Trying to Understand Bell's reasoning

[Shalashaska]

Obviously you are opposing each other so there is no one distinct opposition, you are his opposition, he is yours... I was merely encouraging him to lay it out in detail, I don't have a horse in this race. Most of the arguments of this type (on these forums) end up with people bickering over little details about undefined objects. Define everything from the top, then argue.

Disagreements don't change observations. From what I've read, this is an argument that just happens a lot here, over and over. The outcome appears to be the same, and that is that QM violates Bell Inequalities, and it is the best predictive theory on offer. The rest is details and quibbling because there is no other leg to stand on that I'm aware of.

 

[billschnieder]

JesseM:

I definitely understand what EPR meant by "elements of reality", and I definitely understand that it DOES NOT mean "I can see the moon when I am not looking at it", which is implied if you say outcomes must pre-exist observation. In any case, this is a rabbit trail and distracts from the main issue which I have already explained.

I am only interested in understand Bell's justification for writing

P(AB|H) = P(A|H) * P(B|H)

instead of

P(AB|H) = P(A|H) * P(B|AH)

In all your numerous examples and arguments, the only response relevant to this issue is your claim that, if H is completely specified, A adds no additional information to that already provided by H, therefore P(B|AH) = P(B|H). I don't have to respond to everything else in your rather verbose posts since this is the only point that is relevant to my issue. My response to this point, as I have already pointed out is as follows:

1) The definition of conditional independence is not based on additional information but on any information. In the article I quoted to you, section 2.1 titled "Definitions", page 2, midway down the page, where independence is defined it says:

X [is independent of] Y if any information received about Y does not alter uncertainty about X;

The same article goes on to show in section 3.1, page 3, where conditional independence is defined that just because P(x|yz) = a(x,z), it does not necessarily show that conditional independence applies. Specifically it says:

(2a) P(x|y,z)=P(x|z)
(2b) P(x|y,z)=a(x,z) read, P(x|y,z) is a function of just x and z.

A caution is called for here concerning the use of improper distributions for random variables. It is shown Dawid et al. (1973) that, in such circumstances, it is possible for (2b) to hold and, at the same time, for (2a) to fail. This is referred to as the marginalization paradox.

In any case this is not my main point, so don't focus all your attention here and igore my main point:

2) If P(B|AH) is really equal to P(B|H) as you insinuate, then it shouldn't matter which equation is used. Both should result in the same inequalities right? Do you agree that it should be possible to derive Bell's inequalties from either equation? Now following Bell's logic, try to derive the inequalities from P(AB|H) = P(A|H) * P(B|AH). It can not be done! Can you explain to me why? Just to be clear that you understand this point, let me rephrase it -- P(B|AH) = P(B|H) and P(B|AH) \neq P(B|H) can not both be true at the same time right?

 

[JesseM]

JesseM:

I definitely understand what EPR meant by "elements of reality", and I definitely understand that it DOES NOT mean "I can see the moon when I am not looking at it", which is implied if you say outcomes must pre-exist observation. In any case, this is a rabbit trail and distracts from the main issue which I have already explained.

The definition of "local realism" is not a distraction, it's central to the proof. Nothing I have said implies that "I can see the moon when I am not looking at it", though it does imply that all variables associated with the moon have well-defined variables even when I'm not looking, and therefore we can consider what conclusions could be drawn by a hypothetical omniscient observer who knows the value of all these variables (without assuming anything specific about what the values actually are on any given experimental trial). And all this stuff about variables having well-defined values when I'm not observing them only covers the "realism" aspect of local realism, locality is separate--for example, Bohmian mechanics would be an example of a realist theory that says all physical quantities have well-defined values even when we aren't looking at them, but it's also a non-local theory.

I am only interested in understand Bell's justification for writing

P(AB|H) = P(A|H) * P(B|H)

instead of

P(AB|H) = P(A|H) * P(B|AH)

Obviously this reduction is fine as long as P(B|H) = P(B|AH). And this is guaranteed to be true in a local realist world where A can't have any causal influence on B, and the only reason for correlations between A and B is some set of conditions H1 and H2 in the past light cones of A and B which influence their probabilities in correlated ways.

Do you understand what a "past light cone" is, and why it's essential to the definition of locality?

In all your numerous examples and arguments, the only response relevant to this issue is your claim that, if H is completely specified, A adds no additional information to that already provided by H, therefore P(B|AH) = P(B|H). I don't have to respond to everything else in your rather verbose posts since this is the only point that is relevant to my issue.

No, the rest is quite relevant, since I explicitly show various examples where we have two events A and B such that there is a correlation between A and B, but it is completely due to some other set of conditions H in the past and not due to any causal influence between A and B, and this explains why P(B|H) = P(B|AH).

In general, please don't just assume you know where I am going with a particular line of argument and then say dismissive things like "I don't have to respond to everything else in your rather verbose posts since this is the only point that is relevant to my issue". Consider the possibility that you may not actually understand everything about this issue, and therefore there may be points that you are missing. The alternative, I suppose, is that you have no doubt that you already know everything there is to know about the issue, and are already totally confident that your argument is correct and that Bell was wrong to write that equation, and are just here to pick a fight with Bell's defenders rather than to try to learn anything. If that's your attitude then this isn't really the forum for you--the IMPORTANT! Read before posting sticky in the relativity forum applies to the QM forum too:

This forum is meant as a place to discuss [quantum mechanics] and is for the benefit of those who wish to learn about or expand their understanding of said theory. It is not meant as a soapbox for those who wish to argue [quantum mechanics]'s validity, or advertise their own personal theories.

If on the other hand you have some intellectual humility, and are willing to consider that there's a good chance an argument that has been widely accepted by physicists for decades does not have any obvious holes that only you have been able to spot, then you should also consider that if you seem to see such a hole there is probably something basic missing from your understanding of the argument, and listen to the people who are trying to help guide you through the reasoning rather than immediately dismiss whatever they say if you don't spot the relevance right away. Up to you.

1) The definition of conditional independence is not based on additional information but on any information.

And what does "the definition of conditional independence" have to do with our discussion? I have already said explicitly that A and B are not conditionally independent, and this was true in my examples as well. A and B are causally independent, which is different.

Are you familiar with the phrase "correlation is not causation"? We might find in some study that two variables A and B, such as sugar consumption and heart disease, are correlated--they are not conditionally independent. It might nevertheless be true that this is not because sugar consumption has any causal influence on heart disease, but rather because high sugar consumption tends to be correlated with some other factor C, like a diet with too much salt, that does have a causal influence on heart disease. In this case we would have a conditional dependence between sugar and heart disease, but no causal influence of sugar consumption on heart disease.

Similarly, in the lotto card example, there is definitely a conditional dependence between the probability that Alice finds a cherry when she scratches box 1 of her card, and the probability that Bob finds a cherry when he scratches box 1 of his card--in fact, if the first is true, then we know the second is true with probability 1! But this isn't because Alice's scratching box 1 and finding a cherry had any causal influence on Bob's card. Rather it's because of an event in the past light cone of both these other two events, which exerted a causal influence on both--namely the source picking two lotto cards with an identical pattern of "hidden fruits" behind the respective boxes on each card, with the hidden fruits associated with each card staying constant as the cards travel from the source to the locations of Alice and Bob. This is directly analogous to the way a local-hidden variables theory tries to explain why two experimenters always find the same spin (or opposite spin, depending on the type of particle) when they measure each member of a pair of entangled pair along the same axis.

In the article I quoted to you, section 2.1 titled "Definitions", page 2, midway down the page, where independence is defined it says:

X [is independent of] Y if any information received about Y does not alter uncertainty about X;

I agree 100%, and have never said anything to suggest I was using a different definition of conditional independence. Again, A and B are not conditionally independent, only causally independent. If you are trying to find the probability of B (which could represent an event like 'Bob measured spin-up when measuring along the 180-degree axis), and you don't know anything besides the fact that it was a randomly-selected trial, then you will calculate some probability P(B). But if you are then asked "I want the probability of B on a trial where A also occurred" (where A could represent 'Alice measured spin-up on the 180-degree axis'), this is "information received about A" which does alter your uncertainty about B (now you are calculating P(B|A), which in a Bell type experiment will be different from P(B)), so B is not independent of A. It is nevertheless true that in a local hidden variables theory, if you had God-like knowledge of all the local hidden variables H associated with B, then learning A would give you no additional information about B, so P(B|H) = P(B|AH). But this would not change the fact that A and B are conditionally dependent, not conditionally independent.

2) If P(B|AH) is really equal to P(B|H) as you insinuate, then it shouldn't matter which equation is used. Both should result in the same inequalities right?

No, you're not making any sense. The fact that P(B|AH) is equal to P(B|H) is a specific piece of information about the physics of this problem which would not be true for any arbitrary problem where B and H were defined to mean something different physically. P(AB|H) = P(A|H) * P(B|AH) is a statistical identity which would hold mathematically regardless of the physical definitions of what the variables A, B, and H are supposed to mean; P(B|AH) = P(B|H) is an equation that we derive from specific physical considerations of the meanings of the symbols in Bell's proof. It shouldn't surprise you that in a physics proof, proving the conclusion should require making use of the specific physical assumptions of the proof, and that the conclusion can't be proved solely using general statistical identities which are true regardless of the meanings assigned to the variables!

