, but can't really follow their algorithm far enough to generate values.

I only read the december 2007 paper with the Deterministic Learning Machines. They give two algorithms, the one with DLM (page 16 : deterministic model), and a pseudo-random one, much simpler (page 16 : pseudorandom model). The way to sort results follows (page 17 : time tags / data analysis).

They suggest a possible physical meaning for this bias : "experimental evidence that the time-of-flight of single photons passing through an electro-optic modulator fluctuates consederably can be found in ref 56"

I find this idea interesting, because it is more realistic to suppose that the time-of-flight of a photon can depend of its polarisation in an environnement sensitive to polarisation, than to suppose that the detector purposely discards detections that would comply with Bell's inequality.
This relation is explicitely proposed page 17 in the last paragraph before "5.Data Analysis" (the formula have no number). They later set d=3 in this formula (for 1/2 spin particles).

Zonde's idea to test coincidence efficiency vs relative angle seems good. We could try it on available data (after checking that it works for all possible scenarii of this kind).

DrChinese

Feb16-09, 04:41 PM

They suggest a possible physical meaning for this bias : "experimental evidence that the time-of-flight of single photons passing through an electro-optic modulator fluctuates consederably can be found in ref 56"

I find this idea interesting, because it is more realistic to suppose that the time-of-flight of a photon can depend of its polarisation in an environnement sensitive to polarisation, than to suppose that the detector purposely discards detections that would comply with Bell's inequality.
This relation is explicitely proposed page 17 in the last paragraph before "5.Data Analysis" (the formula have no number). They later set d=3 in this formula (for 1/2 spin particles).

Thanks, I'll look again. There must be a connection between the polarization and detection probability (which is here related to the window size and delay factors) in order to get the desired results. I just couldn't figure out where, and I couldn't figure out why that wasn't highlighted.

mn4j

Feb16-09, 06:39 PM

Let's see if I get this right. The experimental evidence is X, and we are supposed to use that evidence to conclude not-X.

The thing about Bell is that it is more or less independent of whether local reality or QM is correct. It says they both cannot be correct, which was not obvious at the time. So let's get specific.

You do realize that Bell has a definition of local reality which has not been verified experimentally don't you. If you think it has been verified, show me experimental evidence that proves Bell's definition of local reality. Have you not been reading this thread at all? The bulk of the discussion was about this point.

QM says the coincidence rate for entangled photon pairs at 60 degrees is 25%. Local realistic theories say the true coincidence rate is at least 33%. Raedt is saying that the true rate is % but that experiments will always support QM.

NO! Bell's local realist theories say the true coincidence rate is 33%. If you disagree, point me to a reference about a local realist theory which makes that claim. Again you will notice that only Bell makes that claim, which it turns out is a straw-man, because there is no experimental validation of it. Do you know of any local realist theory for which that claim is valid? If not, why do you state it as though it was dogmatically accepted to be the case?

So then we have:
1. What QM predicts
2. What Bell claims (and this is crucial) local realist theories should result in
3. What experiments observe

It turns out (1) agrees with (3) but disagrees with (2). If you are thinking intellectually honestly, you must realize that failure of (3) to agree with (2) can mean that Bell's claim about local realist theories is dubious. Yet, for some reason you'd rather think Bell was a god and every claim he made was dogma, which leads you to conclude that both (1) and (3) are results of non-local realist theories. Why is that, I ask? This is not rocket science.

Now, once again, how are we supposed to conclude there is anything wrong with Bell? Clearly, the entire issue here is Raedt trying to explain why a LR theory, which makes predictions incompatible with QM, actually provides an experimental result compatible with QM.
NO! Your bias is clouding your judgement of Raedt's work. Raedt has developed a model which is unmistakably and convincingly LR, and he shows that it agrees with (1) and (3). Again, if you are thinking intellectually honestly, you must realize that according to [I]Bell's definition (and this is crucial) of what LR means, this is impossible.

If you want to criticize Raedt, you have to show that either:
1) The model he has developed is not LR
2) The model he has developed does not reproduce the results of QM and real experiments
You have done neither.
Now, get serious.
No. YOU get serious!

mn4j

Feb16-09, 06:44 PM

The important thing to realize is that no conclusive test of EPRB has ever been done.
EXACTLY! So on what basis do followers of Bell purport to have proven that local realist theories should produce a certain result???

Here is an experiment to try:
- use a separate set of apparatus for each pair of photons emitted. If you still obtain the QM result, then Raedt's model is wrong.

kobak

Feb16-09, 07:10 PM

> The important thing to realize is that
> no conclusive test of EPRB has ever been done.

EXACTLY! So on what basis do followers of Bell purport to have proven that local realist theories should produce a certain result???

Well, I'm glad that we agree on something. However, your question doesn't relate to my statement that you quote. Let me try to clarify things a bit.

There are two things: Bell's theorem as an abstract theorem, and its experimental tests. Bell's theorem states that local realist theories can't reproduce all the predictions of QM. It doesn't need to be proven by experiment, because the proof is given on a piece of paper. The experiment has to show what is correct: QM or local realism. What I said means that no conclusive proof that QM is right and LR is wrong (i.e. no conclusive violation of Bell's inequalities) has ever been done. This has no relation to the validity of the theorem itself.

Now, you seem to claim that Bell's theorem is wrong. But even if it were wrong, de Raedt's articles about EPR wouldn't prove it wrong (because as I explained, those articles only show that the experiments done so far were not perfect).

Finally, one more point. What exactly does "local realism" mean, is a philosophical question. For his proof Bell used a particular equation (P(A|aBL) = P(A|aL) or something like that) and he gave certain "physical" arguments about why this should be true if we assume local realism.

Famous ET Jaynes (and de Raedt follows Jaynes here) wrote a paper that you cited in this discussion, where he claimed that Bell made a stupid mistake when applying rules of probability (this is not a quote, but that's how it sounds). This is absurd. Bell certainly understood the rules of probability perfectly well and he actually did give the physical arguments for his assumption (that Jaynes seemed to fail to either notice or understand).

You may still say that local realism does not necessarily entail this assumption of Bell. Since "local realism" isn't something defined by a formula, this is in principle a meaningful claim. However I never saw any local realist model that would violate Bell's assumption (in the very particular example that Bell is discussing). de Raedt's models of simulations of Aspect-Weihs experiments have no relation to this issue.

mn4j

Feb16-09, 11:10 PM

The experiment has to show what is correct: QM or local realism. What I said means that no conclusive proof that QM is right and LR is wrong (i.e. no conclusive violation of Bell's inequalities) has ever been done. This has no relation to the validity of the theorem itself.

Why MUST LR and QM contradict each other? Just because Bell says they must? This is what you fail to realize. The issue here is not whether QM is wrong and LR is right! The issue, whether Bells understanding of LR is correct.

Now, you seem to claim that Bell's theorem is wrong. But even if it were wrong, de Raedt's articles about EPR wouldn't prove it wrong (because as I explained, those articles only show that the experiments done so far were not perfect).
If you agree that the experiments were not perfect, then how come those same experiments are still presented as proof of Bell's theorem. Bell's theorem is a negative theorem.

Bell says, "NO LR can reproduce the QM results". Now Bell better be sure that his definition of LR is such that it accounts for EVERY possible LR theory. If even 1 is found that can not be modelled by Bell's equations, Bell's theorem has to be thrown out. Do you agree with this? De Raedt's articles presents just one such models.

Finally, one more point. What exactly does "local realism" mean, is a philosophical question. For his proof Bell used a particular equation (P(A|aBL) = P(A|aL) or something like that) and he gave certain "physical" arguments about why this should be true if we assume local realism.

Famous ET Jaynes (and de Raedt follows Jaynes here) wrote a paper that you cited in this discussion, where he claimed that Bell made a stupid mistake when applying rules of probability (this is not a quote, but that's how it sounds). This is absurd. Bell certainly understood the rules of probability perfectly well and he actually did give the physical arguments for his assumption (that Jaynes seemed to fail to either notice or understand).
I'll take Jaynes over Bell any day when it comes to who understands probability better. Take a look at De Raedts recent article together with Hess (http://arxiv.org/abs/0901.2546) for a succinct explanation of Bell's error.

You may still say that local realism does not necessarily entail this assumption of Bell. Since "local realism" isn't something defined by a formula, this is in principle a meaningful claim. However I never saw any local realist model that would violate Bell's assumption (in the very particular example that Bell is discussing).
Don't you realize that the type of claim Bell is making about LR models requires that he MUST be absolutely sure that he has presented an exhaustive representation of ALL POSSIBLE LR models. I am perfectly happy to accept that Bell's theorem is true ONLY for the LR models narrowly defined by his assumptions.

de Raedt's models of simulations of Aspect-Weihs experiments have no relation to this issue.
Don't forget that you already agreed that de Raedt's model is LR. So they are relevant. Bell's equations do not apply to a deterministic learning machine model like de Raedt's. How then can Bell claim that No LR can reproduce the QM results?

You see, the problem with Bell's theorem is not that his conclusions can not be drawn from his assumptions. The problem is that those conclusions are interpreted by those who don't know better beyond the scope of the assumptions on which they are based. For someone purporting to characterize all LR models, he chose a severely narrow and handicapped subset of LR to base his calculations on.

