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Trying to Understand Bell's reasoning

by billschnieder
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billschnieder
#1
Apr30-10, 11:10 AM
P: 692
In deriving his inequalities, Bell starts his argument by stating the following:

a)- that according to QM, if Alice measures +1 then Bob must measure -1.
b)- if Alice and Bob are remote from each other such that Alices measurement does not influce Bob's measurement, then the results must be predetermined
c)- Since the QM does not predict individual results, it implies that the QM wavefunction is not complete and can be supplemented with "hidden variables" to obtain a more complete state.

He then goes on to calculate what might be expected if such hidden variables are introduced leading to his inequalities. From the above and what I understand so far, the following argument results

1) Bell's ansatz (equation 2 in his paper) correctly represent those local-causal hidden variables
2). Bell's ansatz necessarily lead to Bell's inequalities
3). Experiments violate Bell's inequalities
Conclusion: Therefore the real physical situation of the experiments is not Locally causal.

There is no doubt in my mind that statement (2) has been proven mathematically since I do not know of any mathematical errors in Bells derivation. Similarly, there is very little doubt in my mind that experiments have effectively demonstrated that Bell's inequalities are violated. I say little doubt because no loophole-free experiments have yet been performed but for the sake of this discussion we can assume that loopholes do not matter.

Now the issue I have difficulty understanding is statement (1) and it is fair to say if statement (1) fails, the argument fails with it.

Bell represents local reality by stating the joint probability of the outcome at A and B by as the product of the individual probabilities at each station, essentially the following

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

However, in probability theory,

P(AB|H) = P(A|H)P(B|AH) according to the chain rule, and in the case in which knowledge of A gives us no information about B,
P(B|AH) = P(B|H) and we can then reduce the the equation P(AB|H) = P(A|H)P(B|H).

In the situation Bell is trying to model, he says if Alice gets +1 then Bob MUST get -1. Therefore if we know that Alice already got +1, we therefore now know that Bob MUST have gotten -1. In other words, knowledge of A changes the hypothesis space for calculating the probability of B and P(B|AH) is not equal to P(B|H).

So it appears to me that Bell's ansatz can not even represent the situation he is attempting to model to start with and the argument therefore fails. What am I missing?
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DrChinese
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Apr30-10, 12:01 PM
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Quote Quote by billschnieder View Post
...However, in probability theory,

P(AB|H) = P(A|H)P(B|AH) according to the chain rule, and in the case in which knowledge of A gives us no information about B,
P(B|AH) = P(B|H) and we can then reduce the the equation P(AB|H) = P(A|H)P(B|H).

In the situation Bell is trying to model, he says if Alice gets +1 then Bob MUST get -1. Therefore if we know that Alice already got +1, we therefore now know that Bob MUST have gotten -1. In other words, knowledge of A changes the hypothesis space for calculating the probability of B and P(B|AH) is not equal to P(B|H).

So it appears to me that Bell's ansatz can not even represent the situation he is attempting to model to start with and the argument therefore fails. What am I missing?
There are multiple ways to approach Bell. Sometimes, it is easy to over-focus on the details and miss the big picture. You did summarize the EPR argument in the first part correct, and Bell refers to this as well.

Now, as to his main argument: it is the idea that the QM prediction is incompatible with the LR requirements. I think if you go back to this idea, you will quickly see that your objection is not meaningful. Clearly, the +1/-1 requirement comes from QM and an LR theory must respect this.
RUTA
#3
Apr30-10, 01:16 PM
P: 667
Quote Quote by billschnieder View Post
In deriving his inequalities, Bell starts his argument by stating the following:

a)- that according to QM, if Alice measures +1 then Bob must measure -1.
b)- if Alice and Bob are remote from each other such that Alices measurement does not influce Bob's measurement, then the results must be predetermined
c)- Since the QM does not predict individual results, it implies that the QM wavefunction is not complete and can be supplemented with "hidden variables" to obtain a more complete state.

He then goes on to calculate what might be expected if such hidden variables are introduced leading to his inequalities. From the above and what I understand so far, the following argument results

1) Bell's ansatz (equation 2 in his paper) correctly represent those local-causal hidden variables
2). Bell's ansatz necessarily lead to Bell's inequalities
3). Experiments violate Bell's inequalities
Conclusion: Therefore the real physical situation of the experiments is not Locally causal.

There is no doubt in my mind that statement (2) has been proven mathematically since I do not know of any mathematical errors in Bells derivation. Similarly, there is very little doubt in my mind that experiments have effectively demonstrated that Bell's inequalities are violated. I say little doubt because no loophole-free experiments have yet been performed but for the sake of this discussion we can assume that loopholes do not matter.

Now the issue I have difficulty understanding is statement (1) and it is fair to say if statement (1) fails, the argument fails with it.

Bell represents local reality by stating the joint probability of the outcome at A and B by as the product of the individual probabilities at each station, essentially the following

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

However, in probability theory,

P(AB|H) = P(A|H)P(B|AH) according to the chain rule, and in the case in which knowledge of A gives us no information about B,
P(B|AH) = P(B|H) and we can then reduce the the equation P(AB|H) = P(A|H)P(B|H).