Do you agree that it should be possible to derive Bell's inequalties from either equation?

No, for the reasons above.

Now following Bell's logic, try to derive the inequalities from P(AB|H) = P(A|H) * P(B|AH). It can not be done! Can you explain to me why? Just to be clear that you understand this point, let me rephrase it -- P(B|AH) = P(B|H) and P(B|AH) \neq P(B|H) can not both be true at the same time right?

Of course they can't be true at the same time, they would require different physical assumptions about the meaning of A, B, and H. If you don't understand that proofs in physics show that specific physical conclusions follow from specific physical assumptions, and that you can't necessarily prove the same physical conclusions if you start from completely different physical assumptions, then I don't know what else I can say. We can show that E=mc^2 can be proved if we start from some specific physical assumptions like a definition of energy and the fact that c is a constant velocity which is the same in all reference frames; do you think E=mc^2 could still be proved if we used the same mathematical identities but totally changed the physical definitions of E, m, and c? (or didn't make use of any equations which followed specifically from their physical definitions?)

 

[glengarry]

I think that it is simply incorrect to say that Bell was "really" responding to an actual program of Einstein's. It is more to the point to assume that Bell was, to a degree, putting words into Einstein's mouth by saying that Einstein was an advocate of a "more complete" version of QM, whereas Einstein was simply trying to prove that it is utterly fallacious to speak of QM as any kind of physical theory.

The whole idea of Einstein's advocacy of "local hidden variables," in my view, was just an attempt for certain up-and-comers to make names for themselves by way of "one upping" that most famous and venerable of all theoretical physicists.

In other words, since QM is itself just a theory of the necessarily statistical nature of all possible "real world" measurements, and since Einstein upheld that a "complete" physical theory must necessarily provide a spatio-temporal representation of all aspects of the experimental scenario in question (i.e. all measuring devices and things that are to be measured), then it is senseless to say that Bell showed some kind of flaw in the reasoning of EPR.

EPR, I think, was much more of a medidation on the logical foundations of any possible system of thought that can be called a "physical theory," rather than an attempt to show how an already existing theory can somehow be completed.

When the EPR paper finishes...

While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.

...I do not see any reason to assume that "such a theory" is necessarily identical with "a completed version of QM."

 

[billschnieder]

Nothing I have said implies that "I can see the moon when I am not looking at it", though it does imply that all variables associated with the moon have well-defined variables even when I'm not looking

By saying "observables have definite values even when I am not looking at it", you ARE effectively saying "I can see the moon even when I am not looking at it". Surely you must be aware that there are contextual observables which only have defined values in specific contexts of observation, and clearly it is naive to think those observables pre-exist the act of observation.
The EPR "elements of reality" only states that there exists objective ontological entities apart from the act of observation. EPR does not demand that those entities be passively revealed by observation, or even be definite. The only requirement is that the outcomes of observation ("observables"), be deterministically determined by those elements. EPR does not place any restriction on those entities other than that they be consistent with relativity. In fact, those entities could even be of dynamic nature, and in that case you would not talk of definite values will you.

Obviously this reduction is fine as long as P(B|H) = P(B|AH). And this is guaranteed to be true in a local realist world where A can't have any causal influence on B, and the only reason for correlations between A and B is some set of conditions H1 and H2 in the past light cones of A and B which influence their probabilities in correlated ways.

I do not believe this is accurate for reasons already explained as follows:
1) P(B|AH) = P(B|H) is NOT guaranteed to be true for a local realist world in which there is no causal influence between A and B. Although causal influence necessarily implies logical dependence, lack of causal influde is not sufficient to obtain lack of logical dependence. The example I in the first few posts points this out clearly, as does the article I quoted. It is OK to go from conditional independence to the equation P(B|AH) = P(B|H) due to conditional independence, but it is definitely not OK to go from causal independence to P(B|AH) = P(B|H). By the way I use the term conditional independence because that is exactly what the above equation means. P(B|AH) = P(B|H) means that B is conditionally independent of A with respect to H, or A and B are independent conditioned on H.

2) In Bell's treatment, Hidden variables are supposed to be responsible for the correlation between A and B. However, when Bell write the equation as follows:

P(AB|H) = P(A|H) * P(B|H)

This equation means that conditioned on H, there is no correlation between A and B. Now please take a moment and let this sink in because I have the feeling you have not understood this point. Note on the left hand side, we have a conditional probability, not a marginal probability, so it is irrelevant whether there is a correlation between A and B marginally or not. The equation clearly says conditioned on the hidden variables, there is no correlation between A and B.

And yet, those same hidden variables are supposed to be responsible for the correlation. This is the issue that concerns me. Giving examples in which A and B are marginally dependent but conditionally independent with respect to H as you have given, does not address the issue here at all. Instead it goes to show that in your examples, the correlation is definitely due to something other than the hidden variables! Do you understand this?

No, you're not making any sense. The fact that P(B|AH) is equal to P(B|H) is a specific piece of information about the physics of this problem which would not be true for any arbitrary problem where B and H were defined to mean something different physically.

3) If as you admit, the inequalities can not be derived from P(AB|H) = P(A|H) * P(B|AH), even though you claim the equation is equivalent to P(AB|H) = P(A|H) * P(B|H) in Bell's case, can you tell me why substituting one equation with another which is equivalent, should result in different inequalities, unless they are not really equivalent to start with?
I am not talking about a different problem. I am talking about the same problem in which you claim P(B|AH) = P(B|H) but at the same time claim that P(AB|H) = P(A|H) * P(B|AH) and P(AB|H) = P(A|H) * P(B|H) are not equivalent. Try it using the examples you gave and confirm that you get exactly the same numerical values for both equations, and then explain why Bell's inequalities can not be derived from P(AB|H) = P(A|H) * P(B|AH), using the same definitions for A, B and H.

 

[madhatter106]

Interesting thread, please consider that I do not have any formal education on this subject I find math fascinating in it's relations and patterns.

If I may ask a question is regards to the ratios used?

In the link from Drchinese, which was an interesting read. It is mentioned that the 25% chance is derived from the square of the cosine of 120*. Ok with that wouldn't you need to also take the inverse relationship?

To me the big picture is if your starting with 3 variables the ratios will always be 1/3. Now with the prediction that QM predicts 25% it is because it was derived from the cosine. Am I wrong in thinking that the cosine of one of the variables is simply the ratio against itself? . I know this is nothing new as it's basic trig, what I'm trying to figure is why is the QM ratio is the being taken against itself for A? and the std. is against the whole? Or did I miss something?

 

[madhatter106]

A bit more reading here, so it's related to Malus' Law right?
If the QM model is based off this and this would be the intensity after polarization, it is also the ratio of light percentage that is polarized already to the chosen theta.

So the QM percentage is working from the ratio between percentage of possible existing state to theta. Yet for some reason the follow thru to A,B,C, doesn't seem to be the correct ratio comparison, I can see the 1:3 option for each state but in the case of the polarized beam isn't that based upon the unknown ratio for each A, B, C, state?

 

[madhatter106]

Okay one more ? before I retire for the night. Is this an accurate view?

in the case of polarized light, to assign it a known state it would be pure and no longer an average. then the probability of any other state is forced by that known state.

The QM probability is based on the complete unknown of infinite theta between 0~360. the A,B,C table is already known values that have no internal ratios. I'm assuming the A,B,C values are based on pure polarization? if not is it an accurate comparison? if it's not pure wouldn't the set comparison then have to include the infinite range between 0~120 or the average ratio -0.5? thus simply taking it back to .25?

 

[DrChinese]

By saying "observables have definite values even when I am not looking at it", you ARE effectively saying "I can see the moon even when I am not looking at it". Surely you must be aware that there are contextual observables which only have defined values in specific contexts of observation, and clearly it is naive to think those observables pre-exist the act of observation.
The EPR "elements of reality" only states that there exists objective ontological entities apart from the act of observation. EPR does not demand that those entities be passively revealed by observation, or even be definite.

This is absolutely incorrect, and is somewhat shocking. It is seriously as if you have completely ignored everything important about EPR and Bell to focus on a few things out of context.

The EPR elements of reality have definite values, which can be predicted with 100% certainty prior to observation. That is an experimental fact and has never been in question from 1935 to now. This completely contradicts everything you are saying above.

The relevant question, as I have said in this thread previously, is whether these elements of reality exist simultaneously. Einstein said they do. Why won't you answer a simple question - do you agree with Einstein, yes or no? If you would, then it would clarify your position for the rest of us. If you are unable or unwilling to address this, then please say so and I will be on my way.

