Trying to Understand Bell's reasoning

[billschnieder]

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?

 

[DrChinese]

...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]

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]

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]

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 possesses 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 let's 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]

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]

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]

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 possesses 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]

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]

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 possesses 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 let's 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]

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]

Thanks for your efforts though.

By the way, welcome to PhysicsForums!

 

[billschnieder]

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]

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]

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]

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]

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]

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?

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).

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?

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.

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.

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.

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 $$\lambda$$ stand for the complete set of values of all local physical variables, hidden or non-hidden, which lie within some particular PLCCS of M. Would you agree that in a local realist universe, if we want to know whether the measurement M yielded result A, and B represents some event at a spacelike separation from M, then although knowing B occurred may change our evaluation of the probability A occurred so that P(A|B) is not equal to P(A), if we know the full set of physical facts $$\lambda$$ about a PLCCS of M, then knowing B can tell us nothing additional about the probability A occurred at M, so that P(A|$$\lambda$$) = P(A|$$\lambda$$ B)?

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 $$\lambda_1$$ to label the complete set of physical variables in the PLCCS of M1, and the symbol $$\lambda_2$$ to label the complete set of physical variables in the PLCCS of M2. In this case, if we find that whenever the experimenters chose the same setting they always got the same results at M1 and M2, I'd assert that in a local realist universe this must mean the results each of them got on any such trial were already predetermined by $$\lambda_1$$ and $$\lambda_2$$; would you agree? The reasoning here is just that if there were any random factors between the PLCCS and the time of the measurement which were capable of affecting the outcome, then it could no longer be true that the two measurements would be guaranteed to give identical results on every trial.

 

[DrChinese]

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. ...
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.

OK, so which side of the definition are you taking? Does a particle have definite values for observation outcomes PRIOR to the actual act of measurement, or not? That is the question a realist (such as Einstein) answers as "yes". Others, including Bohr and many/most of the scientific establishment, would answer "no". I am squarely in the "no" camp in case, because I believe there is observer dependence (context) - in case you had not already figured that out. And one of the primary reasons for that belief is Bell.

What is *your* answer? It would be helpful to get a straight answer. If you don't like the question, Bell is not likely to mean much to you - since this is the keystone to the paper.

 

[billschnieder]

Does a particle have definite values for observation outcomes PRIOR to the actual act of measurement, or not? That is the question a realist (such as Einstein) answers as "yes".

Not according to my understanding of EPR. The EPR question is whether it is possible to supplement QM with "elements of reality" such that QM becomes complete and can then predict individual events. That is what I explained in my oppenning post no? Einstein definitely did not say "I can see the moon even if I am not looking at it"! Yet given the elements of reality such as the position of the moon, the sky conditions, the position of a person and his gazing direction, the surface conditions around were the person is standing, Einstein will be able to predict with certainty whether a person in that scenario will see the moon or not, without disturbing any individual. Those are the elements of reality. But it definitely does not mean a person can see the moon without looking at it.

The question is not whether "outcomes pre-exist measurement". As I have explained already, there are many real locally causal situations in which outcomes do not pre-exist the act of measurement. Why are you so bent in insisting that this is the issue. I don't see that question/definition in the EPR paper at all. Maybe I missed it. Could you point me to where EPR says "outcomes must pre-exist measurement"?

 

[RUTA]

Not according to my understanding of EPR. The EPR question is whether it is possible to supplement QM with "elements of reality" such that QM becomes complete and can then predict individual events. That is what I explained in my oppenning post no? Einstein definitely did not say "I can see the moon even if I am not looking at it"! Yet given the elements of reality such as the position of the moon, the sky conditions, the position of a person and his gazing direction, the surface conditions around were the person is standing, Einstein will be able to predict with certainty whether a person in that scenario will see the moon or not, without disturbing any individual. Those are the elements of reality. But it definitely does not mean a person can see the moon without looking at it.