In the situation Bell is trying to model, he says if Alice gets +1 then Bob MUST get -1. Therefore if we know that Alice already got +1, we therefore now know that Bob MUST have gotten -1. In other words, knowledge of A changes the hypothesis space for calculating the probability of B and P(B|AH) is not equal to P(B|H).

So it appears to me that Bell's ansatz can not even represent the situation he is attempting to model to start with and the argument therefore fails. What am I missing?
If there is some way by which information about A can reach B before the B measurement takes place, then we can't know whether B is in some sense "caused by" A or not. He wants his calculation to be applicable to space-like separated A and B, which means in some frames A occurs before B and in other frames B occurs before A. With no frame being the "right" frame, and assuming causes must precede their effects, he does not want his calculation to assume a causal relationship between A and B (this is the "local" part of "local hidden variables"). That means P(B|AH) = P(B|H) and P(A|BH) = P(A|H).

JesseM
#4
Apr30-10, 01:50 PM
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Trying to Understand Bell's reasoning

Quote Quote by billschnieder View Post
Now the issue I have difficulty understanding is statement (1) and it is fair to say if statement (1) fails, the argument fails with it.

Bell represents local reality by stating the joint probability of the outcome at A and B by as the product of the individual probabilities at each station, essentially the following

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

However, in probability theory,

P(AB|H) = P(A|H)P(B|AH) according to the chain rule, and in the case in which knowledge of A gives us no information about B,
P(B|AH) = P(B|H) and we can then reduce the the equation P(AB|H) = P(A|H)P(B|H).

In the situation Bell is trying to model, he says if Alice gets +1 then Bob MUST get -1. Therefore if we know that Alice already got +1, we therefore now know that Bob MUST have gotten -1. In other words, knowledge of A changes the hypothesis space for calculating the probability of B and P(B|AH) is not equal to P(B|H).
Although it's true that knowing A changes your estimate of the probability of B, the idea is that if you already know the full hidden variable state H, then knowing A would give you no additional information about the probability of B. According to a local hidden variables theory there is no direct causal influence between the measurement which gives outcome A and the measurement which gives outcome B because they are spacelike-separated, so to the extent that there is a correlation between the two results, it can only be because there was a correlation in the hidden variables H assigned to each particle at some point in the overlap region of the past light cones of A and B. So, P(B|AH) would indeed be equal to P(B|H), since A can only alter your estimate of the probability of B to the extent that A gives you indirect information about H.
billschnieder
#5
Apr30-10, 06:16 PM
P: 692
RUTA:
But that is the problem. Imagine the following situation

A box contains cards. One card is picked at random and sent in an envelop to Alice and another is sent to Bob. At their remote stations, they both open their envelops and reveal the color of their cards. It is found after numerous repeats of the experiment that their results are always anti-correlated. Whenenver Bob gets red alice gets white and vice versa.

The wise men come together to try and understand the puzzle. One group says, whenever Alice or Bob open their envelops, it instantaneously affects the other envelope so that the results are opposite. Another group says NO, the cards possess a shared hidden property H, right from the source and that is why they are correlated.

To try an figure out if the second group is right, the wise men decide to calculate the probability of the outcome of one such experiment and they write down similar to Bell the following equation

P(AB|H) = P(A|H) * P(B|H) where A = Alice gets red, B = Bob gets white and H = the hidden property, which God already knows but the wise men do not.

The wise men think the above equation is appropriate since if Hidden properties exist, and no instantaneous influences are happening, then A and B should not be dependent.

God observes and smiles because he already knows that the box always contains only two cards one red and one white, which information the wise men do not know, hence "hidden". Therefore the hidden information H = "There are only two cards in the box, one red and one white and the colors of the cards never changes". Obviously, H is completely locally causal.
Now lets look at the situation from God's perpective to see if the equation chosen by the wise men is correct.
According to everything God knows

P(A|H) = 0.5
P(B|H) = 0.5
Therefore the result obtained by the wise men will be P(AB|H) = 0.5 * 0.5 = 0.25!

But this is wrong. There are only two possible outcomes in this experiment,
(A:red, B:white) or (A:white, B:red) Therefore the probability P(AB|H) should be 0.5!

This can be verified using the chain rule of probability theory
P(AB|H) = P(A|H) * P(B|AH)
and since P(B|AH) = 1, (if Alice got red, Bob certainly got white)
therefore P(AB|H) = 0.5 * 1 = 0.5

So the equation chosen by the wisemen is not the correct one even for a situation in which there is not physical influence between A and B. This is my main issue

JesseM:
Note in this case also that P(B|H) is not equal to P(B|AH) even though the situation is completely locally causal. So I don't understand what justified that assumption.
DrChinese
#6
Apr30-10, 07:04 PM
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Quote Quote by billschnieder View Post
RUTA:
But that is the problem. Imagine the following situation

A box contains cards. One card is picked at random and sent in an envelop to Alice and another is sent to Bob. At their remote stations, they both open their envelops and reveal the color of their cards. It is found after numerous repeats of the experiment that their results are always anti-correlated. Whenenver Bob gets red alice gets white and vice versa.

