Bell & CHSH type inequalities and experiments

[minkwe]

I would like to start a focused and detailed discussion of Bell & CHSH type inequalities and their relevance for Aspect-type experiments. I will start by presenting the mathematical derivation of the inequalities and end by asking three questions to focus the discussion.

First Bell's inequality derivation from his original paper:



Note, we have only 3 functions in the above A(a,λ), A(b,λ), A(c,λ). The next step is very important as it affirms and enforces the fact that we have only three functions, by FACTORIZATION:

from which

Since the second term on the right is P(b,c) and the first term is just 1, we have:

This is Bell's inequality, derived as he did it on pages 405-406 of his original paper.
Question 1: Are the three correlations P(a,b), P(a,c), and P(b,c) as they stand in the above inequality independent of each other? Note. I'm not asking if the measurement functions A(a,λ), A(b,λ), A(c,λ) are independent, What I'm asking is this: If you take three independent measurement functions A(a,λ), A(b,λ), A(c,λ) and recombine them in pairs, [A(a,λ)A(b,λ)], [A(a,λ)A(c,λ)], [A(b,λ)A(c,λ)] as was done during the derivation (factorization steps). Are these pairs independent of each other?

Now let us look at the CHSH inequality, we will use the wikipedia derivation which mirrors Bell's derivation above:


Note, we have only 4 functions in the above A(a,λ), B(b,λ), A(a',λ), B(b',λ). The next step is very important as it affirms and enforces the fact that we have only 4 functions, by FACTORIZATION:


From which we obtain:

Using similar arguments as with Bell's inequality, we obtain the CHSH:


Question 2: Are the four correlations E(a,b), E(a,b'), E(a',b') and E(a',b) as they stand in the above inequality independent of each other? Note. I'm not asking if the four measurement functions A(a,λ), B(b,λ), A(a',λ), B(b',λ) are independent, What I'm asking is this: If you take four independent measurement functionsA(a,λ), B(b,λ), A(a',λ), B(b',λ) and recombine them in pairs, [A(a,λ)B(b,λ)], [A(a,λ)B(b',λ)], [A(a',λ)B(b',λ)], [A(a',λ)B(b,λ)] as was done during the derivation (factorization steps). Are these pairs independent of each other?

Now considering the experiments:
Question 3: Are the four correlations E1(a,b), E2(a,b'), E3(a',b') and E4(a',b) from real experiments, each measured on different set of particles from the other dependent on each other in the same way as those in the Bell and CHSH inequalities?

In a genuine CHSH experiment, the terms must be related to each other in the same way as the terms of the CHSH inequality are related to each other. No such experiment has ever been performed and this is not a statement about loopholes, which are completely irrelevant for this discussion.

[minkwe]

To kick off the discussion, here are my own answers:

Question 1: No, the correlations are mutually interdependent
Question 2: No, the correlations are mutually interdependent
Question 3: No, the correlations are independent unlike those in Bell's or CHSH inequalities.

[Heinera]

I see your point.

The answer to question 3 is "yes, if the experiment can be modeled by a LHV model without data rejection." This is a consequence of the fact that the source does not have access to the detector settings in these models, nor can one detector know the setting of the other. So if one assumes a LHV model as a basis of the physics, one can safely do Bell/Aspect-type experiments. The result the the inequalities are experimentally violated only tells us that this assumption was wrong, and nothing more.

[minkwe]

The answer to question 3 is "yes, if the experiment can be modeled by a LHV model without data rejection."

I do not agree. If I have a LHV without data rejection, and I measure each of the correlations from a separate set of particles, they will still not be dependent on each other in the same way as CHSH terms. But if I measure them on the same set of particles, then they will be. Think about the degrees of freedom of the variables. It is not about LHV vs non-LHV, it is about 4 free variables vs 8 free variables, LHV with 4-free variables can not violate the CHSH, LHV with 8-free variables can violate the CHSH since the CHSH demands only 4-free variables (factorization step). Data rejection also does not matter, 4-free variables with or without data rejection can never violate the CHSH.