 

[DrChinese]

2) In Bell's treatment, Hidden variables are supposed to be responsible for the correlation between A and B. However, when Bell write the equation as follows:

P(AB|H) = P(A|H) * P(B|H)

This equation means that conditioned on H, there is no correlation between A and B. Now please take a moment and let this sink in because I have the feeling you have not understood this point. Note on the left hand side, we have a conditional probability, not a marginal probability, so it is irrelevant whether there is a correlation between A and B marginally or not. The equation clearly says conditioned on the hidden variables, there is no correlation between A and B.

I don't necessarily think your statistics is faulty, but I think you are mis-modeling the setup. Bell refers at various times to a and A, and b and B. Sometimes these are interchangeable, and sometimes they are not. I think the A and B should refer to the results of tests at measurement angles a and b. I think if you re-examine the setup, you will see that the above should include a and b as well.

As I have said many times, the reason you are going 'round in circles is because you are missing the point. Bell is trying to say: outcome B is independent of setting a, and vice versa. Write that statement however you like, and then proceed from there.

 

[JesseM]

By saying "observables have definite values even when I am not looking at it", you ARE effectively saying "I can see the moon even when I am not looking at it".

No, I'm not saying I can see it. I'm saying the hypothetical omniscient observer can see it.

Surely you must be aware that there are contextual observables which only have defined values in specific contexts of observation, and clearly it is naive to think those observables pre-exist the act of observation.

Sure, I never said that the omniscient observer might not see the values of various hidden variables change in response to interaction with a measuring device, just that the variables would have well-defined values at all times.

The EPR "elements of reality" only states that there exists objective ontological entities apart from the act of observation. EPR does not demand that those entities be passively revealed by observation, or even be definite.

Again, I didn't say they have to be passively revealed by observation. I'm not sure what you mean by "even be definite" though. What would an indefinite local hidden variable be like? Certainly we could imagine that certain variables which only take integer values when measured could have non-integer values between measurements, but they all must have some well-defined value.

In fact, those entities could even be of dynamic nature, and in that case you would not talk of definite values will you.

Sure I would. A variable that changes dynamically with time still has a definite value at any given point in time. So, we could imagine an omniscient observer who knows these values at each moment, even if we don't know them.

Look, the basic logic of Bell's proof is based on doing the following:
1. note the statistics seen on trials where both experimenters choose the same measurement angle (the simplest case would be if they always get identical results on these trials)
2. imagine what possible sets of local hidden variables might produce these statistics, if we (or a hypothetical omniscient observer) could see them
3. Show that for all possible sets of local hidden variables that give the right statistics on trials where the experimenters chose the same measurement angles, these hidden variables also make certain predictions about the statistics seen when the experimenters choose different measurement angles, namely that the statistics should satisfy some Bell inequalities
4. Show that quantum mechanics predicts that these same Bell inequalities are violated

The proof does not require that we actually know anything about the specifics of what local hidden variables are present in nature (so it doesn't require that we know the hidden variables associated with a particle or the moon when we aren't looking), it's making general statements about all possible configurations of hidden variables that are consistent with the observed statistics when both experimenters make the same measurement.

Do you disagree that this is the logic of the proof? If you are confident you understand the proof and disagree that this is the basic logic, can you explain where my summary is wrong, and what you think the logic is?

1) P(B|AH) = P(B|H) is NOT guaranteed to be true for a local realist world in which there is no causal influence between A and B. Although causal influence necessarily implies logical dependence, lack of causal influde is not sufficient to obtain lack of logical dependence.

Again, there is not a lack of logical dependence between A and B, since P(B|A) is different from P(B). The point is that in a local realist world, if there is a correlation (logical dependence) between two variables A and B that have a spacelike separation and therefore can't causally influence one another, there must be some cause(s) in the past light cones of A and B which predetermined this correlation.

Like I asked earlier, do you know what a "past light cone" is? If not it's really something you need to research in order to follow any discussion about causality in the context of relativity. If you do know what it means, then suppose we have some event B and we look at its past light cone, and we take the complete set of all facts about what happened in its past light cone (including facts about hidden variables) to be L. Do you disagree that if we know L, then whatever our estimate of the probability of B based on L is (i.e. P(B|L)), further information about some event A which lies outside the past or future light cone of B cannot alter our estimate of the probability of B (i.e. P(B|L) must be equal to P(B|LA)), assuming a universe with local realist laws?

If that wasn't true, then learning B would give us some information about the probability that A occurred, beyond whatever information we could have learned by looking at all the events L in the past light cone of B. Here's a proof--

Show: P(A|LB) not equal to P(A|L), given that P(B|L) not equal to P(B|LA).

Proof: P(A|LB) = P(ALB)/P(LB), by the formula for conditional probability.

P(ALB) can be rewritten as P(B|LA)*P(LA), and likewise P(LB) can be rewritten as P(B|L)*P(L). So, substituting into the above:

P(A|LB) = P(B|LA)*P(LA) / (P(B|L)*P(L))

The formula for conditional probability also tells us that P(A|L) = P(LA)/P(L). So substituting that into the above equation, we get:

P(A|LB) = P(A|L)*P(B|LA)/P(B|L)

From the above equation, the only way P(A|LB) can be equal to P(A|L) is if P(B|LA)/P(B|L) = 1. But we know P(B|LA) is not equal to P(B|L), so this cannot be the case; therefore, P(A|LB) is not equal to P(A|L).

If we can learn something about the probability an event A with spacelike separation from us (say, an event happening on Alpha Centauri right now in our frame) by observing some event B over here, and that's some new information beyond what we already could have known from all the prior events L in our past light cone (including past events which might also be in the past light cone of A and thus could have had a causal influence on it), then this is a form of FTL information transfer. Say A was the event of a particular alien horse on Alpha Centauri winning a race, and B was the event of a buzzer going off in my room; then I know that if I hear the buzzer go off, I should place a bet that when reports of the race reach Earth by radio transmission 4 years later, that particular horse will be the winner, and that will be a piece of information that no one who didn't have access to the buzzer could deduce by examining events in my past light cone. If you think this type of scenario is consistent with relativistic causality in a local realist universe, then I don't know what else to tell you, the idea that you can't gain any new information about an event A by observing an event B at a spacelike separation from it, if you already know all possible information about events in the past light cone of B (or just in a cross-section of the past light cone taken at some time after the last moment when the past light cones of A and B intersected, as I imagined in my analysis in posts 61/62 on the other thread, and is also the assumption used in this paper which discusses relativistic causality as it applies to Bell's analysis, which you should probably look through if my own arguments don't convince you) can basically be taken as the definition of relativistic causality. If you disagree, can you propose an alternate one that's stated in terms of what kind of information you can gain about distant events based only on local observations? Or do you think relativity and local realism place absolutely no limits on information you can gain about events outside your past light cone, allowing arbitrary forms of FTL communication?

The example I in the first few posts points this out clearly

The example you quoted doesn't contradict my point about past light cones. If you knew about everything in the past light cone of opening your envelope, including facts about which cards were inserted into the envelopes before they were sent and what happened to your envelope on its journey to you, then you would already know what color card you'd find before you opened it, and if your friend later knew what card was found in the other envelope and was watching a video of you opening your envelope (and the friend also had full knowledge of everything in the past light cone of your opening your envelope), then that additional knowledge of what happened when the second envelope was opened wouldn't change their prediction about what would happen when you opened yours.

It is OK to go from conditional independence to the equation P(B|AH) = P(B|H) due to conditional independence, but it is definitely not OK to go from causal independence to P(B|AH) = P(B|H).

If we're in a local realist universe respecting relativity, and H represents complete knowledge of every physical fact in the past light cone of B (or every fact in a cross-section of the past light cone taken at some time after the last moment the past light cones of A and B intersected), then yes it is OK. If you disagree, you don't understand relativistic causality.

By the way I use the term conditional independence because that is exactly what the above equation means. P(B|AH) = P(B|H) means that B is conditionally independent of A with respect to H, or A and B are independent conditioned on H.

OK, but when the paper you quoted to support your argument said:

X [is independent of] Y if any information received about Y does not alter uncertainty about X;

They weren't talking about X and Y being conditionally independent with respect to some other variable H, they were talking about X and Y being conditionally independent in the absolute sense that P(X and Y) = P(X)*P(Y). If they wanted to talk about conditional independence with respect to some other variable they would have written:

X is independent of Y with respect to H if any information received about Y does not alter uncertainty about X given H

2) In Bell's treatment, Hidden variables are supposed to be responsible for the correlation between A and B. However, when Bell write the equation as follows:

P(AB|H) = P(A|H) * P(B|H)

This equation means that conditioned on H, there is no correlation between A and B. Now please take a moment and let this sink in because I have the feeling you have not understood this point.

Yes, I understand perfectly well that there is no correlation on A and B conditioned on H, given how Bell's theorem defines H in terms of the complete set of information about all physical facts (including facts about hidden variables) in the cross-sections of the the past light cones of A and B, with the cross-sections taken after the last moment that their past light cones intersect. That was the central basis of my argument in posts 61 and 62 on the the other thread, and it's also discussed extensively in the online paper I linked to above.