The wise men come together to try and understand the puzzle...
Have you ever heard of Bertlmann's socks? Pretty much the same example.

You are waaaaay off in your thinking. Your example is no surprise as long as you pick the angles to discuss. But if I pick them (since I am familiar with Bell), it doesn't work.

Simply provide me with a dataset of values for hidden variable values at these angles:

0, 120, 240 degrees.

The dataset should fulfill the requirement that cos^2(theta) is the same as the quantum mechanical prediction. That would be .25 for any pair. Your dataset cannot come closer than .33.

I can ask this, and you know cannot provide such a dataset. Precisely because of Bell (and Mermin!)
billschnieder
#7
Apr30-10, 07:49 PM
P: 692
Dr Chinese wrote:
You are waaaaay off in your thinking.


I did not understand in what way you say I am waaay off. The example I gave does not have any angles. Maybe you did not understand the issue I am struggling with. I am entirely focused on equation (2) of Bell's original paper, also equation (10) of Bell's Bertlmann's socks paper. I am trying to understand his justification for using this equation. That should be a legitimate query no? From reading both articles, it appears to me the only justification given is the assumption that if events at A do not instantaneously influence events at B, then that is the equation we must use. Is that what your understanding is as well?

The issue for me then is that using an example (the one I gave), which I know to be completely locally causal, and events at A do not instantaneously influence events at B, Bell's equation does not work. Therefore as far as I understand the argument, the equation is not justified. I am hoping that someone will help me by pointing out why Bell used P(AB|H) = P(A|H)*P(B|H) (which does not work in locally causal example I gave) instead of P(AB|H) = P(A|H)*P(B|AH), which works in all cases.

I do not see how angles come into the picture.
DrChinese
#8
Apr30-10, 08:27 PM
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Quote Quote by billschnieder View Post
I do not see how angles come into the picture.
That would be why you are off. Not trying to be cutsie, just trying to point you in the right direction.

Looking at [2] is, in my opinion, a waste of time. The meat is right after [14]. There, the 3rd setting is introduced... c.

So imagine you have a classical particle. Its attributes, let's call then A B and C, are well defined at all times. Not so a quantum particle! It is defined by the HUP (among other things) and does not possess well defined values. I am sure you are following me to this point.

Now, Bell discovered that the relationships between A, B and C - cos^2 for photons, cos for electrons - were internally inconsistent. I.e. that they could not have their values AND follow the predictions of QM. This requires no understanding of probability to accept. You simply cannot construct a dataset of a group of SINGLE photons that follows QM.

The entanglement is simply a way to express this in an experimental context. If Alice is a clone of Bob - as EPR imagined (although they may be either symmetric or anti-symmetric), then it is clear that their relationships can be tested and compared. Alice and Bob will follow the predictions of QM. Of course, QM does not postulate that there is a third angle Chris which "could" have been checked as Alice and Bob were.

The best thing you could do for yourself is to work out the 0/120/240 example I gave. Or go to my website.

http://drchinese.com/David/Bell_Theorem_Easy_Math.htm

If you don't follow this, you really can't go any further and you will simply spin in circles. Don't look at [2].
billschnieder
#9
Apr30-10, 08:47 PM
P: 692
Dr Chinese:
[4] is derived from [2], if [2] is not correct how can [4] possibly be correct then? I don't think you have understood my concern. Another way of putting it is

if the equation can not correctly represent the locally causal case in which Alice and Bob use exactly the same angles, why is it a surprise that it fails when more than one angle is introduced?

My main issue is with equation [2] so I hope somebody will help to explain the reasoning behind it. Telling me to stop looking at [2] doesn't help me at all because I can't get past this apparent problem with it.

In school I never liked teachers who told me to just accept what they said even though it did not make sense. I'm hoping someone will explain why equation [2] makes sense.

Thanks for your efforts though.
JesseM
#10
Apr30-10, 09:17 PM
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Quote Quote by billschnieder View Post
RUTA:
But that is the problem. Imagine the following situation

A box contains cards. One card is picked at random and sent in an envelop to Alice and another is sent to Bob. At their remote stations, they both open their envelops and reveal the color of their cards. It is found after numerous repeats of the experiment that their results are always anti-correlated. Whenenver Bob gets red alice gets white and vice versa.

The wise men come together to try and understand the puzzle. One group says, whenever Alice or Bob open their envelops, it instantaneously affects the other envelope so that the results are opposite. Another group says NO, the cards possess a shared hidden property H, right from the source and that is why they are correlated.

To try an figure out if the second group is right, the wise men decide to calculate the probability of the outcome of one such experiment and they write down similar to Bell the following equation

P(AB|H) = P(A|H) * P(B|H) where A = Alice gets red, B = Bob gets white and H = the hidden property, which God already knows but the wise men do not.

The wise men think the above equation is appropriate since if Hidden properties exist, and no instantaneous influences are happening, then A and B should not be dependent.