The CHSH is a constraint on 4 free random variables not 8. Every derivation relies on this fact. LHV models has nothing to do with it.

[Heinera]

The answer to question 3 is "yes, if the experiment can be modeled by a LHV model without data rejection."

I do not agree. If I have a LHV without data rejection, and I measure each of the correlations from a separate set of particles, they will still not be dependent on each other in the same way as CHSH terms. But if I measure them on the same set of particles, then they will be. Think about the degrees of freedom of the variables. It is not about LHV vs non-LHV, it is about 4 free variables vs 8 free variables, LHV with 4-free variables can not violate the CHSH, LHV with 8-free variables can violate the CHSH since the CHSH demands only 4-free variables (factorization step). Data rejection also does not matter, 4-free variables with or without data rejection can never violate the CHSH.

The CHSH is a constraint on 4 free random variables not 8. Every derivation relies on this fact. LHV models has nothing to do with it.

But a LHV model can't have 8 free variables, since the outcome at Alice can't depend on Bob's setting (there goes one degree of freedom), the outcome at Bob can't depend on Alice's setting (there goes another degree), and the hidden variable can't depend on neither Bob's or Alice's settings (there goes the remaining two degrees). So we are left with four. And as you say, "4-free variables with or without data rejection can never violate the CHSH." You are entirely correct.

[minkwe]

But a LHV model can't have 8 free variables, since the outcome at Alice can't depend on Bob's setting (there goes one degree of freedom), the outcome at Bob can't depend on Alice's setting (there goes another degree), and the hidden variable can't depend on neither Bob's or Alice's settings (there goes the remaining two degrees). So we are left with four.

I think you are missing the point. If I measure on A(a) on one set of particles that is one free variable, then A(a) on a different set of particles, that is a different free variable (a total of 2 in this case). The fact that the second measurement does not depend on the first one ensures that the variables remain free. Each of the outcomes is "free" to vary independently of the other. What makes the variables free is the fact that they are measured on disjoint sets of particles.

Degrees of freedom:

In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.[1]

The number of independent ways by which a dynamic system can move without violating any constraint imposed on it, is called degree of freedom. In other words, the degree of freedom can be defined as the minimum number of independent coordinates that can specify the position of the system completely.

Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom

If I throw a single coin, the outcomes of each side are not free variables. P(H) and P(T) are not independent because we are dealing with the same coin. Knowing that the outcome was "heads", tells you that it cannot be "tails". But if I throw two separate coins, an outcome from the first coin and an outcome from the second coin are both free variables. P(H) for the first coin is independent of P(T) for the second coin. Knowing that the first coin gave H tells you absolutely nothing about the second coin. Just because there is no dependence between P(H) and P(T) for the two coins does not mean you can now combine them as a single variable as you appear to be doing above.

And note that the coins are all locally realistic, and governed by the same dynamics.

So contrary to your statement above, a LHV model measured on 8 disjoint sets of particles (4 disjoint sets of pairs) does have 8 free variables not 4. If you still disagree, you will have to explain how the results obtained on one set of particles, constrains the results obtained on another completely disjoint set from the first. That is the only way you can reduce the number of free variables, not by independence. Independence instead ensures that the variables remain free.

[Heinera]

So contrary to your statement above, a LHV model measured on 8 disjoint sets of particles (4 disjoint sets of pairs) does have 8 free variables not 4. If you still disagree, you will have to explain how the results obtained on one set of particles, constrains the results obtained on another completely disjoint set from the first. That is the only way you can reduce the number of free variables, not by independence. Independence instead ensures that the variables remain free.

By degrees of freedom, I meant number of freely picked equations/variables that could be used to influence the correlations (and I assumed you meant the same). If I pick 8 random and uncorrelated variables then obviously I would not violate CHSH or anything else, but would you still say I had 8 degrees of freedom? The point with a LHV model is that you can really only control the correlations of four of the degrees of freedom; the remaining four will just be random noise.

"If you still disagree, you will have to explain how the results obtained on one set of particles, constrains the results obtained on another completely disjoint set from the first."