Pio2001

Feb17-09, 05:09 AM

What about the GHZ proof, then ?

kobak

Feb17-09, 05:12 AM

Dear mn4j,
we are already going in circles. I will try to summarize my points as clear as possible and I would like to ask you to comment on each of them, whether you agree or not. If you still don't listen to what I'm saying, then it's better to stop this discussion.

1. All the experiments that has been done so far to test Bell's inequalities ARE. NOT. PERFECT. This is not something to agree or disagree, it's just a fact, and it's admitted by everybody, including of course the experimenters themselves. Agreed?

2. These experiments definitely can't be presented "as proof of Bell's theorem", because they are not. Please stop asking me why they are presented in such way! If anybody does so, he or she just doesn't understand anything here. Bell's theorem is a theoretical construct, it doesn't need to be proven by experiment. Experiment has to show whether Bell's inequalities are violated or if they are not. Agreed?

3. I quote you: "I am perfectly happy to accept that Bell's theorem is true ONLY for the LR models narrowly defined by his assumptions". OK, let's call all theories that Bell's theorem applies to "Bell local realistic" (BLR). Now, Bell's theorem says and proves that BLR theories should obey Bell's inequalities while QM violates them. Agreed?

4. Your main point seems to be that BLR is only a narrow subclass of LR theories. Well, I repeat: what is "local realism" is a philosophical question. I'm personally quite happy to include in my notion of local realism the assumption that the outcomes of Bob's experiments are statistically independent from Alice's choice of experimental settings (this is Bell's assumption and precise definition of BLR). You are not, right?

5. Since scientific consensus is that BLR and LR are the same thing, and you disagree, the only meaningful way to disagree is to give an example of a LR theory that is not BLR. Agreed?

6. In case you want to say that de Raedt's models are such kind of example, I repeat once again: NO, they ARE NOT. de Raedt showed (as was already known) that the experiments to test Bell's inequalities were not perfect (there are loopholes), and because of these experimental flaws their results can be explained in LR way. Agreed?

7. But (the crucial point!) de Raedt's model is obviously not only LR, but BLR as well!! IF a loophole-free test of Bell's inequalities is ever done and IF Bell's inequalities are still found to be violated, then de Raedt won't be able to explain this with his model (why? because of Bell's theorem). Agreed? Please think a bit before answering.

I'm asking you think, because you wrote that "Bell's equations do not apply to a deterministic learning machine model like de Raedt's". This is just plain wrong. Of course they do apply! De Raedt's model is perfectly BLR.

8. You mentioned the recent de Raedt's article with Hess (http://arxiv.org/abs/0901.2546). I've seen it and I took a brief look, but I didn't read it carefully and I failed to understand the crux of it. I just don't want to investigate 40+ pages of formulas, when I already know that de Raedt's reasoning is often confusing and misleading, and that Hess is well-known for fighting will Bell's theorem, though his claims were long ago shown wrong by people, whose opinion I respect in this issue (see http://arxiv.org/abs/quant-ph/0208187).

If you have read and understood this 40+ pages article, everybody here will be grateful if you give us the arguments in a concise and clear way.

9. For some reason you completely ignored my point about double-slit paper of de Raedt. You were first to mention it! Did you read it? If you did, could you please comment on what I said earlier? If you didn't, how come you use it in the arguments?

Best,
dk

DrChinese

Feb17-09, 10:16 AM

8. You mentioned the recent de Raedt's article with Hess (http://arxiv.org/abs/0901.2546). I've seen it and I took a brief look, but I didn't read it carefully and I failed to understand the crux of it. I just don't want to investigate 40+ pages of formulas, when I already know that de Raedt's reasoning is often confusing and misleading, and that Hess is well-known for fighting will Bell's theorem, though his claims were long ago shown wrong by people, whose opinion I respect in this issue (see http://arxiv.org/abs/quant-ph/0208187).

If you have read and understood this 40+ pages article, everybody here will be grateful if you give us the arguments in a concise and clear way.

Good post. I tried to find the meat in the argument and couldn't either. If anyone had a good counter-argument, they would put their proof up front rather than hide it. Meanwhile, we have the following history of experimental teams seeing violation of Bell Inequalities:

Aspect, 1982: 5 standard deviations.
Kwiat, 1995: 102 standard deviations.
Kurtsiefer, 2002: 204 standard deviations.
Barbieri, 2003 (http://arxiv.org/abs/quant-ph/0303018): 213 standard deviations.

And since entanglement doesn't even exist in any local realistic theory (by definition), it is interesting to note that last year, Vallone et al (http://arxiv.org/abs/0810.4461) were observing hyper-entanglement on photons in 3 independent degrees of freedom. Further, there have been numerous experiments involving time-bin entanglement, including with photons that have never interacted in the past (http://arxiv.org/abs/0704.0758). Just the loophole de Raedt thought to exploit in his later paper (you would think experiments like this would finally end the search for an LR theory).

None of this could be predicted by any local realistic theory. On the other hand, all are predicted by QM. This is why looking for LR theories is a waste of time. It made sense up until the 1970's or so, but not since.

kobak

Feb17-09, 10:24 AM

Meanwhile, we have the following history of experimental teams seeing violation of Bell Inequalities

Sorry, DrChinese, are you saying that some of these experiments were loophole-free? As far as I know, this has so far never been achieved.

If it is true, then exactly how many standard deviations is observed -- doesn't really matter. A strict believer in local realism still can say: there are this and that loopholes, and so the results can be explained in a sophisticated enough local realistic way. It can be 100000 standard deviations, or whatever. What is important (in the sense of putting a full stop in this discussion) is to conduct an experiment, completely free of any loopholes.

DrChinese

Feb17-09, 11:09 AM

Sorry, DrChinese, are you saying that some of these experiments were loophole-free? As far as I know, this has so far never been achieved.

If it is true, then exactly how many standard deviations is observed -- doesn't really matter. A strict believer in local realism still can say: there are this and that loopholes, and so the results can be explained in a sophisticated enough local realistic way. It can be 100000 standard deviations, or whatever.

I disagree. What is being asserted is anti-scientific because in a sense, no experiment is loophole free. As you are undoubtedly aware, there are still experiments going on to test General Relativity. At least there, the competing theories (or versions of GR as you may call them) have key elements in common.

On the other hand, there is no existing candidate LR theory on the table to compare to QM at this time. Stochastic Mechanics (Marshall, Santos) is an example of a field of research in that regard, but every candidate SM model is found to have problems and is quickly modified again. And since such models do not predict anything useful, there is no incentive to study them further. We already have a very useful model - QM - and the experiments supporting it are in the thousands. Something useful from the field of study would go a long way towards convincing the scientific community.

So yes, I think quantity does matter, and I think utility matters. And I think the history of the area does matter as well, including when a theory (QM) is supported by improving technology. That doesn't mean that conventional thinking is right always. I just mean to say that science evolves towards ever more useful theories. I do not see how LR theories can ever hope to fall into that category (useful) since they deny the known phenomena of entanglement. I mean, we are at the point now of entangling particles with no common history. Why don't the local realists acknowledge the obvious hurdle such experiments place on LR theories?

And as a practical matter, I disagree that loophole-free experiments have not been performed. In my opinion, the fair-sampling loophole has been closed (Rowe et al, 2001). In my opinion, the strict locality loophole has been closed (Weihs et al, 1998). Etc. Why should you need to close every loophole simultaneously if you can close each separately? If a prisoner cannot escape from the first lock by itself, and cannot escape from the second lock by itself, how can he escape when both locks are present? I don't disagree with a desire to close all loopholes simultaneously; but I think that is a standard that is being applied to Bell tests which is applied nowhere else in science. Surely you must have noticed this as well.

kobak

Feb17-09, 11:56 AM

I disagree. What is being asserted is anti-scientific because in a sense, no experiment is loophole free. As you are undoubtedly aware, there are still experiments going on to test General Relativity. ... And as a practical matter, I disagree that loophole-free experiments have not been performed. In my opinion, the fair-sampling loophole has been closed (Rowe et al, 2001). In my opinion, the strict locality loophole has been closed (Weihs et al, 1998). Etc. Why should you need to close every loophole simultaneously if you can close each separately? ... I don't disagree with a desire to close all loopholes simultaneously; but I think that is a standard that is being applied to Bell tests which is applied nowhere else in science. Surely you must have noticed this as well.

Three points. First. I'm not an expert in Bell tests and loopholes issue, so can't really comment on that on the detailed level. I know that there's for example "time-coincidence" loophole (http://arxiv.org/abs/quant-ph/0312035), which is apparently exactly the loophole de Raedt is exploiting (http://arxiv.org/abs/quant-ph/0703120, the link I already gave). I'm not sure that all known loopholes were already closed even separately, though this might be true. In particular, I just don't know any details about this "entanglement" studies that you cite (and don't have time at the moment to start reading them). Do they test Bell inequalities after this entanglement "swapping"? Or how else these findings prove LR false?

Second. I guess that I slightly disagree with you about different standards of tests. Of course there are super-precise tests of GR still being done. But to test GR you need to observe something that is predicted by GR, like light deflection or whatnot. When this is observed, nobody claims that there's a "loophole" in this experiment, and the results can be interpreted such that light is not deflected. It's evident: nobody heard of any loopholes in GR tests. On the other hand, to test QM versus LR one needs to show that the Bell's inequalities are violated. And all the attempts to show it still have some loopholes that allow alternative explanations.