God observes and smiles because he already knows that the box always contains only two cards one red and one white, which information the wise men do not know, hence "hidden". Therefore the hidden information H = "There are only two cards in the box, one red and one white and the colors of the cards never changes". Obviously, H is completely locally causal.
Now lets look at the situation from God's perpective to see if the equation chosen by the wise men is correct.
According to everything God knows

P(A|H) = 0.5
P(B|H) = 0.5
Therefore the result obtained by the wise men will be P(AB|H) = 0.5 * 0.5 = 0.25!

But this is wrong. There are only two possible outcomes in this experiment,
(A:red, B:white) or (A:white, B:red) Therefore the probability P(AB|H) should be 0.5!

This can be verified using the chain rule of probability theory
P(AB|H) = P(A|H) * P(B|AH)
and since P(B|AH) = 1, (if Alice got red, Bob certainly got white)
therefore P(AB|H) = 0.5 * 1 = 0.5

So the equation chosen by the wisemen is not the correct one even for a situation in which there is not physical influence between A and B. This is my main issue

JesseM:
Note in this case also that P(B|H) is not equal to P(B|AH) even though the situation is completely locally causal. So I don't understand what justified that assumption.
That's because your version of H is too vague and doesn't actually specify whether the red card was the one that was picked to send to Alice and the white card was the one that was picked to send to Bob, or vice versa. If you completely specified the hidden properties of the envelope that was sent to Bob--namely "the person who picked the cards from the box put the red card in Bob's envelope, and the envelope continued to have that hidden card on its journey to Bob"--then in that case it would be true that P(B|H)=P(B|AH). In fact the only locally realistic way to explain how they always get opposite colors if the events of their opening the envelopes are spacelike-separated is to assume that at some point in the overlap region of their past light cones, there was a process that assigned the envelopes properties such that the colors that would be revealed when they were opened were already predetermined at that point, and predetermined in such a way that the predetermined color of Alice's envelope would always be opposite to the predetermined color of Bob's. If you completely specify these hidden predetermined properties H of each envelope then it's going to be true that P(B|H)=1 or 0, and that knowing A gives you no additional information about the probability of B.

If you still aren't convinced, I gave a more detailed version of this argument in posts #61 and #62 here.
DrChinese
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Apr30-10, 09:34 PM
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Quote Quote by billschnieder View Post
Dr Chinese:
[4] is derived from [2], if [2] is not correct how can [4] possibly be correct then? I don't think you have understood my concern. Another way of putting it is

if the equation can not correctly represent the locally causal case in which Alice and Bob use exactly the same angles, why is it a surprise that it fails when more than one angle is introduced?

My main issue is with equation [2] so I hope somebody will help to explain the reasoning behind it. Telling me to stop looking at [2] doesn't help me at all because I can't get past this apparent problem with it.

In school I never liked teachers who told me to just accept what they said even though it did not make sense. I'm hoping someone will explain why equation [2] makes sense.

Thanks for your efforts though.
You are not in school and I am not your teacher. I am trying to get you to stop looking at the trees and look at the forest. The big picture has nothing to do with [2] and if you keep looking at it, you miss the bigger one.

Ask yourself this: what does it mean for a theory to be realistic? It means that observables have definite values independent of observation. See EPR's definition of "elements of reality".

So if a theory is realistic, and it makes the same predictions as QM, then what does that MEAN? It means that the cos^2 relationship holds AND it holds for all angles - not just those actually observed.

Can you construct a realistic theory with these attributes? No, you cannot. How do I know? That is what Bell tells us. Now, regardless of whether [2] is right or wrong, or [4] is right or wrong, I still know this. When Bell wrote, there were only a few who followed this. They didn't overly dissect the details because they saw the point: QM and realism are not compatible. Now of course there is a way out, through the existence of non-local signaling between Alice and Bob. And a few others, although that simply pulls us further away from the objective.

So I hope you liked the teachers who told you to think outside of the box. Because you are stuck in the box right now. If you don't understand the math from my page, you won't get out either. Take a few minutes to convince yourself that you understand this. Then go re-read Bell. Bell is a road map. Don't take it literally, as everyone reformulates Bell to express it in a way that makes sense to them. I don't consider my formulation different at all. But it is expressed differently.
DrChinese
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Apr30-10, 09:36 PM
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Quote Quote by billschnieder View Post
Thanks for your efforts though.
By the way, welcome to PhysicsForums!
billschnieder
#13
May1-10, 01:24 PM
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Quote Quote by JesseM View Post
That's because your version of H is too vague and doesn't actually specify whether the red card was the one that was picked to send to Alice and the white card was the one that was picked to send to Bob, or vice versa. If you completely specified the hidden properties of the envelope that was sent to Bob--namely "the person who picked the cards from the box put the red card in Bob's envelope, and the envelope continued to have that hidden card on its journey to Bob"--then in that case it would be true that P(B|H)=P(B|AH).
I am not sure I agree with this. In probability theory, when we write P(A|H), we are assuming that we know H but not A. If we knew A already (a certainty), there is no point calculating a probability is there? In this case when Bell obtains cos(theta), his equation will only be valid for two angles when cos(theta)= 0 or 1! How then can this equation apply to other angles? Therefore I don't think that is the reasoning here.