In nature, it doesn't. In a LHV model it does, because the model does not have access to the difference between detector settings. But the model must come up with results so that E(a,b) on one set should give the same value as E(a+phi, b+phi) on the next set (where phi is any angle), since that's what happens in experiments. With a LHV model this constrains the results obtained by the second set in a very strong way, in fact so strong that only a model with linear correlations could achieve this, as long as it doesn't have access to the actual difference between detector settings - i.e. a LHV model.

[minkwe]

By degrees of freedom, I meant number of freely picked equations/variables that could be used to influence the correlations (and I assumed you meant the same). If I pick 8 random and uncorrelated variables then obviously I would not violate CHSH or anything else, but would you still say I had 8 degrees of freedom?

You will have to explain what you mean by freely picked equations and how that is different from the independently measured functions I'm talking about. Like I explained in my last post, A(a) on one set of particles is a different random variable from A(a) on a completely different set of particles. They are each free to vary. In Aspect type experiments, there are two each of A(a), A(a'), B(b), B(b') for a total of 8, each from a different set of particles which you could therefore denote as A1(a), B1(b), A2(a), B2(b'), A3(a'), B3(b), A4(a'), B4(b'), where the numbers represent the set of particle pairs from which each is measured. A1(a) is a different independent random variable from A2(a), each can vary independently of the other, contrary to the CHSH where we have only 4 A(a), A(a'), B(b), B(b'), all drawn from the same set of particle pairs.

An expression with 8 independent random variables has 8 degrees of freedom, whether the CHSH is violated or not.

The point with a LHV model is that you can really only control the correlations of four of the degrees of freedom; the remaining four will just be random noise.

I disagree with this. The point of a LHV model is that in any set of particles you are looking at, at a given moment, all the correlations between pairs of directions for the directions (a, b, a', b') can be explained by recombination of the 4 independent functions A(a), A(a'), B(b), B(b') drawn from that set of particles.

In nature, it doesn't. In a LHV model it does, because the model does not have access to the difference between detector settings.

The difference between detector settings is completely irrelevant to the point I'm making. Angle differences are not used to calculate those terms in experiments, nor are angle differences used in the two derivations I presented above. I still do not see a justification to suggest that in an Aspect-type experiment the E(a,b) from one set of particles will not be independent from the E(a,b') from a completely different set of particles.Just like In my coin toss analogy, I do not see what could possibly make the outcomes on two different coins to not be independent of each other, even if the coins are identical and governed by the same local-realistic dynamics.

[FrediFizzx]

Hi Michel,

I think I have maybe found your problem. The Bell derivation of CHSH shown on wikipedia is wrong. I mentioned to Joy a couple of months ago that I thought it was wrong and he sent me the Clauser-Shimony review paper, "Bell's theorem: experimental tests and implications". They show the 1971 Bell derivation of CHSH and it is correct as far as I can tell. The difference being that the <A's> and <B's> in the C-S review are not the same for the different E(a, b)'s like they show in the wikipedia article. They use <A_a>, <A_a'>, <B_b>, and <B_b'> which means to me that <A_a> is not the same as <A_a'>, etc. So someone got sloppy on the wikipedia article.

However, I think you have been saying that when the <A's> and <B's> are not the same for the E(a, b)'s, that the bound is 4 not 2. So we have to figure out how to clear that up or if not, then you are right and CHSH is a bunch of baloney. It certainly would be nice to have the actual CHSH paper to see their derivation but if you don't have the C-S review paper I can email it to you. They go on after the 1971 Bell derivation of CHSH to sort of show how QM can violate it.

I especially like the last sentence in the C-S review abstract, "The conclusions are philosophically startling: either one must totally abandon the realistic philosophy of most working scientists, or dramatically revise our concept of space-time" (underline mine). Fast-forward 29 years to 2007 and then Dr. Joy Christian discovers that for sure our concept of space needs to be revised and then all the problems and paradoxes go away.

[gill1109]

Good discussion! Key topic.

In a genuine CHSH experiment, the terms must be related to each other in the same way as the terms of the CHSH inequality are related to each other. No such experiment has ever been performed and this is not a statement about loopholes, which are completely irrelevant for this discussion.

