Thoughts about Bell, Bohm, Christian, et al.

[minkwe]

Equation (1) of Bell's paper, has no distinguishing feature that should be different for local/non-local hidden variable theory, other than what we might have in mind about the meaning of the symbols. The form of the expressions is exactly the same for local as for non-local theories. In other words, any suggestion that equation (1) somehow embodies a locality assumption is false.

The whole premise of his paper is based on a locality assumption, as is made very clear in the very next sentence following eq. (1). On the other hand, if the hidden variable would be a function of a and b, eq. (2) no longer holds. Hence, there is no longer a proof, so Bell's theorem does not apply to that case.

I've not contested the premise of the paper, I'm stating simple self-evident uncontroversial facts!

1) A local hidden variable is one which is represents information entirely within Alice's light cone. Let us call this variable λ .
2) A non-local hidden variable is one which represents information outside of Alice's light cone. Let us call this variable γ.
3) A theory which represents Alice's measurement outcome along axis "a" as is non-local, and one which represents the outcome as is local.
4) For a single particle heading towards Alice's SG magnet oriented along an axis "a", with two regions labelled {+1, -1}, a non-local theory would represent the outcome as , and a local theory would represent the outcome as
5) For a pair of spin-half particles heading in opposite directions measured at two stations along the same axis "a", a non-local theory would represent say , while a local theory would represent it as
6) There is nothing in the mathematical form of Bell's equation (1) that restricts it only to local theories. Nothing whatsoever. Bell's vital assumption (2) in the sentence below equation (1) forbids dependence between the outcome and the setting "b". It absolutely does not forbid dependence between and any other non-local information outside of Alice's light-cone. In fact, It forbids dependence between and the setting "b", even if "b" is within Alice's light cone. Therefore it does not amount to a locality assumption, but rather to a "setting independence assumption".

These facts are completely uncontroversial and remain true irrespective of anything else which comes up later in the paper. Which point exactly do you disagree with? If you don't disagree with any of that, then we can examine equation (2).

[Joy Christian]

Correct. However, to be fair to Bell, he did not think that his equation (1) by itself contained the concept of locality. He therefore defines the concept of locality separately, and quite precisely, as follows (and I paraphrase): The measurement outcomes A(a, λ ) = +/-1 and B(b, λ) = +/-1 are said to be locally explicable if A(a, λ) does not depend on either b or B, and likewise B(b, λ) does not depend on either a or A. This of course puts severe restrictions on what the "functions" λ can be.

I think what is missing from the discussion so far is a discussion about the crucial and subtle concept of outcome independence. Please read carefully what I have stated as Bell's locality condition. It has two parts: (1) remote setting independence, and (2) remote outcome independence. Setting independence by itself is not enough. And outcome independence does not depend on whether or not a setting exists at a space-like separated region. So I still think we are being unfair to Bell.

[minkwe]

Joy, non-local outcomes and settings are not the only types of non-local information. Outcome and setting independence does not care if the outcomes and settings are local or not. Therefore, I don't see exactly why you think that is unfair to Bell.

When he writes under equation (1) that:

The vital assumption (2) is that the result B for particle 2 does not depend on setting "a", of the magnet for particle 1, nor A on "b"

Then it does not matter whether the setting "a" is within the light-cone of B, or "b" is within the light cone of A. Setting and outcome independence are separate concepts from locality. Locality implies setting and outcome independence only if the settings and outcomes are outside the light-cone. But setting and outcome independence does not imply locality in a general sense, because settings and outcomes are not the only possible non-local variables; nor does setting and outcome dependence necessarily imply non-locality, because you can have setting and outcome dependence for settings and outcome within the light-cone, and still have locality.

The expression also imply settings and outcome independence, yet are non-local.

Besides, as I've explained, for a single particle, measured at a single setting "a", it makes no sense to talk of "setting and outcome independence" since there is no other setting, or outcome. A definition of locality which does not apply to single particles is quite deficient.

[minkwe]

Now that equation (1) is all cleared up, let us move on to equation (2). Despite wide-spread misunderstanding of equation (2), and Bell's poor choice of notation [P(a,b)], equation (2) is not a definition of a probability. It is simply an expectation value of the product of the two outcomes. As Bell explains in the immediately preceding text, equation two is simply the expectation value of the outcome at Alice multiplied by the outcome at Bob. Which is equivalent to

, or

A few comments:
- The expectation value of the product of two outcomes is the probability weighted average of all possible pairs of outcomes over any valid probability measure. Although Bell used a continuous variable lambda over the space of all hidden variables as the measure, there are others that could have been used. For example the space of all possible outcomes,


In fact, this is one of the the representation that is used often in experiments.

Another valid measure is that of the set of all particle pair emissions.
whereby we get where
This representation is also very popular in experiments. For each particle pair we simply multiply the outcomes together and average the sum, invoking the law of large numbers.

Therefore there is no physical content in equation (2) that distinguishes local theories from non-local ones. The expectation value of a paired-product, is calculated exactly the same way (multiply the results together take the probability weighted average over any valid measure), irrespective of whether the results are produced in a local manner or a non-local manner.