Furthermore, the variables are hidden from the perspective of the wise men, it is not God trying to calculate the probabilities but the wise men, because they do not have all the information. We are only looking from God's perspective to verify that the equation the wise men choose to use corresponds to the factual situation and in the example I gave it does not appear to.

In fact the only locally realistic way to explain how they always get opposite colors if the events of their opening the envelopes are spacelike-separated is to assume that at some point in the overlap region of their past light cones, there was a process that assigned the envelopes properties such that the colors that would be revealed when they were opened were already predetermined at that point, and predetermined in such a way that the predetermined color of Alice's envelope would always be opposite to the predetermined color of Bob's.
But in my example, with the information known by God, this is the case, every box only contains two cards, one red and one white and in each iteration of the experiment one of the cards is sent to Bob and the other to Alice. The equation P(AB|H) = P(A|H)P(A|BH) always works but P(AB|H) = P(A|H)P(A|H) works only in very limited case in which H is no longer hidden and calculating probabilities is pointless.

Note that if we assume that your reasoning is correct, the full equation P(AB|H) = P(A|H)P(A|BH) still works. But in Bell's case, using the full chain rule does not work. So there must be another justification for insisting on P(AB|H) = P(A|H)P(A|H) other than the one you have given.

If you completely specify these hidden predetermined properties H of each envelope then it's going to be true that P(B|H)=1 or 0, and that knowing A gives you no additional information about the probability of B.
But that is not quite true. The justification for reducing P(A|BH) to P(A|H) is not based on whether B gives [i[additional[/i] information but on whether B gives any information. (see Conditional Independence in Statistical Theory, J.R Statist. Soc B, 1979 41, No. 1, pp. 1-31)

if A and B are equal, numerically P(A|BH) = P(A|H), but you could not say in this situation that since B gives you no additional information to H, therefore A and B are independent. They clearly are dependent. Therefore, although it is correct to reduce P(A|H)P(B|AH) to P(A|H)P(B|H) as a result of conditional independence between A and B, it is not correct to go from the fact that their numerical values are the same, to say that they are conditionally independent. X implies Y does not necessarily mean Y implies X.

If you still aren't convinced, I gave a more detailed version of this argument in posts #61 and #62 here.
You re saying pretty much the same thing there, that A gives no additional information to H. But that is not the meaning of conditional independence. Conditional independence means that A gives us no information whatsoever about B.
billschnieder
#14
May1-10, 01:40 PM
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Quote Quote by DrChinese View Post
Ask yourself this: what does it mean for a theory to be realistic? It means that observables have definite values independent of observation. See EPR's definition of "elements of reality".
I am not sure this is true. Unless you have a different meaning for "observable". I can certainly think of many real and locally causal situations in which the outcome of an observation is caused by both the pre-existing properties of an object and the act of observation.

For example, iIf I send Alice a lotto card with two boxes and she scratches one of them to observe an outcome, I certainly can not say the outcome of scratching the card existed independent of Alice's scratching. Won't it be naive to say "The result obtained by Alice through scratching", existed even before Alice ever scratched the card? So I believe the EPR "elements of reality" can pre-exist (eg, contents of boxes on lotto cards) even though the observable -- "the result obtained by Alice through scratching" does not exist until Alice actually scratches. Your definition of realism seems to omit obviously realistic and locally causal situations like this one.
RUTA
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May1-10, 01:45 PM
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Quote Quote by billschnieder View Post
I am not sure this is true. Unless you have a different meaning for "observable". I can certainly think of many real and locally causal situations in which the outcome of an observation is caused by both the pre-existing properties of an object and the act of observation.

For example, iIf I send Alice a lotto card with two boxes and she scratches one of them to observe an outcome, I certainly can not say the outcome of scratching the card existed independent of Alice's scratching. Won't it be naive to say "The result obtained by Alice through scratching", existed even before Alice ever scratched the card? So I believe the EPR "elements of reality" can pre-exist (eg, contents of boxes on lotto cards) even though the observable -- "the result obtained by Alice through scratching" does not exist until Alice actually scratches. Your definition of realism seems to omit obviously realistic and locally causal situations like this one.
Your lotto card example is precisely what Mermin calls an "instruction set," and is an example of DrC's "realism." See "Quantum mysteries revisited," N. David Mermin, Am. J. Phys. 58, #8, August 1990, pp 731-734.
DrChinese
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May1-10, 03:01 PM
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Quote Quote by billschnieder View Post
I am not sure this is true. Unless you have a different meaning for "observable". I can certainly think of many real and locally causal situations in which the outcome of an observation is caused by both the pre-existing properties of an object and the act of observation.

For example, iIf I send Alice a lotto card with two boxes and she scratches one of them to observe an outcome, I certainly can not say the outcome of scratching the card existed independent of Alice's scratching. Won't it be naive to say "The result obtained by Alice through scratching", existed even before Alice ever scratched the card? So I believe the EPR "elements of reality" can pre-exist (eg, contents of boxes on lotto cards) even though the observable -- "the result obtained by Alice through scratching" does not exist until Alice actually scratches. Your definition of realism seems to omit obviously realistic and locally causal situations like this one.
EPR's (Einstein's) definition of an element of reality was that the outcome of experiment could be predicted in advance without disturbing the particle. This was also the definition intended by Bell. In the example: if 2 cards are prepared identically, and Alice scratches hers, then we can predict Bob's with certainty. Therefore, this is an element of reality.