Nevertheless, there is a correlation between A and B in absolute terms--if you do a large collection of trials and just look at incidences of A and B, the probability that B happens is different in the subset of trials where A happened than it is in the complete set of all trials (i.e. P(B|A) is different than P(B)).

Note on the left hand side, we have a conditional probability, not a marginal probability, so it is irrelevant whether there is a correlation between A and B marginally or not. The equation clearly says conditioned on the hidden variables, there is no correlation between A and B.

And yet, those same hidden variables are supposed to be responsible for the correlation. This is the issue that concerns me.

Huh? The hidden variables are responsible for the correlation which exists in absolute terms--you know, the correlation that is seen by actual experimenters doing experiments with entangled particles! Since hidden variables are by definition "hidden" to actual experimenters, we have no experimental data about whether there is a correlation between measurements conditioned on the hidden variables, and thus the idea that there's an absolute correlation but no correlation when conditioned on the hidden variables is perfectly consistent with all real-world observations. And if you understood the nature of relativistic causality you'd see that A and B cannot possibly be correlated when conditioned on H, if H represents the complete set of physical facts about past light cone cross-sections of A and B taken after the last moment when the past light cones of A and B intersected.

Giving examples in which A and B are marginally dependent but conditionally independent with respect to H as you have given, does not address the issue here at all. Instead it goes to show that in your examples, the correlation is definitely due to something other than the hidden variables! Do you understand this?

What? Suppose A is the event of me opening an envelope and finding a red card, and B is the event of you opening an envelope and finding a white card, with these two events happening at a spacelike separation. Let H1 represent the complete set of physical facts about everything in the past light cone of A at some time t after the last moment that the past light cones of A and B intersect, and H2 represent the complete set of physical facts about everything in the past light cone of B at the same time t. H can represent the combination of facts in H1 and H2. Now, H1 necessarily includes the fact that the envelope traveling towards me had a red card in it at that moment, and H2 includes the fact that the the envelope traveling towards you had a white card in it at that moment, so H includes both of these facts. Are you arguing that knowing H is not sufficient to completely determine the fact that we will find opposite colors when we open our respective envelopes and look at the cards? Isn't it true that if we know H on multiple trials like this and in each case H tells us the hidden card in the envelope on its way to me was the opposite color to the hidden card in the envelope on its way towards you, that is sufficient to determine that we will always find opposite colors on opening our envelopes (i.e. knowing H for each trial fully determines the correlation between our results on each trial), and that the probability you will find a white card is conditionally independent of the probability I will find a red card with respect to H? (i.e. if you already know what hidden cards were in the envelopes at some time t when they were on their path to us, your estimate of the probability that you found a white card is not altered by the knowledge that I found a red card when I actually opened my envelope)

3) If as you admit, the inequalities can not be derived from P(AB|H) = P(A|H) * P(B|AH), even though you claim the equation is equivalent to P(AB|H) = P(A|H) * P(B|H) in Bell's case, can you tell me why substituting one equation with another which is equivalent, should result in different inequalities, unless they are not really equivalent to start with?

This is a totally bizarre question. I mean, have you ever seen a proof of anything in physics before? You always start with some physical assumptions, then derive a series of equations, each one derived from previous ones using rules which follow either from your physical assumptions or from mathematical identities. Eventually you reach some final equation which is the conclusion you wanted to prove. Given the assumptions of the problem, each new equation is "equivalent" to a previous equation, or to some combination of previous equations. What you seem to be asking here is, "if all the equations in the proof are equivalent to previous ones, why can't I reach the final conclusion using only mathematical identities like P(AB|H) = P(A|H)P(B|AH), without being allowed to make substitutions that depend specifically on the physical assumptions of the problem like P(B|AH)=P(B|H)?" I don't really know how to respond except by saying "Uhhh, it doesn't work that way, in a physics proof you can't get from your starting equations to your final equation using only transformations of equations that are based on pure math, you have to make use of some actual, y'know, physics in some of your transformations. After all, no one said the final concluding equation was 'equivalent' to the starting equations in a purely mathematical sense, they are only equivalent given the specific physical assumptions you're using in the proof." Really, find me an example of any other proof/derivation in physics (say, a derivation of E=mc^2 from the more basic assumptions in relativity), and I'm sure there'd be some step where some physical assumption is used to transform equation(s) X into equation Y (i.e. X and Y are 'equivalent' given the physical assumptions of the problem), and yet equation X would not suffice to derive the final conclusion if we weren't allowed to make any further transformations based on physical assumptions.

 

[DrChinese]

A bit more reading here, so it's related to Malus' Law right?
If the QM model is based off this and this would be the intensity after polarization, it is also the ratio of light percentage that is polarized already to the chosen theta.

So the QM percentage is working from the ratio between percentage of possible existing state to theta. Yet for some reason the follow thru to A,B,C, doesn't seem to be the correct ratio comparison, I can see the 1:3 option for each state but in the case of the polarized beam isn't that based upon the unknown ratio for each A, B, C, state?

Malus does enter into it, yes. But it is just a bit tricky, as the same formula - cos^2(theta) - comes into play several different ways. Because of that, they look the same but may not be entirely.

 

[yoda jedi]

reality.
Exists objective ontological entities apart from the act of observation. not passively revealed by observation, or even be definite.

i agree.

reality does not need, counterfactual definiteness or indefiniteness, contextuality or non contextuality, determinism or indeterminism etc...
REALITY is:

"Being Qua Being"

 

[madhatter106]

Malus does enter into it, yes. But it is just a bit tricky, as the same formula - cos^2(theta) - comes into play several different ways. Because of that, they look the same but may not be entirely.

Ok,
I see that Bells' inequality would graph out as a straight line and the QM would be a sine wave due to Malus' cosine function. Is it wrong to assume then that X,Y,Z probabilities should also be functions of ratios so that the straight line would approach the sine of QM. When I read over the setup the X,Y,Z states are strict individually to a single plane and due to the polarization eq the cos^2 function that is used to deal with the spherical nature of luminosity.

the source light is entangled states of polarization that average a specific luminosity. the source starts in a spherical output and the measurement is in a single plane, then isn't the unpolarized luminosity that is absorbed pure light? it's and even ratio of all possible states and thus not visible. With that the source light is now part of the probability of one of those states, if the light source had any polarization it then is affecting the probability. pure unpolarized light wouldn't be visible, and using a "created" unpolarized source then it's already a known state. the EM field is then the balanced unpolarized state and the change in energy creates a polarization since theta moves from 0 to ? depending upon the way it was created.

To me then any visible light would need to be in a polarized plane, pure light then is not accessible to measurement as it's absorbed back into the EM field. If that is somehow right, then the probability of certain polarized states depends on how the energy is changed on the quantum level. This doesn't strike me as odd.

I'm I seeing it wrong?

 

[JesseM]

To me then any visible light would need to be in a polarized plane, pure light then is not accessible to measurement as it's absorbed back into the EM field. If that is somehow right, then the probability of certain polarized states depends on how the energy is changed on the quantum level. This doesn't strike me as odd.

Things are only odd if you want to explain the statistics using a local realist theory (like classical electromagnetism from which Malus' law is derived), then you find that the statistics predicted by QM are incompatible with the assumptions of local realism. Are you having trouble understanding why? If so you might take a look at the lotto card analogy I offered in post #18 (where different boxes on the card stand for different detector angles, and getting a cherry or lemon when a given box is scratched stands for getting spin-up or spin-down with a given detector angle), which starts in the paragraph beginning with "Suppose we have a machine that generates pairs of scratch lotto cards"...

 

[DrChinese]

Ok,
I see that Bells' inequality would graph out as a straight line and the QM would be a sine wave due to Malus' cosine function. Is it wrong to assume then that X,Y,Z probabilities should also be functions of ratios so that the straight line would approach the sine of QM. When I read over the setup the X,Y,Z states are strict individually to a single plane and due to the polarization eq the cos^2 function that is used to deal with the spherical nature of luminosity.

the source light is entangled states of polarization that average a specific luminosity. the source starts in a spherical output and the measurement is in a single plane, then isn't the unpolarized luminosity that is absorbed pure light? it's and even ratio of all possible states and thus not visible. With that the source light is now part of the probability of one of those states, if the light source had any polarization it then is affecting the probability. pure unpolarized light wouldn't be visible, and using a "created" unpolarized source then it's already a known state. the EM field is then the balanced unpolarized state and the change in energy creates a polarization since theta moves from 0 to ? depending upon the way it was created.

To me then any visible light would need to be in a polarized plane, pure light then is not accessible to measurement as it's absorbed back into the EM field. If that is somehow right, then the probability of certain polarized states depends on how the energy is changed on the quantum level. This doesn't strike me as odd.

I'm I seeing it wrong?