The question is not whether "outcomes pre-exist measurement". As I have explained already, there are many real locally causal situations in which outcomes do not pre-exist the act of measurement. Why are you so bent in insisting that this is the issue. I don't see that question/definition in the EPR paper at all. Maybe I missed it. Could you point me to where EPR says "outcomes must pre-exist measurement"?

This IS the issue, as DrC has been trying to get you to understand. Read "Quantum mysteries revisited," N. David Mermin, Am. J. Phys. 58, #8, August 1990, pp 731-734. I point you to this paper because you don't have to do any physics to see the problem. He does give you the physics in section III, but the conflict between lhv and QM is illustrated nicely in sections I and II. Let me give you some quotes from that paper (there are three particles traveling from a single source to three detectors, the detectors have two different settings (1 and 2) and there are two different possible outcomes (R and G) for each setting):

"In the absence of connections between the detectors and the source, a particle has no information about how the switch of its detector will be set until it arrives there. Since in each run any detector might turn out to be either the one set to 1 or one of the ones set to 2, to preserve the perfect record of always having an odd number of R flashes in 122, 212, and 221 runs, it would seem to be essential for each particle to be carrying instructions for how its detector should flash for either of the two possible switch settings in might find upon arrival." left hand column, p 732.

"If the instruction sets existed, then 111 runs would always have to produce an odd number of R flashes. But they never do, as I remarked in the third paragraph of this section, ... . Thus, a single 111 run suffices all by itself to give data inconsistent with the otherwise compelling inference of instruction sets." right hand column, p 732.

"Instruction sets require an odd number of R flashes in every 111 run; quantum mechanics prohibits an odd number of R flashes in every 111 run." left hand column, p 733.

 

[billschnieder]

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.

I disagree. God does not play dice. Make up your mind. Either God has complete information such that everything is certain and there are no "probabilities" or he does not.

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

Again it appears it is you who is confused, who is calculating the probabilities, God or "they"? Who is "they" by the way. It can't be Alice because she knows nothing about what is happening at Bob, nor can it be Bob. It can not be God either because he already knows everything so there are no "probabilities". So obviously it must be some external person looking at the data, who does not know everything about the cause of the data.

In any case, this is going off my main issue which I explained in my last post as follows:

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.

The justification for reducing P(A|BH) to P(A|H) is not based on whether B gives additional 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) http://people.csail.mit.edu/tdanford/discovering-causal-graphs-papers/dawid-79.pdf

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. See the paper above. Link added.

Your other example does not address these points.

 

[DevilsAvocado]

... 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?

... Sometimes, it is easy to over-focus on the details and miss the big picture ...

billschnieder here is the BIG PICTURE that you are missing:

 

[JesseM]

I disagree. God does not play dice. Make up your mind. Either God has complete information such that everything is certain and there are no "probabilities" or he does not.

You are definitely confused. Do you even understand what a "local hidden variables" theory is? It says the hidden variables associated with a particle have well-defined values at all times, and they either determine the exact outcomes of each measurement, or they determine the probabilities of different outcomes in a partially random way, with nothing besides the purely local variables associated with a particle (hidden or otherwise) influencing its response to a measurement. In the first case P(A|H) or P(B|H) will always be 1 or 0, so "God does not play dice"; in the second case it may be something else, but in this case there is genuine randomness in nature, so God does play dice (the notion of 'local realism' does not in itself automatically imply determinism). But either way we can show it's impossible to construct a local hidden variables theory that matches the statistics seen by actual human experimenters.

If you don't even understand what a local hidden variables theory is, you really need to go back to basics and try to learn something about the ideas behind the proof, rather than rush to critique it before you even have the first idea of what it's saying.

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

Again it appears it is you who is confused, who is calculating the probabilities, God or "they"?

All probability calculations involving hidden variables are from the perspective of "God" (i.e. from the perspective of an idealized observer who knows the values of all hidden variables). But when I said "that must imply certain things about the statistics they see when they perform different experiments", the statistics are those seen by the human experimenters.