Third. Nobody in his right mind claims that QM is "wrong". For de Raedt, QM is a correct mathematical model working well on the ensemble level only, without saying anything about single events. He is not trying to show that QM is wrong, he is trying to show that it can be completed in a LR way. Well, we know that it's impossible due to Bell. But de Raedt obviously disagrees. And it doesn't make a lot of sense for me to defend de Raedt, but he is most definitely not a crackpot (he has done a huge amount of "real" work in computer simulations of different physical models, including decoherence etc.). I believe (as you do) that his reasoning about Bell is flawed, but he certainly does not try to obscure anything on purpose: I'm quite sure that he is honest.

mn4j

Feb17-09, 12:31 PM

1. All the experiments that has been done so far to test Bell's inequalities ARE. NOT. PERFECT. This is not something to agree or disagree, it's just a fact, and it's admitted by everybody, including of course the experimenters themselves. Agreed?

Agreed! This is a fact. Not a single loophole-free experiment has ever been performed.

2. These experiments definitely can't be presented "as proof of Bell's theorem", because they are not. Please stop asking me why they are presented in such way! If anybody does so, he or she just doesn't understand anything here.
Agreed!

Bell's theorem is a theoretical construct, it doesn't need to be proven by experiment. Experiment has to show whether Bell's inequalities are violated or if they are not. Agreed?

No. I disagree. So long as Bell's inequalities purport to make claims about reality, the correspondence between those inequalities and reality MUST be independently validated by experiments before any claims they make about reality can be said to be proven.

3. I quote you: "I am perfectly happy to accept that Bell's theorem is true ONLY for the LR models narrowly defined by his assumptions". OK, let's call all theories that Bell's theorem applies to "Bell local realistic" (BLR). Now, Bell's theorem says and proves that BLR theories should obey Bell's inequalities while QM violates them. Agreed?
Agreed without prejudice. Note that every loop-hole found to date is a hidden assumption in Bell's proof. I do not claim by agreeing to the above that all loop-holes have been found.

4. Your main point seems to be that BLR is only a narrow subclass of LR theories. Well, I repeat: what is "local realism" is a philosophical question. I'm personally quite happy to include in my notion of local realism the assumption that the outcomes of Bob's experiments are statistically independent from Alice's choice of experimental settings (this is Bell's assumption and precise definition of BLR). You are not, right?

Again remember that every loop-hole is a hidden assumption of Bell's proof. The fact that there are loop holes tells you that BLR is not exhaustive of all LR.

5. Since scientific consensus is that BLR and LR are the same thing, and you disagree, the only meaningful way to disagree is to give an example of a LR theory that is not BLR. Agreed?
No. If you think scientific consensus is that BLR and LR are the same thing, then you have not been paying attention, and this thread does not exist, and the loop-holes do not exist.

6. In case you want to say that de Raedt's models are such kind of example, I repeat once again: NO, they ARE NOT.
If you say Raedt's modes are not examples of LR which are not accounted for by Bell's LR, I repeat once again: YES THEY ARE. You see this kind of discussions takes us no where. Explain why they are not.

de Raedt showed (as was already known) that the experiments to test Bell's inequalities were not perfect (there are loopholes), and because of these experimental flaws their results can be explained in LR way. Agreed?
That is a very narrow reading of de Raedt's work. Did you completely fail to understand the importance of the Deterministic Learning Machine model of de Raedt's?

7. But (the crucial point!) de Raedt's model is obviously not only LR, but BLR as well!!
If de Raedt's model is BLR then how do you explain the fact that the model violates the inequality, when according to Bell it is impossible. Think before you answer. If you want to say that only under certain conditions will violate the inequality, then you still face the question of answering how come some BLR will violate the inequality under certain conditions. There is no escaping here.

IF a loophole-free test of Bell's inequalities is ever done and IF Bell's inequalities are still found to be violated, then de Raedt won't be able to explain this with his model (why? because of Bell's theorem). Agreed? Please think a bit before answering.
This is circular reasoning. A loop-hole free test of Bell's inequality is required to be able to validate the inequality in the first place. Violation of Bell's inequality in any experiment has two possible explanations, not just one.
1) That Bell's inequality is a correct representation of local reality and the experiment is either not real or not local or both
2) That Bell's inequality is not a correct representation of local reality.

Now for some reason, Bell's followers ALWAYS gravitate towards (1). Do you agree that (2) is also a possibility and MUST be considered together with (1) when interpreting the results of these experiments? Please, I need a specific answer to this question.

I'm asking you think, because you wrote that "Bell's equations do not apply to a deterministic learning machine model like de Raedt's". This is just plain wrong. Of course they do apply! De Raedt's model is perfectly BLR.
You have no idea what you are talking about. Even ardent Bell believers have shown that not all LR are accounted for in BLR. See http://arxiv.org/abs/quant-ph/0205016 for one example. Bell's starting equation is the following:

P(AB) = \sum_i P(A|a_i)P(B|b_i )P(\lambda_i)

This experiment does not apply in situations in which \lambda_{i+1} is dependent on \lambda_{i}, like is the case in de Raedt's model. The reason is simple. If case (i) and case (i+1) are not mutually exclusive, you can integrate or as in this case perform a sum the way Bell did.

8. You mentioned the recent de Raedt's article with Hess (http://arxiv.org/abs/0901.2546). I've seen it and I took a brief look, but I didn't read it carefully and I failed to understand the crux of it. I just don't want to investigate 40+ pages of formulas, when I already know that de Raedt's reasoning is often confusing and misleading, and that Hess is well-known for fighting will Bell's theorem, though his claims were long ago shown wrong by people, whose opinion I respect in this issue (see http://arxiv.org/abs/quant-ph/0208187).
This explains why you will never understand him. Apparently, as soon as you see Hess or de Raedt, you put on green goggles. The article you posted as disproving Hess is nothing short of a joke. (See http://arxiv.org/abs/quant-ph/0307092).
If you have read and understood this 40+ pages article, everybody here will be grateful if you give us the arguments in a concise and clear way.
If you are interested in understanding the opposing position, you will make the effort to read and understand their arguments before purporting to refute it. Since it appears you have access to de Raedt personally, why don't you ask him to explain to you concisely what the article is talking about. That will be much better than any of my efforts to explain his work to a hostile audience.
9. For some reason you completely ignored my point about double-slit paper of de Raedt. You were first to mention it! Did you read it? If you did, could you please comment on what I said earlier? If you didn't, how come you use it in the arguments?
You claimed that some photons were lost in their double slit simulation. This is wrong! All photons reach the detector and affect the outcome of the experiment. Maybe what you were trying to say is that in their model, not all photons result in a click. In any case, do you have experimental evidence proving that all photons leaving the source MUST result in a click at the detector in a double slit experiment?

mn4j

Feb17-09, 12:47 PM

On the other hand, there is no existing candidate LR theory on the table to compare to QM at this time.

You are mischaracterizing the debate as one between LR and QM. It is NOT.

Stochastic Mechanics (Marshall, Santos) is an example of a field of research in that regard, but every candidate SM model is found to have problems and is quickly modified again. And since such models do not predict anything useful, there is no incentive to study them further. We already have a very useful model - QM - and the experiments supporting it are in the thousands. Something useful from the field of study would go a long way towards convincing the scientific community.

So yes, I think quantity does matter, and I think utility matters. And I think the history of the area does matter as well, including when a theory (QM) is supported by improving technology.
The utility of a theory says nothing about it's correctness. The system of epicycles was very useful in the dark ages but you won't claim it as a correct theory. Technology always precedes theoretical understanding.

Why should you need to close every loophole simultaneously if you can close each separately? If a prisoner cannot escape from the first lock by itself, and cannot escape from the second lock by itself, how can he escape when both locks are present?
You answer your own question. The papers you mentioned are just prisoners claiming to have escaped because they were able to open one of seven locks. Unless you can open all seven locks you can't reasonably claim to have escaped, even if you change your name to Houdini.

I don't disagree with a desire to close all loopholes simultaneously; but I think that is a standard that is being applied to Bell tests which is applied nowhere else in science. Surely you must have noticed this as well.
It is common sense. The standard is demanded by the claims made by Bell. Extraordinary claims require extraordinary evidence. If you claim there is no green stone on Jupiter, you better get your ducks together and be sure you have combed every micrometer of the planet before you can say your experiment proves the claim. Yet if you claim there is a white stone in Alabama, all you have to do is find one white stone anywhere in Alabama to prove your claim. Bell says NO LR can violate his inequality.

DrChinese

Feb17-09, 01:31 PM

Three points. First. I'm not an expert in Bell tests and loopholes issue, so can't really comment on that on the detailed level. I know that there's for example "time-coincidence" loophole (http://arxiv.org/abs/quant-ph/0312035), which is apparently exactly the loophole de Raedt is exploiting (http://arxiv.org/abs/quant-ph/0703120, the link I already gave). I'm not sure that all known loopholes were already closed even separately, though this might be true. In particular, I just don't know any details about this "entanglement" studies that you cite (and don't have time at the moment to start reading them). Do they test Bell inequalities after this entanglement "swapping"? Or how else these findings prove LR false?

Second. I guess that I slightly disagree with you about different standards of tests. Of course there are super-precise tests of GR still being done. But to test GR you need to observe something that is predicted by GR, like light deflection or whatnot. When this is observed, nobody claims that there's a "loophole" in this experiment, and the results can be interpreted such that light is not deflected. It's evident: nobody heard of any loopholes in GR tests. On the other hand, to test QM versus LR one needs to show that the Bell's inequalities are violated. And all the attempts to show it still have some loopholes that allow alternative explanations.