On the other hand: your definition - in which the "the result obtained by Alice through scratching" does not exist until Alice actually scratches - is ambiguous as to whether this result is also an "element of reality" per EPR. If it has some pre-existing value, which is merely revealed by the process of scratch (observing), then it is an element of reality.

Bell's work concerns these elements of reality. The EPR is generally considered a solid definition, although you can imagine that some will in fact define it differently. Obviously, with a sufficiently different definition than the EPR/Bell one, the Bell result might not hold. I think you will eventually conclude that, in fact, the EPR definition of reality is sufficiently close to your own to use it. After all, it would be difficult to say that there is NOT reality to something that can be predicted in advance.

1. Now, you can see that with a pair of entangled particles (let's use the symmetric case where the spins are the same), Alice can correctly predict a result for Bob at any chosen angle. So let's use the angles 0, 120 and 240. Via experiment, it can be shown that there is an element of reality for these angles according to EPR/Bell. But the question for the realist is: are they simultaneously existing? Einstein felt the answer must be yes, the moon exists even when we are not looking at it.


2. I use these specific angles because any pair of these will have a difference of 120 degrees. If you accept Einstein's conclusion, and then try to model a dataset of photons with values for a polarization at these angles, you cannot get them to have correlations for any adjacent pairs less than 33%. The 3 adjacent pairs are: 0/120, 120/240, 0/240. Try for yourself by modeling the simultaneous values:

-----0-----120-----240
01 + - - (1/3 - do you see how I get this?)
02 - + + (1/3 - do you see how I get this?)
03 + + + (3/3 - do you see how I get this?)
etc. -

You can put any values in you like. Try to have the average as low as possible.


3. QM predicts that in an actual experiment with photons, you would get 25% exactly. That is because the cos^2(120) is 25%. So QM is incompatible with simultaneous reality of photon polarization observables.


I hope you can see from the above there are lots of ways to skin the cat. Or the Bell.
billschnieder
#17
May1-10, 10:16 PM
P: 692
Quote Quote by DrChinese View Post
EPR's (Einstein's) definition of an element of reality was that the outcome of experiment could be predicted in advance without disturbing the particle.
Of course, IF we know the contents of each square and we know the square Alice is going to pick, then we can predict with certainty what Alice will observe without disturbing the card in any way. But the contents of the boxes which are already existing, are elements of reality. However, "Alice scratched box 1" is not an element of reality until Alice actually scratches box 1. "Tomorrow at 2pm Alice will scratch box 1" is also an element of reality if in fact that is the box Alice will scratch even if she has not scratched any box yet. It is not ambiguous at all. This is consistent with the EPR definition of "elements of reality" and clearly, it does not mean that outcomes pre-exist measurement. The reason it is true today that "Tomorrow at 2pm Alice will scratch box 2"is because tomorrow at 2pm Alice will in fact scratch box 1. It would be wrong to conclude that the element of reality "Tomorrow at 2pm Alice will scratch box 1"is what caused Alice to scratch box 1. It is the way the world was, is, and will be that accounts for statements being true, not the other way round.

This was also the definition intended by Bell. In the example: if 2 cards are prepared identically, and Alice scratches hers, then we can predict Bob's with certainty. Therefore, this is an element of reality.
Only if we also know which box Bob is going to scratch. But you get the point I am making, that even if Alice's choice does not instantaneously influence Bob's choice, the outcomes can still be correlated, therefore the correct equation here should have been P(AB|H) = P(A|H)*P(B|AH) not P(AB|H) = P(A|H)*P(B|H) as Bell uses.

In other words, if A and B are correllated, and we are trying to find out if those correlations are caused by H, we can not assume that conditioned on H, A and B are not correlated. By stating the equation as P(AB|H) = P(A|H)*P(B|H), Bell is effectively saying, conditioned on H, A and B are not correlated. Therefore it is not possible to reproduce those correlations using the equation Bell chose and violation of his inequalities isn't very surprising, to me at least.

In yet other words, if conditional independence implies that P(B|AH) = P(B|H) and Bell is claiming conditional independence in this case, then we should obtain the same answer by using either P(AB|H) = P(A|H)*P(B|AH) OR P(AB|H) = P(A|H)*P(B|H).

But Bell's inequalities can only be derived for P(AB|H) = P(A|H)*P(B|H). Using P(A|H)*P(B|AH) gives different inequalities. This tells me that Bell's assumption of conditional independence appears not to be correct as I have explained above.

Einstein felt the answer must be yes, the moon exists even when we are not looking at it.
What I have explained is that, it is not reasonable to translate this statement to "Einstein can see the moon, even if he is not looking at it." If "seeing the moon" is an outcome of an experiment, you can not claim that realism means that "seeing the moon" pre-existed the act of actually "seeing".