Who is "they" by the way. It can't be Alice because she knows nothing about what is happening at Bob, nor can it be Bob.

It's the statistics they find in retrospect once they get together (or send signals) and compare their results, potentially long after they actually perform the measurements at a spacelike separation. For example, for some choice of detector settings they might find that on all trials where they happened to set their detectors at the same angle, they always got the same result (if Alice got spin-up then Bob got spin-up too, and same with spin-down), but on all trials where they happened to set their detectors at different angles, they only got the same result on 1/4 of all these trials. The point of my lotto card example was to show that these statistics are impossible under a local hidden variables theory--if they always get the same result when they scratch the same box on their cards, that implies that on the subset of trials where they scratched different boxes, they should have gotten the same result on 1/3 or more of the trials in that subset. Did you read my lotto card example and consider the math behind it? If we're going to continue to this discussion, you really need to thoughtfully consider the examples and arguments people give you rather than just giving knee-jerk argumentative responses and acting as though you are totally confident that you are right and that every physicist since Bell has missed something obvious that only you were smart enough to discover.

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.

The justification for reducing P(A|BH) to P(A|H) is not based on whether B gives additional 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) http://people.csail.mit.edu/tdanford/discovering-causal-graphs-papers/dawid-79.pdf

Where in the paper do you think it says or implies that it's incorrect to say that if B gives no additional information about the probability of A beyond what H gives you, then we can reduce P(A|BH) to P(A|H)? (I'm pretty confident the paper says no such thing and you are misunderstanding it somehow) Can you give a specific quote and page number? And just as you completely ignored my lotto card example, it appears here you completely ignored my flashlight example, and just repeated your original objection almost verbatim. Again, if we're going to have an actual discussion you need to be willing to give thoughtful consideration to examples and arguments, otherwise this will go nowhere.

if A and B are equal, numerically P(A|BH) = P(A|H)

What do you mean "if A and B are equal"? A and B aren't numbers, they're events. Do you mean "if P(A) and P(B) are equal"? But that doesn't make sense either since the conditional probability of A given some other facts can be different than the absolute probability of A, I could easily come up with an example where P(A)=P(B) but numerically P(A|BH) is not equal to P(A|H) (this would have to be an example where B does give some additional information about the probability of A beyond what H alone gives).

but you could not say in this situation that since B gives you no additional information to H, therefore A and B are independent.

I never claimed A and B were independent, so this is irrelevant to my argument. In my flashlight example A and B are clearly statistically dependent (though not causally dependent--do you understand the difference?), since P(A) = 0.55 but P(A|B) = 0.50909..., so B does give you some information about the probability of A. But if you already know H, then B gives you no additional information about the probability of A beyond what you already knew from H, so P(A|H and B) = P(A|H). Do you disagree that this is true in the example I gave?

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.

Again, no one claimed that A and B are conditionally independent, neither I nor Bell. In my flashlight example they are clearly conditionally dependent--did you read that example and think about it? If not please do so, in particular tell me if you disagree that the following probabilities would be correct in the example:

P(A) = 0.55
P(A|B) = 0.50909...
P(A|H1) = 0.7
P(A|H1 and B) = 0.7

 

[Prologue]

Don't get too discouraged Bill, there is always an opposition. Think of it as a test of your argument, not a personal attack. Now, I have not studied probability theory or local/nonlocal stuff much, so I am in no position to comment on that, but I can see that there are a lot of posts that aren't addressing his concern. I think there needs to be a well-defined list of axioms laid out for all to use in this argument. It would help a lot. Again, Bill's problem is with [2], not the outcome of Bell's stuff. He agrees that if [2] is true then the other stuff is all true, but the question is all about [2], not about the stuff that follows it. I don't think he is getting lost amongst the trees, I think he is questioning whether one of the trees in the forest really is a tree.

Edit: JesseM points out in the next post that what I really meant was [1] everywhere there is a [2] above, which is true.