Somewhat. It is easier to follow some of the issues if you remember that there are several ways to determine the polarization of light. The "best" way (of course depends on the situation ) involves using a polarizing beam splitter, a PBS. You can orient this at any angle, and it will split the beam into an H component and a V component. Of course, that is relative to its axis. In this manner, you can see than the PBS does not itself change the light in some manner that you consider to be "active".

Not sure all of what you are asking, but you should definitely check out some of the traditional sources on optical physics. With entangled pairs, you will be looking at single photons but much of the same rules apply. But some do not.

 

[billschnieder]

No, I'm not saying I can see it. I'm saying the hypothetical omniscient observer can see it.

Doesn't matter, there are blind people on Earth who will never see the moon. "seeing the moon" is not a variable that belongs to the moon and has a definite outcome. Seeing the moon is contextual, for a blind person it does not exist at all. An omniscient being can not "see the moon" if they are not looking at it, neither can they know that "Tom can see the moon" if Tom is not looking at the moon. Simply being aware that the moon exists is a different observable from "seeing the moon". And the latter, does not have a definite outcome prior to observation. So I'm tired of trying to explain over and over that "realism" does NOT mean observables have definite values prior to observation, I have given you one clear example that does not.

Again, I didn't say they have to be passively revealed by observation. I'm not sure what you mean by "even be definite" though. What would an indefinite local hidden variable be like? Certainly we could imagine that certain variables which only take integer values when measured could have non-integer values between measurements, but they all must have some well-defined value.

Consider a very simplistic example, the color of the sun, does not have a definite value. Although based on the context, which includes sky conditions, time of day, type of goggles the person is wearing, the person will observe a specific color. You can definitely not say in this case that the sun has a definite color even when nobody is looking at it can you? However, you can say there are objective "elements of reality" which deterministically result in whatever the person observed. The latter is the EPR definition of realism, the former is definitely not. We will have to agree to disagree here.

Look, the basic logic of Bell's proof is based on doing the following:
...
Do you disagree that this is the logic of the proof? If you are confident you understand the proof and disagree that this is the basic logic, can you explain where my summary is wrong, and what you think the logic is?

I already explained the logic in the first post, what about that logic which started this thread is unclear or wrong to you? I believe it is clear from that post that if premise (1) fails, the whole logic fails with it. Premise (1) defines how local hidden variable theories consistent with QM and EPR should behave. That premise is my focus and that is why I keep trying to focus the discussion on that point because it is easy to get off-topic without addressing that central issue.

I do not see in your responses so far a convincing reason why we should use
P(AB|H) = P(A|H)*P(B|H) and not P(AB|H) = P(A|H)*P(B|AH)

That is not to say you have not given reasons, just that they are not convincing for reasons I have outlined already.

Again, there is not a lack of logical dependence between A and B, since P(B|A) is different from P(B). The point is that in a local realist world, if there is a correlation (logical dependence) between two variables A and B that have a spacelike separation and therefore can't causally influence one another, there must be some cause(s) in the past light cones of A and B which predetermined this correlation.

Like I asked earlier, do you know what a "past light cone" is?

I ignored that question , because it is an irrelevant distraction from the central issue, and it is so obvious I don't even understand why you bring it up.

If we can learn something about the probability an event A with spacelike separation from us (say, an event happening on Alpha Centauri right now in our frame) by observing some event B over here, and that's some new information beyond what we already could have known from all the prior events L in our past light cone (including past events which might also be in the past light cone of A and thus could have had a causal influence on it), then this is a form of FTL information transfer.

Herein lies the crux of the misunderstanding. In the situation being modeled by Bell, we are not calculating the probability of an event a Alice, we are calculating the probability of a joint event or coincidence between Alice and Bob. Again, note that it is not possible to determine that there is a coincidence unless you jointly consider both outcomes at Alice and Bob. This is the reason why you MUST still use
P(AB|H) = P(A|H)*P(B|AH)

Look at the left hand side, it says the probability of the joint event AB conditioned on H. You have probably heard it asked, "why can't we send information by FTL if it really possible?" The answer comes back to this equation. It is not possible to determine that a coincidence has occurred unless you have access to the results from each side. That is why you need the P(B|AH) because it ensures that the coincidences can be accounted for. However, as I have pointed out already. Therefore by writing the equation as
P(AB|H) = P(A|H)*P(B|H)
Bell has effectively restricted his model to only those situations in which there is no correlation conditioned on H. And in that case, to perform an experiment exactly according to what Bell modeled will require that the experimenters know exactly the nature of H, in order to effectively screen it out.

P(AB|H) = P(A|H)*P(B|H)
Clearly means that conditioned on H, there is no correlation between A and B. It is therefore impossible to for H to cause any correlations whatsoever with this equation. Now can you explain how it is possible for an experimenter to collect data consistent with this equation, without knowing the exact nature of H?

If we're in a local realist universe respecting relativity, and H represents complete knowledge of every physical fact in the past light cone of B (or every fact in a cross-section of the past light cone taken at some time after the last moment the past light cones of A and B intersected), then yes it is OK. If you disagree, you don't understand relativistic causality.

So long as you are trying to link an event about A and B such as coincidences, the simple act of trying to calculate a joint probability forces you to use P(AB|H) = P(A|H)*P(B|AH) and not P(AB|H) = P(A|H)*P(B|H).

It is only possible for A and B to be marginally correlated while at the same time uncorrelated conditioned on H, if H is NOT the cause of the correlation.

Yes, I understand perfectly well that there is no correlation on A and B conditioned on H, given how Bell's theorem defines H in terms of the complete set of information about all physical facts

Are you sure you understand that it? Can you explain how the hidden variables H are supposed to be responsible for the correlation between A and B, and yet conditioned on H there is no correlation between A and B. I do not see anything you have written so far in this thread or the other one answers this question.

Nevertheless, there is a correlation between A and B in absolute terms--if you do a large collection of trials and just look at incidences of A and B, the probability that B happens is different in the subset of trials where A happened than it is in the complete set of all trials (i.e. P(B|A) is different than P(B)).

Huh? The hidden variables are responsible for the correlation which exists in absolute terms--you know, the correlation that is seen by actual experimenters doing experiments with entangled particles! Since hidden variables are by definition "hidden" to actual experimenters, we have no experimental data about whether there is a correlation between measurements conditioned on the hidden variables, ...

In case you are not sure about the terminology, in probability theory, P(AB) is the joint marginal probability of A and B which is the probability of A and B regardless of whether anything else is true or not. P(AB|H) is the joint conditional probability of A and B conditioned on H, which is the probability of A and B given that H is true. There is no such thing as the absolute probability.

... and thus the idea that there's an absolute correlation but no correlation when conditioned on the hidden variables is perfectly consistent with all real-world observations.

I agree, there are cases in which a correlation may exist between A and B marginally, but will not exist when conditioned on another variable, like in some of the example you have give. However, the EPR case being modeled by Bell is not one of such, precisely because Bell is trying to introduce hidden variables H which should be responsible for the correlation between A and B. Do you understand this?

This is a totally bizarre question. I mean, have you ever seen a proof of anything in physics before? You always start with some physical assumptions, then derive a series of equations, each one derived from previous ones using rules which follow either from your physical assumptions or from mathematical identities.

The question is very clear. Let me put it to you in point form and you can give specific answers to which points you disagree with.

1) You say in the specific example treated by Bell, P(B|AH) = P(B|H). It is not me saying it. Do you disagree?
2) The above statement (1) implies that in the specific example treated by Bell, where the symbols A, B and H have identical meaning, P(B|AH) and P(B|H) are mathematical identities. Do you disagree?
3) The above statement (2) implies that in the specific example treated by Bell, where the symbols A, B and H have identical meaning, P(A|H)*P(B|AH) and P(A|H)*P(B|H) are mathematical identities. Do you disagree?
4) The above statement (3), implies that if using P(A|H)*P(B|H) results in one set of inequalities, the mathematically identical statement P(A|H)*P(B|AH) should result in the same set of inequalities where the symbols A, B and H have identical meaning. Do you disagree?
5) Given the above (1-4). Explain to me why it is not possible to obtain the same inequalities by using either P(A|H)*P(B|AH) or P(A|H)*P(B|H).

Eventually you reach some final equation which is the conclusion you wanted to prove. Given the assumptions of the problem, each new equation is "equivalent" to a previous equation, or to some combination of previous equations. What you seem to be asking here is, "if all the equations in the proof are equivalent to previous ones, why can't I reach the final conclusion using only mathematical identities like P(AB|H) = P(A|H)P(B|AH), without being allowed to make substitutions that depend specifically on the physical assumptions of the problem like P(B|AH)=P(B|H)?" I don't really know how to respond except by saying "Uhhh, it doesn't work that way

Bell himself says in his original paper that it is not difficult to reproduce the QM correlations using an equation like P(A|H)*P(B|AH). If two equations are mathematically equivalent, they should give the same numerical result, no?

 

[DrChinese]

So I'm tired of trying to explain over and over that "realism" does NOT mean observables have definite values prior to observation, I have given you one clear example that does not.