I think you are missing a crucial step in the chain of reasoning. It is indeed usually skated over lightly, or even completely forgotten in the litetature.

Consider a CHSH style experiment with a total of N runs. You'll use about N/4 of the runs to calculate each of the four correlations. A different N/4 for each. However, if the runs on which any particular setting pair is observed are a random sample of all N runs, then the observed correlation based on the subsample will be close to the unobserved but theoretically (because of LHV) existing correlation based on all N.

Counting degrees of freedom and numbers of independent variables is not the whole story. We need statistics and probability to connect sub-sample, sample and population. I'm afraid many physicists have little appreciation of this side of the story.

I can email the CHSH original to anyone who would like to see it. We could also set up a shared dropbox folder of key papers. Anyone interested?

[FrediFizzx]

Yes, send me the CHSH paper that shows their derivation.

[Mikko]

I would like to start a focused and detailed discussion of Bell & CHSH type inequalities and their relevance for Aspect-type experiments. I will start by presenting the mathematical derivation of the inequalities and end by asking three questions to focus the discussion.

First Bell's inequality derivation from his original paper:



Note, we have only 3 functions in the above A(a,λ), A(b,λ), A(c,λ). The next step is very important as it affirms and enforces the fact that we have only three functions, by FACTORIZATION:

from which

Since the second term on the right is P(b,c) and the first term is just 1, we have:

This is Bell's inequality, derived as he did it on pages 405-406 of his original paper.
Question 1: Are the three correlations P(a,b), P(a,c), and P(b,c) as they stand in the above inequality independent of each other? Note. I'm not asking if the measurement functions A(a,λ), A(b,λ), A(c,λ) are independent, What I'm asking is this: If you take three independent measurement functions A(a,λ), A(b,λ), A(c,λ) and recombine them in pairs, [A(a,λ)A(b,λ)], [A(a,λ)A(c,λ)], [A(b,λ)A(c,λ)] as was done during the derivation (factorization steps). Are these pairs independent of each other?

There is a problem in this question. The meaning of your "independent" is not clear. Do you mean logical or statistical dependence? Do you mean their variation when the experiment is repeated several times or uncertainty of the predicted measurement? If you vary ρ, P's vary, and hardly independently.

Now let us look at the CHSH inequality, we will use the wikipedia derivation which mirrors Bell's derivation above:


Note, we have only 4 functions in the above A(a,λ), B(b,λ), A(a',λ), B(b',λ). The next step is very important as it affirms and enforces the fact that we have only 4 functions, by FACTORIZATION:


From which we obtain:

Using similar arguments as with Bell's inequality, we obtain the CHSH:


Question 2: Are the four correlations E(a,b), E(a,b'), E(a',b') and E(a',b) as they stand in the above inequality independent of each other? Note. I'm not asking if the four measurement functions A(a,λ), B(b,λ), A(a',λ), B(b',λ) are independent, What I'm asking is this: If you take four independent measurement functionsA(a,λ), B(b,λ), A(a',λ), B(b',λ) and recombine them in pairs, [A(a,λ)B(b,λ)], [A(a,λ)B(b',λ)], [A(a',λ)B(b',λ)], [A(a',λ)B(b,λ)] as was done during the derivation (factorization steps). Are these pairs independent of each other?

This question has the same problem as the first one.

Now considering the experiments:
Question 3: Are the four correlations E1(a,b), E2(a,b'), E3(a',b') and E4(a',b) from real experiments, each measured on different set of particles from the other dependent on each other in the same way as those in the Bell and CHSH inequalities?

This question has a similar problem, too. It is even not clear that the same kind of dependence is meant as the context different. Perhaps we could understand you better if you could answer, using your interpretation of the words, the following question: Is E1(a,b) from a real experiment independent of E1(a,b) from another real experiment that uses the same setup including the same a and b?

In a genuine CHSH experiment, the terms must be related to each other in the same way as the terms of the CHSH inequality are related to each other. No such experiment has ever been performed and this is not a statement about loopholes, which are completely irrelevant for this discussion.