Third. Nobody in his right mind claims that QM is "wrong". For de Raedt, QM is a correct mathematical model working well on the ensemble level only, without saying anything about single events. He is not trying to show that QM is wrong, he is trying to show that it can be completed in a LR way. Well, we know that it's impossible due to Bell. But de Raedt obviously disagrees. And it doesn't make a lot of sense for me to defend de Raedt, but he is most definitely not a crackpot (he has done a huge amount of "real" work in computer simulations of different physical models, including decoherence etc.). I believe (as you do) that his reasoning about Bell is flawed, but he certainly does not try to obscure anything on purpose: I'm quite sure that he is honest.

A couple of comments, and by the way I doubt our positions are very different overall.

Any theory, including GR, can be attacked as lacking loophole free experimental support by a sufficiently motivated scientist. The concept would be to deny an essential element of the theory, and then try to show that somehow the experiment "could" be wrong even if the evidence is convincing by normal scientific standards. What if GR readings are not a fair sample? Maybe Newtonian physics is correct instead because there is a built-in sample bias. (I am only kidding of course.)

The entanglement swapping issue is really just another aspect of the hurdles any LR theory must explain. Two independently created photon pairs A1/A2 and B1/B2 are created. By performing a suitable partial Bell State Measurement (BSM) on one of each pair (A1 & B1), their partners A2 and B2 are now partially entangled (as to time bin). (So in this particular case, polarization is not swapped but that has been done as well by Pan, Zeilinger et al.) So the question is: how do the local hidden variables guiding A2 and B2 - which have never been in causal contact - manage to be correlated? That's not even a process that a non-local Bohmian (dBB) type theory has an easy time with.

As to intellectual honesty: no assertion being made to the negative. I just ask why someone in that position wouldn't make the source of the delta between LR model and QM be obvious? That is the first thing we all look for. And yet I always find myself reading a lecture on the wrongs of Bell while looking for that little detail I know is there somewhere. The author, I would think, would know what that detail is.

I always ask myself: what would Einstein have thought about Bell or Aspect? If he were alive today, I think he would be well convinced and would cede the essential point.

DrChinese

Feb17-09, 01:39 PM

The utility of a theory says nothing about it's correctness. The system of epicycles was very useful in the dark ages but you won't claim it as a correct theory. Technology always precedes theoretical understanding.

That's wrong: how are theories judged correct? There is no such standard. Theories can have experimental support, and theories can make predictions that can be tested. And that is how they are judged. Correctness implies black and white, right or wrong. Theories can be better or worse depending on their application. But I cannot meaningfully say a theory is correct.

As to technology preceding theory: that makes no sense at all. Sometimes it does, sometimes it doesn't. There is no historical absolute on this. So again, meaningless.

kobak

Feb17-09, 04:19 PM

Thanks for replying. In the beginning I started answering and addressing all the points where we disagree, but this is getting too huge and difficult to handle (so I'll concentrate on the main issue). But let me first say one thing. I'm not a "hostile audience", because I sincerely try to understand what de Raedt is saying. But it's extremely difficult to talk with you because you're constantly being very sloppy. Here's an example:

If you say Raedt's modes are not examples of LR, I repeat once again: YES THEY ARE. You see this kind of discussions takes us no where. Explain why they are not.

Are you joking? I have been saying all the way that I think that de Raedt's models are LR. Well, I think that you just miswrote something here, but it's quite difficult to try to decipher you sometimes. Another example:

2) That Bell's inequality is not a correct representation of local reality.
Now for some reason, Bell's followers ALWAYS gravitate towards (1). Do you agree that (2) is also a possibility

What on Earth is this second option supposed to mean at all? "Bell's inequality is not a representation of local reality"? Eh? I guess that what you mean here is that BLR is not LR. But it's really a pain to guess what you meant all the time.

Now, it's clear that the *MAIN POINT* is that you think there are LR theories that are not BLR. And you think that de Raedt's model is an example. I don't see why it's not BLR, I think it's absolutely BLR, and I think that I never saw any LR-but-not-BLR suggestion. And without an example I won't believe that that's possible. Here's your objection:

If de Raedt's model is BLR then how do you explain the fact that the model violates the inequality, when according to Bell it is impossible.

Well, my answer is simple: it does not violate the inequality. The inequality "seems" to be violated in the particular experimental setup because the certain post-selection procedure is applied. It's possible to create correlation by post-selecting, that's what this whole coincidence loophole is about!!

You have no idea what you are talking about. Even ardent Bell believers have shown that not all LR are accounted for in BLR. See http://arxiv.org/abs/quant-ph/0205016 for one example.

Thanks for giving this link, it's actually interesting. I don't see though how it proves your point. Authors clearly write that "memory loophole" that they're describing can be avoided in experiment. If it's avoided along with other loopholes -- goodbye LR.

You claimed that some photons were lost in their double slit simulation. This is wrong! All photons reach the detector and affect the outcome of the experiment. Maybe what you were trying to say is that in t heir model, not all photons result in a click. In any case, do you have experimental evidence proving that all photons leaving the source MUST result in a click at the detector in a double slit experiment?

Yes, I meant exactly this: not all photons result in a click. I don't have an evidence, but it could be obtained. I described two experiments that could check this model. Take a look at the second. De Raedt says that if the screen is moved back and forth, the interference picture will get smeared. If this experiment is done and interference is NOT smeared, then de Raedt himself said that he would "retire", which I guess means that he will admit that his models are totally wrong and give up. And what would you say in this case?

kobak

Feb17-09, 04:23 PM

DrChinese, yes, I think that our position regarding the main points here is the same. And I must say that I myself have also wondered many times about what "would Einstein have thought about Bell or Aspect"...

Pio2001

Feb17-09, 04:55 PM

No. I disagree. So long as Bell's inequalities purport to make claims about reality, the correspondence between those inequalities and reality MUST be independently validated by experiments before any claims they make about reality can be said to be proven.

In photons' polarisation experiments, the correspondance between Bell's inequality and reality are that a detection is noted 1 and an absence of detection is noted 0, and also that nothing that is done outside the past light-cone of an event has any observable consequence on this event, which corresponds to the fact that in Bell's theorem, A does not depend on beta and that B does not depend on alpha.

The second correspondance is validated by experiments that show that nothing can go faster than light.
The first correspondance has not to be experimentally validated. You don't have to prove that you set 1 for a detection and 0 otherwise. We believe you !

Agreed without prejudice. Note that every loop-hole found to date is a hidden assumption in Bell's proof. I do not claim by agreeing to the above that all loop-holes have been found.

Action of the detector on the source, disproven by Aspect with ultra-fast switch, was not a hidden assumption, it was the explicit assumption that A did not depend on beta and conversely.
Fair sampling loophole was not either. Bell's theorem applies to the means of all measurments, not only some of them.
Statistics loophole have been filled with the GHZ evidence.

I've not studied all this, but not all loopholes were hidden assumptions in Bell's theorem. Actually, it seems to me that most loopholes claimed to be found in Bell's theorem rather than in experiments were unfounded. The CHSH generalisation of Bell's theorem makes things more clear : it takes into account anything that can happen around the measurment as hidden variable.

If you say Raedt's modes are not examples of LR which are not accounted for by Bell's LR, I repeat once again: YES THEY ARE. You see this kind of discussions takes us no where. Explain why they are not.

They violate Bell's inequality because Cxy depends on both t(n,1) and t(n,2) (equation 3), which is not the case in Bell's theorem. In Bell's theorem, Cxy depends only on the product of the measurments results (the Kronecker deltas in equation 3).

The role of t(n,1) and t(n,2) is to introduce a measurable individual dependance on the measurment angles, while they have no effect on the individual spin results.

Technically, it makes Cxy not being Bell's coincidence rate anymore. It has more to do with "what we measure" than with "what is locality".

That is a very narrow reading of de Raedt's work. Did you completely fail to understand the importance of the Deterministic Learning Machine model of de Raedt's?

De Raedt's pseudorandom model works without any Deterministic Learning Machine, and perfectly predicts Bell's inequality violation ! DLM are not involved in this step.
DLM are there to restore determinism, after the prevous step has restored locality.

Moreover, I'm not sure of it, but it seems to me that DLM would be accounted for as hidden variables in the general CHSH proof of Bell's theorem of 1969 :
This generalisation attributes hidden variables not only to the particles, but also to the measurment devices. For this purpose, the result A, function of the hidden variable lambda, and of the angle alpha, with the value -1 or +1, is replaced by the average value of A, function of alpha and lambda, on all hidden variables of the measurment device, and we start with
|average of A| <= 1. (respectively for B...)

Violation of Bell's inequality in any experiment has two possible explanations, not just one.
1) That Bell's inequality is a correct representation of local reality and the experiment is either not real or not local or both
2) That Bell's inequality is not a correct representation of local reality.

Now for some reason, Bell's followers ALWAYS gravitate towards (1). Do you agree that (2) is also a possibility and MUST be considered together with (1) when interpreting the results of these experiments? Please, I need a specific answer to this question.

I myself agree, but case 2 deals with what we do, while case 1 deals with what we get.