Consider a very simplistic example, the color of the sun, does not have a definite value. Although based on the context, which includes sky conditions, time of day, type of goggles the person is wearing, the person will observe a specific color. You can definitely not say in this case that the sun has a definite color even when nobody is looking at it can you? However, you can say there are objective "elements of reality" which deterministically result in whatever the person observed. The latter is the EPR definition of realism, the former is definitely not.

You haven't given such an example. And you keep ignoring the EPR definition: if the sun's color can be predicted with 100% certainty WITHOUT first observing it or disturbing it in any way... then there is an element of reality to its color.

If that is not your definition - and it sounds like it isn't because of the first paragraph: then really, why would anyone care? Why should people care about Schneider's* definition? Wouldn't we want to discuss something with a shared meaning here at a public forum? Bell is all about demonstrating that QM is inconsistent with observer independent elements of reality. That is the shared vision of Bell. So if you want to understand his reasoning, try looking there. It has nothing to do with statistical notation.

*A cryptic play on words

 

[billschnieder]

You haven't given such an example. And you keep ignoring the EPR definition: if the sun's color can be predicted with 100% certainty WITHOUT first observing it or disturbing it in any way... then there is an element of reality to its color

Consistent with EPR, I can predict the observed color in a specific context if I know everything about all the elements of reality that are part of the specific context. Yet I can not say the color of the sun exists prior to realization of the specific context. Therefore observables having definite values prior to observation is definitely not the EPR definition. The EPR definition, is "existence of elements of reality which deterministically result in the observables" such that it is possible to predict in advance, what would obtain given all the parameters of a specific context.

 

[JesseM]

Doesn't matter, there are blind people on Earth who will never see the moon.

But "hidden" means hidden to the entire community of human experimenters who can share information with one another. Of course in a local realist universe it might be that future experiments could allow us to observe formerly hidden variables, but it's not important to the proof one way or another.

"seeing the moon" is not a variable that belongs to the moon and has a definite outcome.

I don't know what "belongs to the moon" means. In a local hidden variables theory every basic variable should be associated with a particular point in spacetime. If you want to talk about a human "seeing the moon", that would presumably be shorthand for a certain combination of states of variables in the volume of spacetime where the human was making that observation (a macrostate corresponding to a particular 'microstate' involving the exact values of all the local variables in that region), including variables associated with the location of photons arriving at the human's location from the moon.

Seeing the moon is contextual, for a blind person it does not exist at all.

In a local realist theory there is an objective truth about which variables are associated with a given point in spacetime (and the values of those variables). This would include any variables associated with the region of spacetime occupied by the moon, and any associated with the region of spacetime occupied by a human. The variables associated with some humans might correspond to a state that we could label "observing the moon", and the variables associated with other humans might correspond to a state we could label "not observing the moon", but the variables themselves are all assumed to have an objective state that does not depend on whether anyone knows about them.

A "contextual" hidden variables theory is one where knowledge of H is not sufficient to predetermine what results the particle will give for any possible measurement of a quantum-mechanical variable like position or momentum, the conditions at the moment of measurement (like the exact state of the measuring device at the time of measurement) can also influence the outcome--see p. 39 here on google books, for example. This doesn't mean that all fundamental variables (hidden or not) associated with individual points in spacetime don't have definite values at all times, it just means that knowing all variables associated with points in the past light cone of the measurement at some time t does not uniquely determine the values of variables in the region of spacetime where the measurement is made (which tell you the outcome of the measurement).

An omniscient being can not "see the moon" if they are not looking at it,

As I said before, "omniscient being" is just a cute way of describing what it would be like if all hidden variables were known. You're taking the metaphor way too seriously if you imagine an omniscient being who only knows the values of hidden variables if he is "looking at" them; the only reason for invoking such a being is so we can talk about the objective states of all hidden variables that might influence observable experimental results.

neither can they know that "Tom can see the moon" if Tom is not looking at the moon.

In a local realist universe there is an objective truth about all local variables, and descriptions of macroscopic facts like "Tom seeing the moon" are just shorthand for certain combinations of local variables, much like macrostates vs. microstates in statistical mechanics. So there's an objective truth about whether "Tom sees the moon" is true or false in some particular region of spacetime containing Tom, and the omniscient observer knows whether it's true or false.

Simply being aware that the moon exists is a different observable from "seeing the moon". And the latter, does not have a definite outcome prior to observation. So I'm tired of trying to explain over and over that "realism" does NOT mean observables have definite values prior to observation, I have given you one clear example that does not.

I never once said that observables have definite values prior to observation. I said that all the fundamental physical variables, hidden or otherwise, have definite values at all times. But an "observable" is the outcome of a particular measurement, and it's certainly possible that fundamental physical variables associated with points on the particle's worldline don't uniquely determine this, that fundamental physical variables associated with the measuring device also influence the outcome.

Consider a very simplistic example, the color of the sun, does not have a definite value. Although based on the context, which includes sky conditions, time of day, type of goggles the person is wearing, the person will observe a specific color. You can definitely not say in this case that the sun has a definite color even when nobody is looking at it can you?

Not unless "color" is one of the fundamental physical variables associated with particular points in spacetime. But whatever these variables are, they do have objective values at every single point.

However, you can say there are objective "elements of reality" which deterministically result in whatever the person observed.

Yes, local "elements of reality" associated with particular points in spacetime, such that all macroscopic facts can be reduced to combinations of facts about these fundamental facts, are what I have been talking about all along.

Look, the basic logic of Bell's proof is based on doing the following:
1. note the statistics seen on trials where both experimenters choose the same measurement angle (the simplest case would be if they always get identical results on these trials)
2. imagine what possible sets of local hidden variables might produce these statistics, if we (or a hypothetical omniscient observer) could see them
3. Show that for all possible sets of local hidden variables that give the right statistics on trials where the experimenters chose the same measurement angles, these hidden variables also make certain predictions about the statistics seen when the experimenters choose different measurement angles, namely that the statistics should satisfy some Bell inequalities
4. Show that quantum mechanics predicts that these same Bell inequalities are violated

The proof does not require that we actually know anything about the specifics of what local hidden variables are present in nature (so it doesn't require that we know the hidden variables associated with a particle or the moon when we aren't looking), it's making general statements about all possible configurations of hidden variables that are consistent with the observed statistics when both experimenters make the same measurement.

Do you disagree that this is the logic of the proof?

I already explained the logic in the first post, what about that logic which started this thread is unclear or wrong to you?

Obviously any short description of "the logic" leaves some stuff out. I am not saying there was anything incorrect about your points 1-3 summarizing Bell's logic, I'm just saying it leaves out any discussion of how Bell arrived at his equation which you mention in 1. My point above is that he does this by imagining we (or a hypothetical omniscient observer) know what the hidden-variables state is, and considering all possible hidden-variables states that could lead to the observed statistics when the experimenters choose to make the same measurement. Do you think this is an incorrect characterization of what Bell is doing?

I do not see in your responses so far a convincing reason why we should use
P(AB|H) = P(A|H)*P(B|H) and not P(AB|H) = P(A|H)*P(B|AH)

That is not to say you have not given reasons, just that they are not convincing for reasons I have outlined already.

But you have not explained whether you disagree with my statements about complete knowledge of all physical variables in the past light cone of some measurement-event, or if so, why. Perhaps this is because you were misunderstanding me and thinking I was talking about "observable", even though I never suggested this. Now that you (hopefully) understand that I am talking about the fundamental local physical variables that must completely determine all macroscopic physical facts in a universe obeying local realist laws, I will re-ask the question, and if you are actually making a good-faith argument here rather than just trying to rhetorically discredit me, I hope you will give me a straight answer:

suppose we have some event B and we look at its past light cone, and we take the complete set of all facts about what happened in its past light cone (including facts about hidden variables) to be L. Do you disagree that if we know L, then whatever our estimate of the probability of B based on L is (i.e. P(B|L)), further information about some event A which lies outside the past or future light cone of B cannot alter our estimate of the probability of B (i.e. P(B|L) must be equal to P(B|LA)), assuming a universe with local realist laws?

Yes or no, agree or disagree that P(B|L) = P(B|LA) given the definition of L as encompassing all facts about fundamental physical variables (local 'elements of reality') in the past light cone of B?

If we can learn something about the probability an event A with spacelike separation from us (say, an event happening on Alpha Centauri right now in our frame) by observing some event B over here, and that's some new information beyond what we already could have known from all the prior events L in our past light cone (including past events which might also be in the past light cone of A and thus could have had a causal influence on it), then this is a form of FTL information transfer.

Herein lies the crux of the misunderstanding. In the situation being modeled by Bell, we are not calculating the probability of an event a Alice, we are calculating the probability of a joint event or coincidence between Alice and Bob.