[Heinera]

In nature, it doesn't. In a LHV model it does, because the model does not have access to the difference between detector settings.

The difference between detector settings is completely irrelevant to the point I'm making. Angle differences are not used to calculate those terms in experiments, nor are angle differences used in the two derivations I presented above. I still do not see a justification to suggest that in an Aspect-type experiment the E(a,b) from one set of particles will not be independent from the E(a,b') from a completely different set of particles.

Here you confuse the results of an experiment, and the output of a model, which are two different things. In an experiment, the E(a,b) from one set of particles is independent from the E(a,b’) from a completely different set of particles, as you say. The point is that it is impossible to construct a LHV model so that the same is true for the model.

Earlier you wrote that Joy’s experiment couldn’t possibly violate the CHSH-inequality if all correlations were computed on the entire N set of data. That is true, and it means that you are very close to understanding Bell’s theorem. But then you say that if each correlation is computed on separate sets of N/4 observations, we would see a violation. Now, in a classical model, why should that make a difference? Say you measure height and weight of one million persons, and compute the correlation. Then you randomly pick a subset of 250 000 of the data pairs, and compute the correlation again. Would you expect this correlation to be significantly different from the previous one? Obviously not.

[minkwe]

Hi Michel,

I think I have maybe found your problem. The Bell derivation of CHSH shown on wikipedia is wrong. I mentioned to Joy a couple of months ago that I thought it was wrong and he sent me the Clauser-Shimony review paper, "Bell's theorem: experimental tests and implications". They show the 1971 Bell derivation of CHSH and it is correct as far as I can tell. The difference being that the <A's> and <B's> in the C-S review are not the same for the different E(a, b)'s like they show in the wikipedia article. They use <A_a>, <A_a'>, <B_b>, and <B_b'> which means to me that <A_a> is not the same as <A_a'>, etc. So someone got sloppy on the wikipedia article.

The wikipedia derivation is correct. You will have to point me to your favourite derivation and I will use it to show you that it dies not escape the point I'm making.

[minkwe]

Here you confuse the results of an experiment, and the output of a model, which are two different things. In an experiment, the E(a,b) from one set of particles is independent from the E(a,b’) from a completely different set of particles, as you say. The point is that it is impossible to construct a LHV model so that the same is true for the model.

No I'm not. I asked 3 questions, 2 about Theoretical Bell and CHSH constructs and one about experiments. You objected only to my answer about experiments. That's what I'm clarifying to you that the same is true for a LHV model of the experiment.

The experiment uses a completely separate set of particles, why would you insist on using a single set if particles when modelling it? Even the QM predictions are for separate sets of particles.

[minkwe]

There is a problem in this question. The meaning of your "independent" is not clear. Do you mean logical or statistical dependence?

Question 1 & 2 are not about an experiment. Will your answers to the question change based on what kind of dependence? I mean both. Im asking a very specific question about the relationship between the measurement functions and the PS, not between P's and lambda.

This question has a similar problem, too. It is even not clear that the same kind of dependence is meant as the context different.

It makes no difference.

Perhaps we could understand you better if you could answer, using your interpretation of the words, the following question: Is E1(a,b) from a real experiment independent of E1(a,b) from another real experiment that uses the same setup including the same a and b?

This makes no sense. The number after E represents a specific set of particles. It is impossible to repeat the experiment on the same set of particles. Your question contains a contradiction.

EDIT:
However E(a,b) from a real experiment (ie, E1(a,b)) is independent of E(a,b) from another real experiment (ie E2(a,b)) that uses the same setup including the same a and b. There is a subtlety in how you relate symbols in a mathematical expression to values you calculate from an experiment that most people are missing. That is why I bring up degrees of freedom and the coint-toss analogy because it illustrates the issue clearly in a simple example. P(H) and P(T) from a single coin toss are related to each other in a different manner than P(H) from one coin toss and P(T) from a different coin toss, even if the two coins are physically identical.

[minkwe]

Good discussion! Key topic.
I think you are missing a crucial step in the chain of reasoning. It is indeed usually skated over lightly, or even completely forgotten in the litetature.