In De Raedt's simulation, Cxy is not the coincidence rate defined in Bell's theorem. That's how Bell's inequality does not represents what's going on in the simulation.
If the simulation is a good representation of reality, then the experiment can be modified so as to make W big enough compared to |t(n,1) - t(n,2)| in equation 3, so that the Heaviside function is always equal to 1, and Cxy tends to Bell's definition of the measurments product.
This way, we get back the experiment in adequation with Bell's theorem (case 2 is discarded), and we can test local determinism.

Another, sad, example : Joy Christian's use of Clifford algebra to prove Bell wrong ( http://arxiv.org/abs/quant-ph/0703179 ). Christian uses the half spin model, where Bell's theorem is applied setting spin down = -1, and spin up = +1.
He starts from the hypothesis that spin down and spin up are not real numbers, but numbers from Clifford algebra. He then shows that S can be equal to more than 2.

Since Bell's theorem says nothing else than if the possible results are -1 or 1, then S<=2, Christian's result is trivial and useless !

mn4j

Feb17-09, 10:54 PM

That's wrong: how are theories judged correct? There is no such standard. Theories can have experimental support, and theories can make predictions that can be tested. And that is how they are judged. Correctness implies black and white, right or wrong. Theories can be better or worse depending on their application. But I cannot meaningfully say a theory is correct.

As to technology preceding theory: that makes no sense at all. Sometimes it does, sometimes it doesn't. There is no historical absolute on this. So again, meaningless.
I guess then you believe the system of epicycles is an accurate representation of the solar system and the motion of the planets!

mn4j

Feb17-09, 11:49 PM

What on Earth is this second option supposed to mean at all? "Bell's inequality is not a representation of local reality"? Eh? I guess that what you mean here is that BLR is not LR. But it's really a pain to guess what you meant all the time.
If you have thought it through clearly enough you will know what the second option means. Let me put it to you in layman terms.

A man was depressed to the point he believed he was dead. No matter the efforts of his family he kept saying he was dead. A smart doctor tried to convince him that dead men do not bleed. After a significant effort he accepted. But at that moment, the doctor pierced him with a needle and he started bleeding. Can you guess what his next statement was? The doctor had hoped he would say "I am alive". Instead he said "Oops, I guess dead men bleed afterall".

The violation of Bell's inequality only proves that that the assumptions used in deriving the inequality do not apply to the experiment in question. Do you agree? Please give me a specific answer to this.

Those assumptions include assumptions about the way probabilities of local realist variables are supposed to be calculated. So in effect, Bell has an untested presentation of how local realist theories are supposed to behave. Yet when the inequalities are violated, instead of re-evaluating those assumptions, Bell proponents screem "I guess dead mean bleed after all".

Now, it's clear that the *MAIN POINT* is that you think there are LR theories that are not BLR. And you think that de Raedt's model is an example. I don't see why it's not BLR, I think it's absolutely BLR, and I think that I never saw any LR-but-not-BLR suggestion. And without an example I won't believe that that's possible.
I already mentioned why this is not a constructive criticism. I also explained in my previous post to you why de Raedt's mode is not accounted for by Bell. The article I presented by a pro-Bellist clearly states that Bell's model does not account for models like de Raedt's which have memory effects, notwithstanding the conclusion of that paper. Also your claim that you never saw any suggestion that Bell's representation of LR was not exhaustive is surprising because it is numerous in the literature. This thread was started by one such, Hess has presented a few, Joy Christian has presented a few. Are you serious?

Well, my answer is simple: it does not violate the inequality. The inequality "seems" to be violated in the particular experimental setup because the certain post-selection procedure is applied. It's possible to create correlation by post-selecting, that's what this whole coincidence loophole is about!!
What experiment setup, it is a simulation. What aspect of the simulation do you claim deviates from how real experiments are actually performed.
Thanks for giving this link, it's actually interesting. I don't see though how it proves your point.
Do you agree that according to this article, Bell's theory does not account for models with memory effects. The authors state as much even though they end up dismissing it's importance. At least they were honest to admit that Bell's theory does not account for such local realist theories, which you apparently are still unwilling to do.
Authors clearly write that "memory loophole" that they're describing can be avoided in experiment. If it's avoided along with other loopholes -- goodbye LR.
That is beside the point. Do they or do they not state that Bell's model of LR does not apply to situations in which there are memory effects? Please answer this question.

Yes, I meant exactly this: not all photons result in a click. I don't have an evidence, but it could be obtained. I described two experiments that could check this model.
You don't have evidence, yet you are ready and willing to proclaim proudly that de Raedt's model is wrong on this basis? Isn't it more prudent to wait until you have obtained such evidence before you make such claims?

Take a look at the second. De Raedt says that if the screen is moved back and forth, the interference picture will get smeared. If this experiment is done and interference is NOT smeared, then de Raedt himself said that he would "retire", which I guess means that he will admit that his models are totally wrong and give up. And what would you say in this case?
Do you believe, the interference will get smeared if the slits are moved back and forth? What about the source?

DrChinese

Feb17-09, 11:55 PM

I guess then you believe the system of epicycles is an accurate representation of the solar system and the motion of the planets!

The map is not the territory, my friend. Theory is always a model (map). And some are better than others.

kobak

Feb18-09, 07:49 AM

Hello again, mn4j,
I suggest that if we continue this discussion at all, let's try to concentrate on the most important points only and also try avoid nitpicking each other (like what "scientific consensus" really means etc.). I'm quite sure that we won't reach an agreement, but we can at least pinpoint our disagreements.

The violation of Bell's inequality only proves that that the assumptions used in deriving the inequality do not apply to the experiment in question. Do you agree? Please give me a specific answer to this.

Yes, certainly.

Those assumptions include assumptions about the way probabilities of local realist variables are supposed to be calculated. So in effect, Bell has an untested presentation of how local realist theories are supposed to behave.

Well, let me put it a bit differently. I've been always repeating here that "local realism" is not a well-defined term. So strictly speaking you're right: it might not be fully correct to say (without any additional clarifications) that Bell's theorem proves that all LR theories should obey Bell's inequality. I hope you'll be happy that I agree with you here.

However, here's my main point: Bell derives his technical assumption about probabilities distribution by providing some particular *physical* intuition. This technical assumption that he uses is certainly always true and absolutely uncontroversial in all areas of classical (meaning non-quantum) physics. It's just the plain fact, that in all classical physics the outcome of Bob's experiment can never be statistically dependent on Alice's choice of experimental setup etc., so Bell's assumption holds. So let me drop the issue of "local realism" and make the following claim instead: Bell's theorem shows that *assuming "classicality"* -- his inequality has to be true.

Do you agree to such a statement? Note, that even if de Raedt model turns out to (a) be not BLR, (b) violate Bell's inequalities, (c) be just right (I still strongly disagree that it's possible, but even if it's like that), -- then this is certainly not a "classical" model. In classical physics apparata do not learn. The same goes for Christian's models: if they are right, then ok, spin measurements form a Clifford algebra (whatever this means), and this is again certainly not a "classical" model.

So, to recapitulate: do you agree that Bell's technical assumptions about correlations are completely well motivated, if we change the assumption of "local realism" to assumption of "classicality"? I very much hope that you will agree to that.

---------

Now, except of this, I see two main issues. First: I claim that de Raedt's model is BLR (and hence has to obey Bell's inequalities, and hence will not be able to hold anymore, after a loophole-free violation of inequalities is observed). You're saying that his model is not BLR, and your argument is that it's even stated in the Darrett-Popescu article. Well, I have to read it more carefully to answer you here. I will try to find the time for that and get back then, that's important.

Second point is that you also say that de Raedt model is not BLR because it violates Bell's inequalities. Here I disagree strongly, and I think that this shows that you don't really understand what a "coincidence loophole" is. Here's again, how I see it: 1. de Raedt's simulation does not deviate from real experiment of Weihs et al (you asked me, how it deviated; well, it doesn't). 2. The coincidence loophole (which this experiment did not avoid) means that it's possible to explain the apparent Bell's inequality violation by the fact that events are post-selected, and because of this post-selection the correlation is created out of nothing. 3. This is exactly what de Raedt is exploiting. 4. Bottomline of this analysis: de Raedt's model is BLR, it obeys Bell's inequalities, but in the non-perfect loopholed experiment it can LOOK like it violates them. This is the view expressed here: http://arxiv.org/abs/quant-ph/0703120 (see also http://arxiv.org/abs/quant-ph/0312035 about coincidence loophole). Do you understand this argumentation? You may disagree (please tell where exactly), but do you understand it?

Let me also ask for a clarification of your point of view. Do you think that Bell's inequalities are in reality NOT violated (and all experimental violations are only due to loopholes)? Or do you think that his inequalities in reality ARE violated (so that even ideal perfect experiment will find violations), but these violations can be explained by some LR theory which is not accounted by Bell's theorem? I guess that yours is the latter opinion. Which means that even a perfect loophole-free experiment will not prove anything to you, right? Why are you then arguing about loopholes and this "prisoners who can get out in different ways" at all?

---------

Finally, about double-slit paper.

You don't have evidence, yet you are ready and willing to proclaim proudly that de Raedt's model is wrong on this basis? Isn't it more prudent to wait until you have obtained such evidence before you make such claims?

It is. I don't proclaim that his model is wrong on this basis, I'm proposing a bet (let's put it that way). Imagine this experiment is done exactly as de Raedt himself proposed it (screen is jittered from left to right parallel to itself). Question: what will happen? My bet: interference pattern doesn't change. "de Raedt's" bet: interference pattern gets smeared, because "detectors" on the screen won't have enough time to "learn". Your bet?