We are calculating multiple things. In particular, when we go from the mathematical identity P(AB|H) = P(A|H)*P(B|AH) to the equation P(AB|H) = P(A|H)*P(B|H), we are doing the substitution P(B|AH) = P(B|H), and P(B|H) clearly refers to the conditional probability of the single event B at Bob's location given the hidden-variables state H, not to a joint event between Alice and Bob. And you may notice that the equation P(B|H) = P(B|AH) looks a lot like the equation in my question about the light cone above, P(B|L) = P(B|LA). And the section of my post you quoted above was simply about the fact that if P(B|L) was not equal to P(B|LA), this would imply FTL information transmission which is inconsistent with locality, which is trying to show why in a local realist universe it must be true that P(B|L) = P(B|LA). So again, I am hoping for a simple answer for you on the question of whether you agree that local realism does imply that equation must be true in the scenario I described (if you do, then we can go on to examine how well Bell's assumptions about the physical meaning of H resemble my assumptions about the physical meaning of L).

Again, note that it is not possible to determine that there is a coincidence unless you jointly consider both outcomes at Alice and Bob. This is the reason why you MUST still use
P(AB|H) = P(A|H)*P(B|AH)

Look at the left hand side, it says the probability of the joint event AB conditioned on H. You have probably heard it asked, "why can't we send information by FTL if it really possible?"

When I talked about FTL information transmission I wasn't talking about a joint event. Again, I was saying that if P(B|L) was different from P(B|LA), that would imply the possibility of FTL transmission, so we can be confident that in a local realist universe P(B|L) must be equal to P(B|LA). Again, I need an answer to this simple question before I can proceed with the argument.

The answer comes back to this equation. It is not possible to determine that a coincidence has occurred unless you have access to the results from each side. That is why you need the P(B|AH) because it ensures that the coincidences can be accounted for. However, as I have pointed out already. Therefore by writing the equation as
P(AB|H) = P(A|H)*P(B|H)
Bell has effectively restricted his model to only those situations in which there is no correlation conditioned on H. And in that case, to perform an experiment exactly according to what Bell modeled will require that the experimenters know exactly the nature of H, in order to effectively screen it out.

No, the experimenter doesn't know H, H represents variables that can be hidden from the experimenter. Again, see my statement above about the logic of the proof involving imagining we (or an omniscient observer) could know the state of all fundamental physical variables, and derive some statements that logically would have to be true in a local realist universe for any possible state of those fundamental variables which are consistent with the observed results when the experimenters choose the same angles. Please tell me whether you agree or disagree with this statement about Bell's logic in the proof.

Have to go now, will respond to the rest later...

 

[DrChinese]

Consistent with EPR, I can predict the observed color in a specific context if I know everything about all the elements of reality that are part of the specific context. Yet I can not say the color of the sun exists prior to realization of the specific context. Therefore observables having definite values prior to observation is definitely not the EPR definition. The EPR definition, is "existence of elements of reality which deterministically result in the observables" such that it is possible to predict in advance, what would obtain given all the parameters of a specific context.

I don't know if you are talking about semantics or substance.

According to EPR, if I can predict with certainty the result of a measurement of Bob without first observing or disturbing Bob, then there is an element of reality in Bob's observable. There need be no determinism involved regarding Bob, and the outcome could be completely random with no apparent cause. It only needs to be predictable in advance. I would say, by most standards, that means it has a specific value. I don't need to know anything about the context other than what it takes to predict, either. Now, how do you read EPR differently? My reading is about as exact as can be short of quoting EPR, and I assume you have read it. What is there to question here?

 

[DrChinese]

I never once said that observables have definite values prior to observation...

This was the EPR conclusion. And where Bell started from. So it is relevant.

 

[madhatter106]

Tell me if I got this right, from what I understand this isn't about the probability of permutations from an unknown group but the permutations of a 'known' group.

I take it like this, I have a ball that is 1/2 black 1/2 white and when split it will form 1 black and 1 white ball. according to classical physics you are always to going to end up with that arrangement measured or not, whereas QM states that until measured you could have 2 of the same color and that by measuring the one it automatically sets the other.

I'm a bit confused as to why this is problem? by having the 'envelope' of the experiment a known contained value it doesn't remove the probability that you'll need to measure at least one variable to know the other.

Is the confusing bit that there is a possibility that when measuring one value that it still does not mean that what you measured will determine the other, so that if let's say you measure a white ball and assume the other is black but upon receiving the information that the other is white as well it changes your previously measured value since you've become aware of the other state? and that change would have to alter the measurement in 'negative' values.

strange as it seems that makes sense to me, I though probably do not have a conventional view of photons, which allows me to accept that possibility however odd. I think in the classical form that assumption can be made but not from looking at the individual eq. but the whole picture to infer the possibility. I probably sound crazy, I'm just going on how my meager view of physics is to look at the entire range together instead of separately.

 

[DrChinese]

Tell me if I got this right, from what I understand this isn't about the probability of permutations from an unknown group but the permutations of a 'known' group.

I take it like this, I have a ball that is 1/2 black 1/2 white and when split it will form 1 black and 1 white ball. according to classical physics you are always to going to end up with that arrangement measured or not, whereas QM states that until measured you could have 2 of the same color and that by measuring the one it automatically sets the other.

If Alice is one color, Bob is always the expected color. That is not what is in question. And as long as you look at the issue that way - as EPR did - there is the possibility of a classical solution.

The issue has to do with when you look at shades, i.e. angles that are not 90 or 180 degrees apart. At various settings, 0/120/240 being a great one to study, things stop making sense. You must look at that example in detail (or one like it) to understand anything. Or go to the "DrChinese Easy Math" page (just google that) and it lays it out. You already follow the 1/3 bit, so the next part is to realize that is an upper limit and that QM (and experiment) give a value of 1/4. As a result, the EPR logic (elements of reality) is refuted.

 

[madhatter106]

If Alice is one color, Bob is always the expected color. That is not what is in question. And as long as you look at the issue that way - as EPR did - there is the possibility of a classical solution.

The issue has to do with when you look at shades, i.e. angles that are not 90 or 180 degrees apart. At various settings, 0/120/240 being a great one to study, things stop making sense. You must look at that example in detail (or one like it) to understand anything. Or go to the "DrChinese Easy Math" page (just google that) and it lays it out. You already follow the 1/3 bit, so the next part is to realize that is an upper limit and that QM (and experiment) give a value of 1/4. As a result, the EPR logic (elements of reality) is refuted.

I did went through most of it last night and thank you, it was a good read. When I see the example of 0/120/240 I instantly go back to trig and the periodic function of those values. the cos^2 value ratio in respect to theta is integral to the outcome.

does the question become why at ratios other than 1:1:sqrt2 do things stop making sense? graphing that ratio will always be a straight line by it's definition and the other a wave with periodic rates. so anything other than right angles will have anomalous results, esp as cosine or sine theta approaches infinity right?

So fundamentally the EM field has some hidden attribute that when the charge is not perpendicular there are strange results. this would be akin to saying that there is another 'variable' between b and e on the EM field that affects those states, yes?

 

[ThomasT]

... Bell's ansatz can not even represent the situation he is attempting to model to start with and the argument therefore fails.

I agree with this statement, but not necessarily for the reason you gave. (I don't fully understand it yet, having read through the thread quickly.)

Bell's formulation is sufficient to rule out a certain set of lhv theories, but it doesn't imply anything about Nature except that the disparity between Bell's ansatz (and thus Bell inequalities) and the experimental situations does make violations of Bell inequalities useful as indicators of entanglement.

Here's some observations:

1. The hidden variable, H in your notation, is irrelevant in the joint context. Coincidence rate, P(A,B), is solely a function of Theta, the angular difference of the polarizers.

You wrote (replying to another poster):
And yet, those same hidden variables are supposed to be responsible for the correlation. This is the issue that concerns me. Giving examples in which A and B are marginally dependent but conditionally independent with respect to H as you have given, does not address the issue here at all. Instead it goes to show that in your examples, the correlation is definitely due to something other than the hidden variables! Do you understand this?

I think I understand this.

2. The relevant hidden variable is the relationship (wrt some common motional property, usually spin and polarization because of the relative frequency of optical Bell tests) between the entangled entities. This relationship is the physical entanglement, and it is the deep cause of the observed correlations. This relationship, the entanglement, varies so slightly from pair to pair that it's , effectively, a constant, and can only be produced via quantum processes -- and this is accounted for in the QM treatment via an emission model applied to a particular preparation. In other words, QM assumes a local common cause for the entanglement.

3. Bell's locality condition reduces to P(A,B) = P(A)P(B) , which is the definition of statistical independence.

4. The observed statistical dependence is essentially due to three local (c-limited) processes: a) the production of entanglement via emission, b) the filtration of the entangled entities by a global measurement parameter, the angular difference of the crossed polarizers, and c) the data matching process, the final link in a local causal chain that ultimately produces the statistical dependence.

The requirements set forth by Bell for an lhv theory of entanglement seem to be at odds with the reality of the experimental situation(s) that produce the correlations that allow the conclusion that entanglement has been produced. So, it shouldn't be surprising that inequalities based on Bell's formulation are violated by Bell tests as well as the predictions of QM.