Consider a CHSH style experiment with a total of N runs. You'll use about N/4 of the runs to calculate each of the four correlations. A different N/4 for each. However, if the runs on which any particular setting pair is observed are a random sample of all N runs, then the observed correlation based on the subsample will be close to the unobserved but theoretically (because of LHV) existing correlation based on all N.

I'm not missing anything, but I think you are confusing terminology. We are talking about the 4 correlations E(a,b), E(a,b'), E(a',b') and E(a',b) and how the are derived in the CHSH inequality from 4 measurement functions A(a,λ), B(b,λ), A(a',λ), B(b',λ). Nothing about N.

For question 3 referring to experiments, we are talking about how the corresponding 4 correlations E(a,b), E(a,b'), E(a',b') and E(a',b) are calculated. Nobody contests or denies the fact that in Aspect-type Each of the correlations are measured on a distinct set of particles so that using my suggested notation, we have E1(a,b), E2(a,b'), E3(a',b') and E4(a',b) where the numbers represent the distinct set of particles used to calculate the correlation. The number of particles in each set is not necessarily the same and no particle from one set contributes to another set. Nobody denies that we therefore have 8 measurement functions, not 4. So I don't see why talking about "N" here is even relevant to the discussion. Perhaps if you would answer the 3 questions I asked, we could have a focused discussion about the issue being discussed in this thread; N has nothing to do with it.

Counting degrees of freedom and numbers of independent variables is not the whole story.

It is the story we are interested in, in this thread. Degrees of freedom is very important in statistics.

[minkwe]

Earlier you wrote that Joy’s experiment couldn’t possibly violate the CHSH-inequality if all correlations were computed on the entire N set of data. That is true, and it means that you are very close to understanding Bell’s theorem.

I understand Bell's theorem very well thank you . If you think Bell's theorem is about correlations calculated on a single set of particles then I'm afraid you do not understand it. The QM predictions are for distinct sets of particles.

But then you say that if each correlation is computed on separate sets of N/4 observations, we would see a violation. Now, in a classical model, why should that make a difference?

Isn't that the whole point I've been explaining? It is like asking, why should it make a difference if I derive the inequality by assuming a single toss of one coin, and use results from two different tosses to verify if the inequality is valid, and then concluding later that since the inequality is violated, it means my single coin does not have two sides.

It makes a difference precisely because a constraint derived from 4 free variables should not be expected to hold on 8 free variables.
It makes a difference because Bell's theorem states: "No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.", and the predictions made by QM are for distinct sets of particles.

Say you measure height and weight of one million persons, and compute the correlation. Then you randomly pick a subset of 250 000 of the data pairs, and compute the correlation again. Would you expect this correlation to be significantly different from the previous one? Obviously not.

You are missing the point completely. The issue is not about the difference between population, means and sample means. The issue is between free variables and non-free variables. The height of the 250000 people is not a independent free variable from the weight of the same 250000 people. However, the height of the 250000 people is an independent free variable from the weight of another 250000 completely different people drawn from the same population.

[Heinera]

Say you measure height and weight of one million persons, and compute the correlation. Then you randomly pick a subset of 250 000 of the data pairs, and compute the correlation again. Would you expect this correlation to be significantly different from the previous one? Obviously not.

You are missing the point completely. The issue is not about the difference between population, means and sample means.

But it is. Given some large N, you are suggesting that restricting the correlation computation on N pairs of numbers to an N/4 random subset of those pairs somehow gives you a significantly different correlation, are you not?

[minkwe]

Yes, send me the CHSH paper that shows their derivation.

Here is the derivation from CHSH(1969) page881:

Suppose for some b' and b we have P(b',b) = 1-δ, where 0 <= δ <= 1. Experimentally interesting cases will have δ close to but not equal to zero. Here we avoid Bell's experimentally unrealistic restriction that for some pair of settings b' and b there is perfect correlation (ie, δ = 0). Dividing Γ into two regions Γ+ and Γ- such that Γ± = {λ|A(b',λ) =± B(b,λ)} we have





Therefore,

Notice the crucial factorization step is still there on line 3.