Final comment: Bell makes a mistake as far as he suggests in the preceding sentence that represents the outcome of the measurement in QM. His previous description mandates that the outcomes be limitted to +1 or -1. does not qualify. He repeats the mistake in equation (3).

[AnotherGuest]

In(3), and in the text just before, Bell is using conventional quantum mechanical terminology. is called an "observable". Mathematically, it is a Hermitean operator on a complex Hilbert space. It has eigenvalues and eigenvectors. The possible values of the observable which are actually observed when the observable is measured are the eigenvalues. The state of the system then is supposed to collapse to the eigenvector. The probability of each particular eigenvalue is given by the Born law.

If we have an observable X and the quantum system is in the state , then the expectation value of the probability distribution I just described can be quite simply calculated as and is often denoted simply by .

Bell is just using completely standard quantum mechanics notation. Lots of textbooks tell you what operators to use, what quantum state (a unit vector in the Hilbert space), and how to do the computations. The story is a little bit more complicated when we have to talk about eigenspaces instead of eigenvectors. Which is the case here. The Hilbert space is a tensor product of two spaces, the spin operators act on each space separately ...

Anyway, and are two observables, which commute, so their product is also an observable. The eigenvalues of the product are the products of the eigenvalues. And so on and so forth ...

[AnotherGuest]

I have been searching the internet for the orthodox quantum mechanical calculation of the singlet state correlation function. It's Exercise 6.8 in Peres' book. Ballentine does it in his book, in Section 20.2.

Bell uses conventional quantum mechanics notation whereby stand for the expectation value of an observable. You observe an observable. Quantum mechanics (Born's rule) tells you the probability distribution of the possible outcomes (the eigenvalues of the operators). If you do things with density operators, then you get the pretty formula . For a pure state, . Now we just need to write down the Hilbert space, the operators, and the state, and do the calculations. Ballentine says "the result can be calculated ... by brute force. Alternatively, we can invoke the rotational invariance of the singlet state ...". But then, of course, you first should verify the rotational invariance. Either way, there is quite a lot of work involved.

[Heinera]

- The expectation value of the product of two outcomes is the probability weighted average of all possible pairs of outcomes over any valid probability measure. Although Bell used a continuous variable lambda over the space of all hidden variables as the measure, there are others that could have been used.

lambda is not a probability measure ( is).

Therefore there is no physical content in equation (2) that distinguishes local theories from non-local ones. The expectation value of a paired-product, is calculated exactly the same way (multiply the results together take the probability weighted average over any valid measure), irrespective of whether the results are produced in a local manner or a non-local manner.

The RHS integral in (2) presupposes that a and b are kept fixed in the integration, this means that neither lambda nor rho can depend on a or b, in addition to the requirement that A does not depend on b and B does not depend on a. That is also what we mean by a local realistic theory. If (2) instead was e.g.



or



the proof would not go through.

[minkwe]

lambda is not a probability measure ( is).

, and so is , as I explained.

The RHS integral in (2) presupposes that a and b are kept fixed in the integration, this means that neither lambda nor rho can depend on a or b, in addition to the requirement that A does not depend on b and B does not depend on a. That is also what we mean by a local realistic theory. If (2) instead was e.g.



or



the proof would not go through.

"a" and "b" are not variables on the RHS of equation (2), there kept fixed as you say yourself. They are labels. There is absolutely nothing wrong with


and even if you still see any problem in the above integral, there should be absolutely no complain about
, where is a non-local hidden variable.

Therefore contrary to your claim, the integral is just fine for non-local hidden variables. Note, the integral is simply representing the probability weighted average of the product of Alice and Bob's outcomes, which is easily done for any theory, local or non-local, so long as outcomes are present.

[Heinera]

"a" and "b" are not variables on the RHS of equation (2), there kept fixed as you say yourself. They are labels. There is absolutely nothing wrong with


and even if you still see any problem in the above integral, there should be absolutely no complain about
, where is a non-local hidden variable.

Therefore contrary to your claim, the integral is just fine for non-local hidden variables. Note, the integral is simply representing the probability weighted average of the product of Alice and Bob's outcomes, which is easily done for any theory, local or non-local, so long as outcomes are present.

The expressions you wrote can be converted to the form



by a suitable choice of . Just ensure that if . It is easy to show that Bell's proof no longer works in that case.

[AnotherGuest]

There is absolutely nothing wrong with

and even if you still see any problem in the above integral, there should be absolutely no complain about
, where is a non-local hidden variable.

What is important is that there is a fixed set over which these integrals are taken.

Minkwe is right that it doesn't matter (for Bell's subsequent analysis) what *name* you give to the variable of integration, as long as it varies over a fixed set (fixed means: not depending on a or b).

And Minkwe is right that it doesn't matter (for Bell's subsequent analysis) whether you *think* of the variable as being local or non-local. The only thing that is essential for Bell's subsequent analysis, is that

where the functions A and B take values in {-1, +1} and where rho is a probability density on Gamma.