And additionally: imagine that experiment is made and I win. What would that mean? I think that de Raedt thinks that it will prove his model false, and he certainly does not believe that this outcome is possible AT ALL. What do you think?

Please don't ask questions about different experimental setups, I'm interested only in this one, that is defined absolutely precisely.

sirchasm

Feb18-09, 08:01 AM

A comment on "the map is not the territory".
I believe this implies we can only simulate "territory" as in QM simulations, fliud-dynamics on supercomputers. supernova and nuclear weapons simulations etc.

So the question has to be: when is a simulation 'exact'? And can we simulate the territory = construct a map, that is indistinguishable = identical to "the territory"?

mn4j

Feb18-09, 08:58 AM

The map is not the territory, my friend. Theory is always a model (map). And some are better than others.
Yes you are right, all theories are valid :rolleyes:, everything is relative, the map of China is a valid map of the US, just worse than the US map and the earth is the center of the solar system, depending on your perspective.

DrChinese

Feb18-09, 09:23 AM

So the question has to be: when is a simulation 'exact'? And can we simulate the territory = construct a map, that is indistinguishable = identical to "the territory"?

It is impossible, in principle, for any map to precisely model a territory other than for a limited (by assumption) scope. (The only way is if the map *IS* the territory, in which case it really doesn't qualify as a map anymore.) The reason I bring this up is that our natural desire is to use a map (i.e. a model) as a convenience. When you use a map, you assume it is accurate enough to be useful for your purposes. But most people know a map is a model; yet they have trouble seeing a theory as a model of reality too. Yet it is, and this is not merely a philosophical issue. As you can see from the discussion, some people think their model is "true" or "correct". Well, a map can be more useful or less useful (as in a relative way) but I don't see how any map can be absolutely and finally "true".

Take gravity, as an example: the acceleration due to gravity on earth is about 9.8 meters per second. And yet there are only a few objects on the planet that are actually accelerating (relative to the earth) at that rate. Most are at rest or moving according to other forces acting on the object. So obviously, no theory of gravity can describe the movement of objects on earth. You must instead expand your description to include numerous other variables on a case by case basis.

...Which defeats the point of our original map, and shows it to be absolutely wrong - if we insist on calling the map itself absolutely correct. If we acknowledge it as only a useful tool, the problem disappears. That is why I believe it is unfair to criticize QM as a tool. Because as a tool, it is useful and fulfills the reasonable demands we place on it. Only when someone tries to ask if it is "correct" does a problem arise. Instead, I think we need only ask if a better tool can be found. EPR asked if a more complete specification of a quantum system was possible, and they believed it was. But the subsequent evidence (Bell+Aspect et al) is that it is not. But I do not believe that makes QM true, it just means it is a good - or perhaps best - map.

DrChinese

Feb18-09, 09:29 AM

Yes you are right, all theories are valid :rolleyes:, everything is relative, the map of China is a valid map of the US, just worse than the US map and the earth is the center of the solar system, depending on your perspective.

Sorry, didn't mean to divert you from your distinctly more interesting discussion with kobak. But I think your statement pretty well proves my point of what happens when people think their model IS reality. They freak out because they find out they are not in Kansas any more. I will bow out for a while on this thread and let you continue. Later,

mn4j

Feb18-09, 09:52 AM

Well, let me put it a bit differently. I've been always repeating here that "local realism" is not a well-defined term. So strictly speaking you're right: it might not be fully correct to say (without any additional clarifications) that Bell's theorem proves that all LR theories should obey Bell's inequality. I hope you'll be happy that I agree with you here.
Yes I'm happy, thank you.
However, here's my main point: Bell derives his technical assumption about probabilities distribution by providing some particular *physical* intuition. This technical assumption that he uses is certainly always true and absolutely uncontroversial in all areas of classical (meaning non-quantum) physics.
And here is my main point: The above statement is False for the following reasons:
1) Bell's particular *physical* intuition does not account for the most interesting class of local hidden variables, the types Einstein and Schrödinger would have liked to see.
2) The technical assumptions he uses are not always true, for reasons I have explained here. The effect is that this introduces further hidden assumptions -- at the very least, the assumption that those technical assumptions are always true for local hidden variables. Without independent validation of this assumption, the possibility that this assumption is false will never go away, even if 99% of the scientists believe it.
3) Those technical assumptions are not uncontroversial in all areas of classical physics. In fact violation of Bell's inequalities is not limited to quantum systems. Take a look at de Raedt's recent paper for an example in which there is violation of Bell's inequality for a voting game with three human players.

It's just the plain fact, that in all classical physics the outcome of Bob's experiment can never be statistically dependent on Alice's choice of experimental setup etc., so Bell's assumption holds. So let me drop the issue of "local realism" and make the following claim instead: Bell's theorem shows that *assuming "classicality"* -- his inequality has to be true.
This is false. I have already explained in this thread that any two time varying harmonic systems are correlated and as such their probabilities are not disjoint. Unless you want to claim that two pendulums or clocks on opposite sites of the globe are not classical.
Note, that even if de Raedt model turns out to (a) be not BLR, (b) violate Bell's inequalities, (c) be just right (I still strongly disagree that it's possible, but even if it's like that), -- then this is certainly not a "classical" model. In classical physics apparata do not learn.

This is false. Let's take two classical systems
1) Heat transfer in a gas: How does the heat go from one end to another? The molecules learn the velocity of the "hotter" molecules they collide with, and transfer this "knowledge" to other molecules they themselves collide with. This is exactly what a DLM is.
2) A billiard ball. The stationary ball, on collision with the moving ball, learns the momentum of the on-coming ball. That is a DLM.

DLMs are classical!

So, to recapitulate: do you agree that Bell's technical assumptions about correlations are completely well motivated, if we change the assumption of "local realism" to assumption of "classicality"? I very much hope that you will agree to that.
Sorry, I can not agree that. Because as I have explained repeatedly, I know of systems for which Bell's inequality does not apply, which can not be classified as non-classical. But also because I am not quite sure what you mean by classical.
Now, except of this, I see two main issues. First: I claim that de Raedt's model is BLR (and hence has to obey Bell's inequalities, and hence will not be able to hold anymore, after a loophole-free violation of inequalities is observed).
So then, even if you believe it is BLR, since de Raedt's model is also a simulation, not much different from a theoretical derivation, the fact that it violates Bell's inequalities either means Bell's mathematics is wrong, or de Raedt's mathematics is wrong. So again the issue boils down to whether "a bleeding man is alive" or "dead men bleed".

I will reply to your second point in a separate post as this is getting too long.

mn4j

Feb18-09, 10:54 AM

Second point is that you also say that de Raedt model is not BLR because it violates Bell's inequalities. Here I disagree strongly, and I think that this shows that you don't really understand what a "coincidence loophole" is.
I think maybe you are the one who is not quite sure what it means. Do you deny the fact that de Raedt's model reproduces the QM result? Doesn't that mean it also violates the inequality. Please answer this question.
Here's again, how I see it: 1. de Raedt's simulation does not deviate from real experiment of Weihs et al (you asked me, how it deviated; well, it doesn't).
Does the real experiment deviate from Bell's inequality? Does the real experiment agree with QM? Does QM violate also violate the coincidence time loophole? (see http://arxiv.org/abs/0801.1776)

2. The coincidence loophole (which this experiment did not avoid) means that it's possible to explain the apparent Bell's inequality violation by the fact that events are post-selected, and because of this post-selection the correlation is created out of nothing.
Again you focus only on the "dead men bleed" part and completely ignore the fact that the coincidence time loophole can also mean Bell's inequality does not model the behaviour of all real local systems.

3. This is exactly what de Raedt is exploiting. 4. Bottomline of this analysis: de Raedt's model is BLR, it obeys Bell's inequalities but in the non-perfect loopholed experiment it can LOOK like it violates them. This is the view expressed here: http://arxiv.org/abs/quant-ph/0703120 (see also http://arxiv.org/abs/quant-ph/0312035 about coincidence loophole). Do you understand this argumentation? You may disagree (please tell where exactly), but do you understand it?
http://arxiv.org/abs/quant-ph/0703120 has been refuted by de Raedt (see http://arxiv.org/abs/0706.2957).
The bottom line is this: de Raedt's model satisfies the Einstein's conditions of local causality and exactly reproduce the single particle and two-particle expectation values of the singlet state.

Let me also ask for a clarification of your point of view. Do you think that Bell's inequalities are in reality NOT violated (and all experimental violations are only due to loopholes)? Or do you think that his inequalities in reality ARE violated (so that even ideal perfect experiment will find violations), but these violations can be explained by some LR theory which is not accounted by Bell's theorem?
My viewpoint is that it is possible to find a model that satisfies the Einstein's conditions of local causality and exactly reproduce the expectation values of the singlet state, contrary to Bell's claims. My viewpoint is that constructing Bell's inequalities in a manner which accounts for all possible real experiments like the ones performed so far will result in inequalities that are never violated. My viewpoint is that no experiment has ever been performed exactly as Bell modelled in his equations. Therefore Bell's theorem is currently an untested theorem, and when such such an experiment is performed, it will not violated the inequalities.
Which means that even a perfect loophole-free experiment will not prove anything to you, right?Show me a loophole free experiment which violates Bell's inequalities and I will concede. My view is there will never be a loophole free experiment because the problem is not with the experiments but with the inequality. If a theory is so restrictive in scope that it has taken many talented experimentalists several decades to test it in vain, then maybe the answer is not that "dead men bleed afterall" or rather, that "we need more perfect experiments". The answer is that "the bleeding man is alive" or rather, that Bell's inequalities do not accurately represent real experiments that can be performed.