 

[JesseM]

It would help a lot. Again, Bill's problem is with [2], not the outcome of Bell's stuff. He agrees that if [2] is true then the other stuff is all true, but the question is all about [2], not about the stuff that follows it. I don't think he is getting lost amongst the trees, I think he is questioning whether one of the trees in the forest really is a tree.

He said his problem was with (1), not (2). Specifically he said that he didn't understand why, if H represents the full set of variables that influence the outcome of a given measurement in a local way, then if A is some possible outcome for that measurement, while B is some outcome for a different measurement performed at a spacelike separation from the first, then P(A|H) = P(A|BH). The point I'm making is that this equation is not incompatible with the idea of a statistical correlation between A and B, i.e. the idea that P(A) is different from P(A|B). I gave specific examples like the lotto example and the flashlight example where this would be true. And if the concrete examples don't suffice to show intuitively why the equation should hold in a local hidden variables theory, my argument in post #61 and #62 on the other thread was trying to give a fairly detailed argument as to why P(A|H) = P(A|BH) must be true in a local realist theory, provided we let H represent the complete information about all physical variables (hidden and otherwise) in a past light cone cross-section (PLCCS) of the measurement M1 which might yield result A, with the PLCCS chosen so that no event in it has the measurement M2 which might yield result B in its future light cone (i.e. the cross-section of M1's past light cone is taken at a time after the last time that the past light cones of M1 and M2 intersect).

 

[DrChinese]

Don't get too discouraged Bill, there is always an opposition. Think of it as a test of your argument, not a personal attack. Now, I have not studied probability theory or local/nonlocal stuff much, so I am in no position to comment on that, but I can see that there are a lot of posts that aren't addressing his concern. I think there needs to be a well-defined list of axioms laid out for all to use in this argument. It would help a lot. Again, Bill's problem is with [2], not the outcome of Bell's stuff. He agrees that if [2] is true then the other stuff is all true, but the question is all about [2], not about the stuff that follows it. I don't think he is getting lost amongst the trees, I think he is questioning whether one of the trees in the forest really is a tree.

Edit: JesseM points out in the next post that what I really meant was [1] everywhere there is a [2] above, which is true.

A simple review of what I posted previously will show that it is Bill who is missing the train. You don't need to consider separability if you look at the underlying argument. Instead of splitting hairs over semantics, why not address the meat? Just try to answer the following 2 questions: a) do you believe observables have well defined values independent of observation, as Einstein supposed? b) if yes, please present a set of values for angle settings 0, 120 and 240 for some group of photons that you believe is representative.

If you don't get this concept, you are missing the forest. It's cute you pretend that someone is the "loyal opposition" but actually Bill is coming off more as as a craggly contrarian. I am not actually sure craggly is a word, by the way. The meaning of Bell is what is important.

 

[DevilsAvocado]

billschnieder, I think DrChinese, JesseM and RUTA did a good job in trying to explain, but you still seem a little 'skeptical'.

The real problem is that you are making an assumption on completely wrong premises, almost like – "I can prove that the probability for this car making 100 mph doesn’t make sense" – when the car is actually making 200 mph...

And I show you where your assumption goes wrong:

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 let's look at the situation from God's perpective to see if the equation chosen by the wise men is correct.

Your "God" is clearly misinformed. Yes there are "two cards in the box", but then all goes wrong. One card is not red and the other white, both cards are red on one side and white on the other!

This is called http://en.wikipedia.org/wiki/Spin_(physics)" , and is absolutely fundamental in QM. Bell of course knew this when writing his ansatz.

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!

Run this Java applet:

"[URL Mermin's EPR gedanken experiment animated[/B]

[/URL]

...and you see why this is also wrong...

 

[DevilsAvocado]

... there is always an opposition ...

Correct, and in this case billschnieder is in opposition, and 'we' belong to the 99.99% majority...

 

[Prologue]

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.

 

[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.