However, despite the problematic nature of lhv accounts of entanglement, the foundation of a c-limited, locally causal understanding of entanglement is at hand.

 

[DrChinese]

So fundamentally the EM field has some hidden attribute that when the charge is not perpendicular there are strange results. this would be akin to saying that there is another 'variable' between b and e on the EM field that affects those states, yes?

That would be an attempt to restore local realism, which just won't be possible. Recall that you can entangle particles at other levels as well, such as momentum/position or energy/time. Although it shows up as one thing for spin, you cannot explain it in the manner you mention.

Even for spin, if you look at it long enough, you realize that there is no solution to the mathematical problems. Bell's Inequality is violated because there is no local realistic solution possible.

 

[billschnieder]

JesseM:
Yes or no, agree or disagree that P(B|L) = P(B|LA) given the definition of L as encompassing all facts about fundamental physical variables (local 'elements of reality') in the past light cone of B?

The equation
P(AB|L) = P(A|L)P(B|L)
Is NOT NECESSARILY true given the definition of L as encompassing all facts about fundamental physical variables in the past light cones of A and B. Note the emphasis! So don't give me an example in which it is true and claim that it is always true. In case you are not aware, your claim that the above equation is ALWAYS true, for local elements of reality is what is well known as the prinicple of common cause (PCC). There are numerous treatments showing the problems with it and I don't need to go into that here. Look up Simpson's paradox and Bernstein's Paradox.

This is discussed in the Stanford Encyclopedia of Philosophy available online here: http://seop.leeds.ac.uk/entries/physics-Rpcc/

The simple reason the above is not always true is because it is not always possible to specify L such that it "screens off" the correlation as those paradoxes mentioned above indicate.

ThomasT:
The requirements set forth by Bell for an lhv theory of entanglement seem to be at odds with the reality of the experimental situation(s) that produce the correlations that allow the conclusion that entanglement has been produced.

I agree.
P(AB|L) = P(A|L)P(B|L)
Means that there is no longer any correlation between A and B conditioned on L, because it has been screened-off by L. Deriving Bell's inequalities using the above equation implies that the only data (A, B) capable of being compared with the inequalities must be uncorrelated. In order to collect such data will require the experimenters to know exactly the nature of the hidden variables in order to collect it. Therefore Bell's inequalities apply only to independent or uncorrelated data.

 

[JesseM]

JesseM:
Yes or no, agree or disagree that P(B|L) = P(B|LA) given the definition of L as encompassing all facts about fundamental physical variables (local 'elements of reality') in the past light cone of B?

The equation
P(AB|L) = P(A|L)P(B|L)
Is NOT NECESSARILY true given the definition of L as encompassing all facts about fundamental physical variables in the past light cones of A and B. Note the emphasis! So don't give me an example in which it is true and claim that it is always true.

Given local realism, yes it is. Do you disagree that if P(B|L) was not equal to P(B|LA), that would imply P(A|L) is not equal to P(A|BL), meaning that learning B gives us some additional information about what happened at A, beyond whatever information we could have learned from anything in the past light cone of B (proof in post #41)? Do you disagree that this is a type of FTL information transmission, since we're learning about an event outside our past lightcone that can't be derived from any information in our past light cone? If you do disagree that this is FTL information transmission, can you explain how you would define FTL information transmission which presumably must be forbidden in a local relativistic theory?

In case you are not aware, your claim that the above equation is ALWAYS true, for local elements of reality is what is well known as the prinicple of common cause (PCC). There are numerous treatments showing the problems with it and I don't need to go into that here. Look up Simpson's paradox and Bernstein's Paradox.

This is discussed in the Stanford Encyclopedia of Philosophy available online here: http://seop.leeds.ac.uk/entries/physics-Rpcc/

Can you find any sources that claim the principle of common cause would be violated in a relativistic universe with local realist laws, where the "cause" can stand for every possible local microscopic fact in the past light cone of one of the two events? Most of the problems discussed in the article you link to above arise from the fact that they are trying to find "causes" that are vague macro-descriptions which don't specify all the precise microscopic details which might influence the correlations. Note in the "conclusions" section where they say:

One should also not be interested in common cause principles which allow any conditions, no matter how microscopic, scattered and unnatural, to count as common causes. For, as we have seen, this would trivialize such principles in deterministic worlds, and would hide from view the remarkable fact that when one has a correlation among fairly natural localized quantities that are not related as cause and effect, almost always one can find a fairly natural, localized prior common cause that screens off the correlation. The explanation of this remarkable fact, which was suggested in the previous section, is that Reichenbach's common cause principle, and the causal Markov condition, must hold if the determinants, other than the causes, are independently distributed for each value of the causes. The fundamental assumptions of statistical mechanics imply that this independence will hold in a large class of cases given a judicious choice of quantities characterizing the causes and effects. In view of this, it is indeed more puzzling why common cause principles fail in cases like those described above, such as the coordinated flights of certain flocks of birds, equilibrium correlations, order arising out of chaos, etc. The answer is that in such cases the interactions between the parts of these systems are so complicated, and there are so many causes acting on the systems, that the only way one can get independence of further determinants is by specifying so many causes as to make this a practical impossibility. This, in any case, would amount to allowing just about any scattered and unnatural set of factors to count as common causes, thereby trivializing common cause principles.

So, the types of problems with the "principle of common cause" when we are restricted to these sorts of macroscopically describable causes don't apply to the "principle of common cause" when we are talking about every microscopic physical fact in the past light cone of a particular event. Something similar seems to be true with Simpson's paradox and Bernstein's paradox--for example, look at the last two pages of http://scistud.umkc.edu/psa98/papers/uffink.pdf (presented at http://scistud.umkc.edu/psa98/papers/abstracts.html#uffink), which says:

Also, in order to evade the Simpson paradox, it seems that one can save the principle by specifying that the cause C is a sufficient causal factor with respect to a class of events. It would be reasonable to take this class to include at least all events in the past of C, perhaps also those outside of C's causal future. However, this means one needs to introduce concepts from the space-time background in the principle.

...

Remarkably, a variant of the principle of the common cause taking explicity account of relativistic space-time has been around for a long time, although it is seldom discussed in the philosophical literature. It is Penrose and Percifal's (1962) principle of conditional independence.

These authors consider two spacelike separated bounded regions A and B in spacetime, and let C be any region which dissects the union of the past-light cones of A and B into two parts, one containing A and the other containing B. The P(A&B|C) = P(A|C)*P(B|C) where A, B, C are the histories of the regions A, B and C, i.e. complete specifications of all events in those regions.

For our discussion, the salient points in which this formulation differs rom other formulations are, first, in this version only non-local correlations are to be explained ... Thirdly, conditional independence is demanded only upon conditionalizing upon the entire history of a region C. This entails that the problems such as Simpson's paradox connected with incomplete specifications of the factors cannot appear.

The paper is titled The Principle of the Common Cause faces the Bernstein Paradox, so presumably when the author says "problems such as Simpson's paradox" this is meant to apply to Bernstein's paradox as well.

Also note that the Stanford Encyclopedia of Philosophy article actually discusses Penrose and Percifal's argument about picking a region C which divides the past light cones of A and B, in section 1.3, 'the law of conditional independence'. They state the conclusion of the "law of conditional independence", namely P(A&B|C) = P(A|C)*P(B|C), without attempting to dispute that it should hold in a classical relativistic universe. But they treat this "law of conditional independence" as a different claim from "Reichenbach's common cause principle" which is the main topic of the article (again seemingly because Reichenbach's principle is based on distinct macroscopically identifiable 'causes'), saying "This is a time asymmetric principle which is clearly closely related to Reichenbach's common cause principle and the causal Markov condition ... one cannot derive anything like Reichenbach's common cause principle or the causal Markov condition from the law of conditional independence, and one therefore would not inherit the richness of applications of these principles, especially the causal Markov condition, even if one were to accept the law of conditional independence."

So again, if you think that my version of the common cause principle could fail in a relativistic universe with local realist laws, even if the "common cause" is defined as the complete set of microscopic physical facts in one measurement's past light cone (or in a region of spacetime which divides the overlap of the two past light cones of each measurement as with the 'law of conditional independence' formulation by Penrose and Percifal above), you need to either find authors who specifically talk about such detailed specifications, or else actually make the argument yourself rather than trying to dismiss it with vague references to the literature.

 

[JesseM]

I never once said that observables have definite values prior to observation...

This was the EPR conclusion. And where Bell started from. So it is relevant.

It is relevant, yes. And once you realize why, in a local realist universe, it must be true that P(AB|H)=P(A|H)P(B|H) for the right choice of H, then you can also show that if there is a perfect correlation between measurement results when the experimenters choose the same detector angles, then in a local realist universe the only way to explain this is if H predetermines what measurement results they will get for all possible angles. But the general conclusion that P(AB|H)=P(A|H)P(B|H) for the right choice of H doesn't require us to start from that assumption, so if bill is unconvinced on this point it's best to try to show why the conclusion would be true even if we don't assume identical detector settings = identical results.