I don't proclaim that his model is wrong on this basis, I'm proposing a bet (let's put it that way). Imagine this experiment is done exactly as de Raedt himself proposed it (screen is jittered from left to right parallel to itself). Question: what will happen? My bet: interference pattern doesn't change. "de Raedt's" bet: interference pattern gets smeared, because "detectors" on the screen won't have enough time to "learn". Your bet?
Isn't it common sense that a moving camera takes smeared images? In case you did not know, this experiment has been performed many times over by lay people and you lost the bet already.

kobak

Feb18-09, 05:39 PM

Well, I fear that I don't see how this discussion can lead to anything more. From what you've just said, I'm now sure that you don't understand what a coincidence time loophole is. This is crucial, so there's unfortunately no point in this discussion anymore.

I will briefly react to some points you made.

And here is my main point: The above statement is False for the following reasons:
1) Bell's particular *physical* intuition does not account for the most interesting class of local hidden variables <...>
2) The technical assumptions he uses are not always true, for reasons I have explained here. The effect is that this introduces further hidden assumptions -- at the very least, the assumption that those technical assumptions are always true for local hidden variables <...>
3) Those technical assumptions are not uncontroversial in all areas of classical physics. In fact violation of Bell's inequalities is not limited to quantum systems. Take a look at de Raedt's recent paper for an example in which there is violation of Bell's inequality for a voting game with three human players.

My statement was this: "in classical physics Bell's technical assumption was always uncontroversial and generally considered true". You're saying that this is false and give as an example some theories of hidden variables. I'm sorry, I was talking about CLASSICAL physics. So I disregard your points (1) and (2) here. The only meaningful response is your point (3). Well, I guess I should take a look on this de Raedt's example. But what I did already see, is the "Bernoulli urn" example from Jaynes, also repeated by de Raedt, -- and this example totally and completely misses the point. So I'm quite fed up by his examples.

This is false. I have already explained in this thread that any two time varying harmonic systems are correlated and as such their probabilities are not disjoint. Unless you want to claim that two pendulums or clocks on opposite sites of the globe are not classical.

I have seen your example with pendula. I don't see how this relates to the situation when two space-like separated measurements are performed and one result is statistically dependent on the choice of other experiment. I'm talking about this very situation (as described by EPR and Bell), not about some abstract correlation between something.

I think maybe you are the one who is not quite sure what it means. Do you deny the fact that de Raedt's model reproduces the QM result? Doesn't that mean it also violates the inequality. Please answer this question.

This is exactly the point where it becomes clear that you don't properly understand this loophole issue. de Raedt's model reproduces the QM result for THIS PARTICULAR experiment (Weihs et al.) which has loopholes. Because of this experimental flaws, it's possible for a BLR theory to give the impression that it violates Bell's inequalities. In the loophole-free experiment de Raedt's model will obey Bell's inequalities, unlike QM.

I really can't express this clearer (without giving technical arguments, that are anyway contained in the articles I cited).

http://arxiv.org/abs/quant-ph/0703120 has been refuted by de Raedt (see http://arxiv.org/abs/0706.2957).

Sure, I've seen this. I think his reply misses the point completely. Let me ask you something: did you actually read this critique of de Raedt and his reply? Or are you saying that it has been successfully refuted just because there exists an article claiming to be a refutation? I'm just curious.

Isn't it common sense that a moving camera takes smeared images? In case you did not know, this experiment has been performed many times over by lay people and you lost the bet already.

It's amazing how you keep avoiding answering my simple question for several postings already. I asked you what do you think about the outcome of double-slit interference when the screen is moved from left to right, and you're telling me something about cameras.

What on Earth has it to do with my question? If I project a movie on a white (ideal) screen with a projector and then start moving the screen from left to right with any frequency I want, the picture won't change, and all the viewers can still enjoy the movie. Now we're talking about double-slit interferometer instead of beamer. Screen is moved. Question: will the interference picture get smeared? If you say "yes" (like de Raedt does), here's the second question: assume, just assume for the sake of argument, that it's found NOT to change, exactly like the movie in the example above. What would be your conclusion? I remind here, that de Raedt said that in this case he would "retire".

I'm wondering whether you are again going to skip answering these two direct questions.

mn4j

Feb18-09, 11:24 PM

My statement was this: "in classical physics Bell's technical assumption was always uncontroversial and generally considered true". You're saying that this is false and give as an example some theories of hidden variables. I'm sorry, I was talking about CLASSICAL physics. So I disregard your points (1) and (2) here. The only meaningful response is your point (3). Well, I guess I should take a look on this de Raedt's example. But what I did already see, is the "Bernoulli urn" example from Jaynes, also repeated by de Raedt, -- and this example totally and completely misses the point. So I'm quite fed up by his examples.
You did not define what you mean by classical physics.

I have seen your example with pendula. I don't see how this relates to the situation when two space-like separated measurements are performed and one result is statistically dependent on the choice of other experiment. I'm talking about this very situation (as described by EPR and Bell), not about some abstract correlation between something.
Is it unclassical for photons and electrons which make up all experimental apparatus to exhibit time-varying harmonic oscillation? I take it you do not understand the difference between logical dependence and physical causation. You see, when you calculate probabilities of systems which are known to be correlated, like harmonic systems, you MUST consider them to be logically dependent even if there is no physical effect transfered between them, otherwise you get paradoxical results.

This is exactly the point where it becomes clear that you don't properly understand this loophole issue. de Raedt's model reproduces the QM result for THIS PARTICULAR experiment (Weihs et al.) which has loopholes. Because of this experimental flaws, it's possible for a BLR theory to give the impression that it violates Bell's inequalities. In the loophole-free experiment de Raedt's model will obey Bell's inequalities, unlike QM.
You seem to have stuck on this one loophole and you can't get passed it. Your statement is wrong. For all your claims about talking to de Raedt, you have no idea what his model is all about evidently. In case you did not know, de Raedt's model agrees with QM in single-photon beam-splitter and Mach-Zehnder interferometer experiments, wheeler’s delayed choice experiment, quantum eraser, EPRB experiments with photons, EPRB experiments with non-orthogonal detection planes etc.

You did not tell me if you believe the QM formalism also suffers from the coincidence time loop-hole. I wonder why?
Sure, I've seen this. I think his reply misses the point completely. Let me ask you something: did you actually read this critique of de Raedt and his reply? Or are you saying that it has been successfully refuted just because there exists an article claiming to be a refutation? I'm just curious.

I have read every article I point you to. Have you read the ones you point me to? Did you bother reading de Raedt's reply? What exactly do you believe misses the point. Spell it out and I will explain why it is you who missed the point. We can go into technical detail if you prefer.
It's amazing how you keep avoiding answering my simple question for several postings already. I asked you what do you think about the outcome of double-slit interference when the screen is moved from left to right, and you're telling me something about cameras.
The answer is so blindingly obvious I did not expect that you were serious. De Raedt gave you the answer even. The pattern gets smeared!!! It is not an experiment you need to bother doing. It has been done many times!!! Do I really need to answer the second question? If you think I'm wrong, go do the experiment and bring your results, then we can talk.
What on Earth has it to do with my question? If I project a movie on a white (ideal) screen with a projector and then start moving the screen from left to right with any frequency I want, the picture won't change
That is a very naive look at the situation. You can't be serious? A movie screen is not a detector. A photographic film is. Try projecting a still image on a photographic film while moving the film from left to right and then develop the film and see if it is not smeared. Do you seriously believe the same will not happen if the detector in front of a double-slit is jiggled? Isn't the interference image whatever is recorded on the detector. Or maybe let's jiggle your head real fast, while you watch, since your eyes are the real detector in this case. Are you telling me you will enjoy the same quality of movie as an unjiggled pair of eyes? You can perform this experiment right now. Jiggle your head while you read this page and tell me whether the image does not get smeared.

I'm wondering whether you are again going to skip answering these two direct questions.
I can point to umpteen questions of mine you have not answered but then again I'm not keeping score.

Are you ready to concede that classical systems DO learn just like in de Raedt's model? Are you ready to concede that Bell's model does not account for systems which learn, like de Raedt's?

kobak

Feb19-09, 03:19 PM

You did not define what you mean by classical physics.

That's easy: let's just say, that by classical physics in this case I mean all physics known before 1920. To make this comment self-contained, I repeat my claim: in classical physics Bell's technical assumption (that you and Jaynes and de Raedt keep saying is unjustified) is always true. You disagree. Well, it seems that to refute my claim one example would suffice. This example must be of the following (just to spell it out): two space-like separated experiments are done and the result of Alice's experiment "A" is NOT statistically independent from Bob's choice of experimental setup "b" or his outcome "B".

Maybe I miss something here, but I don't see how your harmonic oscillators can provide a counterexample to that. Please explain, if you think it can. And here's another consideration: if it were THAT easy to provide a counterexample to Bell (by just taking two harmonic oscillators), then why would all this story about deterministic learning machines be necessary? It seems to me that you make an oversimplification here. It's impossible to replicate QM correlations by just considering two oscillators or clocks, is it?