Once I understand Bell's justification for "skinning the cat" the way he did in the original paper, I will move to the others. But for now I am only interested in understand his original paper. Thanks for the links to your website. I will check it out.
JesseM
#18
May2-10, 12:31 AM
Sci Advisor
P: 8,470
Quote Quote by JesseM
That's because your version of H is too vague and doesn't actually specify whether the red card was the one that was picked to send to Alice and the white card was the one that was picked to send to Bob, or vice versa. If you completely specified the hidden properties of the envelope that was sent to Bob--namely "the person who picked the cards from the box put the red card in Bob's envelope, and the envelope continued to have that hidden card on its journey to Bob"--then in that case it would be true that P(B|H)=P(B|AH).
Quote Quote by billschnieder View Post
I am not sure I agree with this. In probability theory, when we write P(A|H), we are assuming that we know H but not A. If we knew A already (a certainty), there is no point calculating a probability is there?
I think I explained clearly in the text above that I was calculating the probability of B, not A, given either just H or given both H and A. If you want to calculate the probability of A rather than B, then you can easily modify the paragraph above:

That's because your version of H is too vague and doesn't actually specify whether the red card was the one that was picked to send to Alice and the white card was the one that was picked to send to Bob, or vice versa. If you completely specified the hidden properties of the envelope that was sent to Alice--namely "the person who picked the cards from the box put the red card in Alice's envelope, and the envelope continued to have that hidden card on its journey to Alice"--then in that case it would be true that P(A|H)=P(A|BH).

Quote Quote by billschnieder View Post
In this case when Bell obtains cos(theta), his equation will only be valid for two angles when cos(theta)= 0 or 1! How then can this equation apply to other angles? Therefore I don't think that is the reasoning here.
Huh? The argument is about what probabilities would be calculated by an ideal observer if they had access to the hidden variables H (which are assumed to have well-defined values at all times in a local realist theory), not just what probabilities are calculated by normal observers who don't know the values of the hidden variables. How could it be otherwise, when H explicitly appears in the conditional probability equations?
Quote Quote by billschnieder View Post
Furthermore, the variables are hidden from the perspective of the wise men, it is not God trying to calculate the probabilities but the wise men, because they do not have all the information. We are only looking from God's perspective to verify that the equation the wise men choose to use corresponds to the factual situation and in the example I gave it does not appear to.
No, you're completely confused, the argument is about taking a God's-eye-view and saying that no matter how we imagine God would see the hidden variables, in a local realist theory God would necessarily end up making predictions about the statistics of different correlations that are different from what we humans actually observer in QM.
Quote Quote by billschnieder View Post
But in my example, with the information known by God, this is the case, every box only contains two cards, one red and one white and in each iteration of the experiment one of the cards is sent to Bob and the other to Alice. The equation P(AB|H) = P(A|H)P(A|BH) always works but P(AB|H) = P(A|H)P(A|H) works only in very limited case in which H is no longer hidden and calculating probabilities is pointless.
It's not "pointless" if you can use this hypothetical God's-eye-perspective (where nothing is hidden) to show that if the hidden variables are such that Alice and Bob always get the same result when they perform the same measurement, that must imply certain things about the statistics they see when they perform different measurements--and that these statistical predictions are falsified in real quantum mechanics! This is a reductio ad absurdum argument showing that the original assumption that QM can be explained using a local realist theory must have been false.

Perhaps you could take a look at the scratch lotto analogy I came up with a while ago and see if it makes sense to you (note that it's explicitly based on considering how the 'hidden fruits' might be distributed if they were known by a hypothetical observer for whom they aren't 'hidden'):

Suppose we have a machine that generates pairs of scratch lotto cards, each of which has three boxes that, when scratched, can reveal either a cherry or a lemon. We give one card to Alice and one to Bob, and each scratches only one of the three boxes. When we repeat this many times, we find that whenever they both pick the same box to scratch, they always get the same result--if Bob scratches box A and finds a cherry, and Alice scratches box A on her card, she's guaranteed to find a cherry too.

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

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

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

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

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

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

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

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

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

But now suppose Alice and Bob look at all the trials where they picked different boxes, and found that they only got the same fruits 1/4 of the time! That would be the violation of Bell's inequality, and something equivalent actually can happen when you measure the spin of entangled photons along one of three different possible axes. So in this example, it seems we can't resolve the mystery by just assuming the machine creates two cards with definite "hidden fruits" behind each box, such that the two cards always have the same fruits in a given box.

And you can modify this example to show some different Bell inequalities, see post #8 of this thread for one example.
Quote Quote by billschnieder View Post
But that is not quite true. The justification for reducing P(A|BH) to P(A|H) is not based on whether B gives [i[additional[/i] information but on whether B gives any information. (see Conditional Independence in Statistical Theory, J.R Statist. Soc B, 1979 41, No. 1, pp. 1-31)
I don't have access to that reference (could you quote it?) but I'm confident it doesn't say what you think it does. In a situation where the probability of A is completely determined by H and the probability of B is also completely determined by H, then it would naturally be true that P(A|BH) would be equal to P(A|H), even if the P(A) was not equal to P(A|B) (i.e. if you don't know H, knowing B does give some information about the probability of A). Do you claim the reference somehow contradicts this?