I take it you do not understand the difference between logical dependence and physical causation.

Just a short note: I think I do. It also seems to me that "logical dependence" is a term that you took from Jaynes (and de Raedt), because it's never being used in modern probability/statistics treatments. The precisely defined term that is usually used is "statistical dependence". Try googling this term, and your one. "Logical dependence" is used only in mathematical logic, but that's different.

You seem to have stuck on this one loophole and you can't get passed it.

Well, I'm sorry, but I believe that this is the most important and crucial point here.

You did not tell me if you believe the QM formalism also suffers from the coincidence time loop-hole. I wonder why?

Because I don't understand what your question means! How can a "formalism" suffer from a "loophole"? I think that this question gives another indication that you fail to understand the meaning of this loophole issue.

Let me give an analogy (it will be a rather silly one). Imagine we're trying to prove general relativity by observing gravitational waves. The opposing theory is Newtonian gravity, where there's no waves. So it the waves are observed, Newtonian theory is proved to be false. The experiment is done and experimental setup for some reason has to be located on the surface of the ocean. The gravitational waves are indeed observed. It may seem that GR is proven, but there's a subtlety: the whole apparatus was located in water and there were waves in water, so in principle it's possible to explain the apparent gravitational waves that were observed by just some influence from the water waves. So folks that don't believe in GR take this view. This experimental problem is called "water loophole" and experimenters are working hard to avoid it putting the setup on a hard ground, but they didn't succeed so far.

To spell it out: GR corresponds to QM, gravitational waves to the violation of Bell's inequalities, water loophole to coincidence-time loophole, explanation of how this faulty experiment can result in apparent detection of gravitational waves to de Raedt's explanation of how the faulty experiments can result in apparent violation of Bell's inequalities.

And now you come and ask, whether "the QM formalism also suffers from the coincidence time loop-hole". Analogy: does the general relativity suffer from the water loophole? This question just doesn't make any sense to me.

I have read every article I point you to. Have you read the ones you point me to? Did you bother reading de Raedt's reply? What exactly do you believe misses the point. Spell it out and I will explain why it is you who missed the point. We can go into technical detail if you prefer.

OK, I'm sorry for suspecting that you didn't read them. Yes, I also did. We could go into technical details, but it doesn't make any sense before we understand the issue of loophole in a similar way. Actually, when I read de Raedt's reply I was just lost several times. I think it misses the point, because it's unclear, and it is unclear because de Raedt apparently also does not understand this loophole issue. It's just that his reply doesn't really reply to what was said by Seevinck and Larsson. The critique is almost one page long and makes a clear point, while de Raedt writes a lengthy reply with all kind of beating around the bush.

Here's one particular example (maybe it's not the most important one. I don't remember any more, but I just have this written down anyway). Seevinck and Larsson say that in de Raedt's model \gamma is less then \gamma_0 which would be necessary to violate the inequality modified to take a loophole into account (and this equality is still violated by QM). de Raedt triumphally replies that they made a mistake in derivation, and then presents his own version of this derivation, which results in \gamma -> 0. Zero is clearly still less than \gamma_0, but he never comments on it.

The answer is so blindingly obvious I did not expect that you were serious. De Raedt gave you the answer even. The pattern gets smeared!!! <...> That is a very naive look at the situation. You can't be serious?

Oh, now I see that that's just a misunderstanding. Probably I didn't explain my experiment well enough, and now I checked and found that it's not described in arXiv version of the paper (http://arxiv.org/abs/0809.0616). It's just that when I asked de Raedt about this paper, he sent me a draft with a fuller version, and the experiment is described there. We're just talking about different experiments. You're right about yours, that's clear (and that "experiment" really is stupid and obvious, I agree).

Roughly speaking, what I meant is that screen is jittered, detection events are counted in individual detectors and then relocated with a computer program, that takes into account where the screen was at the moment of detection. Does it make more sense now? The crucial point is that de Raedt's detectors have first to learn before they start reproducing QM probabilities, and by jittering we prevent them from learning correctly (I quote de Raedt: "In other words, the experiment should address a time scale that is sufficiently short such that our detector models have not yet reached the stationary state").

He actually proposes a bit different setup, where only *single* detector is used, and it's slided back and forth along the "screen" line. The detection events are counted and then plotted versus the position of the detector at the moment of detection. If this detector is moved slowly (so that it reaches stationary state at every point) then the resulting plot will show the usual interference pattern. However if it's moved fast, then it would get smeared. Not because of this silly "moving camera" effect, but because detector doesn't have time to learn afresh at every single point.

Please tell me if this experiment now makes sense to you. If it does, then my questions remain the same. Do you think that interference pattern will change? What do you think is the prediction of the usual QM? What would you say if it actually doesn't change?

I can point to umpteen questions of mine you have not answered but then again I'm not keeping score. Are you ready to concede that classical systems DO learn just like in de Raedt's model? Are you ready to concede that Bell's model does not account for systems which learn, like de Raedt's?

First question: well, in the certain sense, yes. If the billiard ball getting kicked by another billiard ball counts for "learning", then yes. It's just that nobody in the times of classical physics would think that parts of the screen "learn" the phase of the incoming photons.

Second question: no, not yet. As I already said, here I should take a better look on the Popescu article. I didn't have time so far.

mn4j

Feb21-09, 10:49 AM

Hi Kobak,
Sorry for the late response. I will try to be brief because every response just seems to be getting longer and longer. So I will not attempt to respond to every teeny-weenie point.

That's easy: let's just say, that by classical physics in this case I mean all physics known before 1920. To make this comment self-contained, I repeat my claim: in classical physics Bell's technical assumption (that you and Jaynes and de Raedt keep saying is unjustified) is always true. You disagree. Well, it seems that to refute my claim one example would suffice. This example must be of the following (just to spell it out): two space-like separated experiments are done and the result of Alice's experiment "A" is NOT statistically independent from Bob's choice of experimental setup "b" or his outcome "B".
I have given one example already in this thread. Bob and Alice each have a pendulum, and they are free to adjust the length of the string as they like. Coincidences are said to occur if both Bob and Alice's pendulum have swung to the same angular position at the same time. Bob's selection of the length is not ontologically dependent on Alice's length selection and they are free to select any length they want. In fact they are not even aware of the existence of each other until the experiment is complete and we are looking at the results. The results were recorded as a time tagged values of the deviation of the pendulum from the target position for a given duration after every change in settings.
1) Do you agree that these two experiments are local realist in the classical sense?
2) Do you agree that there will be a correlation between the results obtained by Bob and Alice?
3) If you agree to (2). Do you agree that this correlation or (statistical dependence as you call it) is not due to spooky-action at a distance or conspiracy?
4) Do you then agree that given a coincidence, if I (the external observer doing the calculations) were to know Bob's string length, the result Bob obtained and the result Alice obtained, I should be able to calculate a probability for Alice's string length which is higher than the maximum entropy value?

In case you don't understand the last point, let me explain: Given a fair coin, the maximum entropy probability for heads or tails is 0.5. This value tells you nothing about the ACTUAL result of the experiment you just performed by throwing a coin. However, if the coin throw was part of a bet in which you chose heads and I see you going of to buy a bear, I will be able to say the probability that the result was heads is higher than 0.5. Even though I did not see the actual result and I have no determinative proof that the coin used in the particular case was fair. As you can see from this example, probability is NOT JUST about frequencies of outcomes but about state of knowledge (Read up on Harold Jeffreys). Frequencies of outcomes is just a subset of the ways of updating the state of knowledge.

So then going back to point (4), you see that it is possible to obtain a probability for Alice's string length that is higher than the maximum entropy value, if I know Bob's string length. This is logical dependence and exists even for situations in which there is no communication or physical influence between Alice and Bob. If this doesn't make sense to you, then you can not even begin to understand my arguments in this thread, or Jayne's for that matter.

Maybe I miss something here, but I don't see how your harmonic oscillators can provide a counterexample to that.Please explain, if you think it can.

Consider the following harmonic oscillator equation.

y(t) = A sin(\omega t + \theta)

in your favourite plotting package, generate a range of t values. Any range you choose. Using two random number generators, pick two sets of triplets (A, \omega, \theta)
use one set to calculate the corresponding y(t)_1 and the other set to calculate the corresponding y(t)_2 with the previously generate t values.
Plot y(t)_1 vs y(t)_2 and confirm that indeed there is a correlation between the two, no matter what values of (A, \omega, \theta) you used for each, however randomly you generated them.

In case you still think I did not understand what the coincidence time loophole is, let me point out to you that in this case, it has to do with which point in y(t)_1 is plotted against which point in y(t)_2 . I won't go into much further detail than that here other than to say you can even introduce a constant offset (or time delay) between t values used in both cases without eliminating the correlation.

Because I don't understand what your question means! How can a "formalism" suffer from a "loophole"? I think that this question gives another indication that you fail to understand the meaning of this loophole issue.
Bell's inequality is a mathematical formulation also. And it does suffer from many loopholes. Your question tell's me you believe loopholes are problems in the experiments rather than Bell's inequalities. How come then that in every paper trying to address a loophole, they always derive new inequalities which account for the loophole cases? See the original paper about coincidence-time loophole which confirms this. The loophole is in Bell's inequality not the experiment and Quantum mechanics also makes assumptions about coincident events.