For example, suppose we have two identical-looking flashlights X and Y that have been altered with internal mechanisms that make it a probabilistic matter whether they will turn on when the switch is pressed. The mechanism in flashlight X makes it so that there is a 70% chance it'll turn on when the switch is pressed; the mechanism in flashlight Y makes it so there's a 40% chance when the switch is pressed. The mechanism's random decisions aren't affected by anything outside the flashlight, so whether or not flashlight X turns on doesn't change the probability that flashlight Y turns on.

Now suppose we do an experiment where Alice is sent one flashlight and Bob is sent the other, by a sender who has a 50% chance of sending X to Alice and Y to Bob, and a 50% chance of sending Y to Alice and X to Bob. Let H1 and H2 represent these two possible sets of "hidden" facts (hidden to Alice and Bob since the flashlights look identical from the outside): H1 represents the event "X to Alice, Y to Bob" and H2 represents the event "Y to Alice, X to Bob". Let A represent the event Alice's flashlight turns on when she presses the switch, B represents the event that Bob's flashlight turns on when she presses the switch.

Here, P(A) = P(A|H1)*P(H1) + P(A|H2)*P(H2) = (0.7)*(0.5) + (0.4)*(0.5) = 0.55
and P(B) = P(B|H1)*P(H1) + P(B|H2)*P(H2) = (0.4)*(0.5) + (0.7)*(0.5) = 0.55

Since P(A|B) = P(A and B)/P(B), we must have P(A|B) = (0.7)*(0.4)/(0.55) = 0.5090909...
So you see that P(A|B) is slightly lower than P(A), which makes sense since if Bob's flashlight lights up, that makes it more likely Bob got flashlight X which had a higher probability of lighting, and more likely A got flashlight Y with a lower probability of lighting.

But despite the fact that B does give some information about the probability of A, it is still true that P(A|B and H1) = P(A|H1) = 0.7, since H1 tells us that Alice got flashlight X, and that alone completely determines the probability that Alice's flashlight lights up when she presses the switch, the fact that Bob's flashlight lit up won't alter our estimate of the probability that Alice's lights up. Likewise, P(A|B and H2) = P(A|H2) = 0.4.

I'm sure that whatever the reference you gave says, it doesn't imply that this reasoning is incorrect.
Quote Quote by billschnieder View Post
You re saying pretty much the same thing there, that A gives no additional information to H. But that is not the meaning of conditional independence. Conditional independence means that A gives us no information whatsoever about B.
I gave a pretty detailed argument in posts #61 and 62 on that thread, starting with the paragraph towards the end of post #61 that says "Let me try a different tack". If you aren't convinced by my comments so far in this post, perhaps you could identify the specific point in my argument on the other thread where you think I say something incorrect? For example, do you disagree with this part?
I'd like to define the term "past light cone cross-section" (PLCCS for short), which stands for the idea of taking a spacelike cross-section through the past light cone of some point in spacetime M where a measurement is made; in SR this spacelike cross-section could just be the intersection of the past light cone with a surface of constant t in some inertial reference frame (which would be a 3D sphere containing all the events at that instant which can have a causal influence on M at a later time). Now, let [tex]\lambda[/tex] stand for the complete set of values of all local physical variables, hidden or non-hidden, which lie within some particular PLCCS of M. Would you agree that in a local realist universe, if we want to know whether the measurement M yielded result A, and B represents some event at a spacelike separation from M, then although knowing B occurred may change our evaluation of the probability A occurred so that P(A|B) is not equal to P(A), if we know the full set of physical facts [tex]\lambda[/tex] about a PLCCS of M, then knowing B can tell us nothing additional about the probability A occurred at M, so that P(A|[tex]\lambda[/tex]) = P(A|[tex]\lambda[/tex] B)?
In case we are dealing with a local realist universe that is not deterministic, I think I should add here that the PLCCS of M is chosen at a time after the last moment of intersection between the past light cones of M and B, so that no events that happen after the PLCCS can have any causal influence on B. Continuing the quote:
If so, consider two measurements of entangled particles which occur at spacelike-separated points M1 and M2 in spacetime. For each of these points, pick a PLCCS from a time which is prior to the measurements, and which is also prior to the moment that the experimenter chose (randomly) which of the three detector settings under his control to use (as before, this does not imply the experimenter has complete control over all physical variables associated with the detector). Assume also that we have picked the two PLCCS's in such a way that every event in the PLCCS of M1 lies at a spacelike separation from every event in the PLCCS of M2. Use the symbol [tex]\lambda_1[/tex] to label the complete set of physical variables in the PLCCS of M1, and the symbol [tex]\lambda_2[/tex] to label the complete set of physical variables in the PLCCS of M2. In this case, if we find that whenever the experimenters chose the same setting they always got the same results at M1 and M2, I'd assert that in a local realist universe this must mean the results each of them got on any such trial were already predetermined by [tex]\lambda_1[/tex] and [tex]\lambda_2[/tex]; would you agree? The reasoning here is just that if there were any random factors between the PLCCS and the time of the measurement which were capable of affecting the outcome, then it could no longer be true that the two measurements would be guaranteed to give identical results on every trial.


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