The set Gamma can be anything you like (quaternions, octonions, ... ; as fancy as you wish), and you may think about it, physically, in any way you like.

Moreover, Minkwe is right that the assumptions Bell makes ensure that A(a, gamma) = - B(a, gamma) for all a and gamma.

[Heinera]

There is absolutely nothing wrong with

and even if you still see any problem in the above integral, there should be absolutely no complain about
, where is a non-local hidden variable.

What is important is that there is a fixed set over which these integrals are taken.

Minkwe is right that it doesn't matter (for Bell's subsequent analysis) what *name* you give to the variable of integration, as long as it varies over a fixed set (fixed means: not depending on a or b).
.

But since the intregation variable now depends on a and b, the integration is no longer over a fixed set, but over a subset that depends on a and b. This is equivalent to saying that the probability measure depends on a and b.

If the hidden variable depends on a and b, you can just as well say that the source (which produces the hidden variable) knows about a and b, and adjusts its distribution of generated lambdas accordingly. Bell's proof does not work in such a case.

[minkwe]

But since the intregation variable now depends on a and b, the integration is no longer over a fixed set, but over a subset that depends on a and b.

Again, does not depend on "a" and "b", if you've been following what exacly represents.
Secondly, "a" and "b" are not variables on the RHS of equation (2). They are fixed, they do not vary. The notation was a poor choice on Bell's part. It should have been something like
Thirdly, I've given you other valid probability measures which do not involve any hidden variables, local or non-local, with expressions to calculate the same expectation value. The point being that, there is nothing in equation (2) that should be unique for local hidden variables. It is simply a weighted average of Alice's outcome multiplied by the corresponding Bob's outcome for all the unique possibilities in any measure. In experiments, which some nowadays claim as evidence of non-locality, such averages are routinely calculated without fanfare. I've not see any complains that their calculations are wrong.

There is no physical content in equation (2), it is a mathematical expression for the expectation value of the product of two outcomes irrespective of whether those outcomes are obtained locally or not.

Are you suggesting that Bell's derivation will fail to proceed if we replaced equation (2) with:

, or

This expression is theory-agnostic.

[Heinera]

Are you suggesting that Bell's derivation will fail to proceed if we replaced equation (2) with:

, or

Since there is no nor any in that expression, Bell's derivation would obviously fail to proceed.

[AnotherGuest]

There is absolutely nothing wrong with

and even if you still see any problem in the above integral, there should be absolutely no complain about
, where is a non-local hidden variable.

What is important is that there is a fixed set over which these integrals are taken.

Minkwe is right that it doesn't matter (for Bell's subsequent analysis) what *name* you give to the variable of integration, as long as it varies over a fixed set (fixed means: not depending on a or b).
.

But since the intregation variable now depends on a and b, the integration is no longer over a fixed set, but over a subset that depends on a and b. This is equivalent to saying that the probability measure depends on a and b.

If the hidden variable depends on a and b, you can just as well say that the source (which produces the hidden variable) knows about a and b, and adjusts its distribution of generated lambdas accordingly. Bell's proof does not work in such a case.

As far as I know mathematics, if is some function defined on a set , then in the expression you can give the variable any other name you like, as long as it is not simultaneously in use for something else. You can give it the name if you like. So you could also write if you want to. Nothing has changed. What's in a name? The important thing is that the function is the same, and the set over which we integrate, , is the same. One could also just write . It stands for the same thing.

[minkwe]

Are you suggesting that Bell's derivation will fail to proceed if we replaced equation (2) with:

, or

Since there is no nor any in that expression, Bell's derivation would obviously fail to proceed.

Needless to say, your statement is obviously and demonstrably false. Do you want me to derive Bell's inequalities from that expression or are you ready to withdraw your statement.

[Heinera]

Are you suggesting that Bell's derivation will fail to proceed if we replaced equation (2) with:

, or

Since there is no nor any in that expression, Bell's derivation would obviously fail to proceed.

Needless to say, your statement is obviously and demonstrably false. Do you want me to derive Bell's inequalities from that expression or are you ready to withdraw your statement.

Please go ahead.

[Joy Christian]

Needless to say, your statement is obviously and demonstrably false. Do you want me to derive Bell's inequalities from that expression or are you ready to withdraw your statement.

No need to waste your time. The trivial derivation can be found in Eqs. (11) to (14) of this paper: http://arxiv.org/abs/1501.03393.

[AnotherGuest]

Minkwe is right, yet again!

Take rho to be the probability measure putting equal mass 1/N on each i = 1, ..., N

So we define Lambda = {1, ..., N}

A^i_a = +/- 1 is now a function of i and a.

The "local hidden variable" lambda is now the index "i".

P_N(a, b) = 1/N sum A^i_a B^i_b is yet another instance of Bell's integral expression for P(a, b).

So you can derive Bell's inequality for it. Then let N go to infinity.

By the way, I don't see any difference at all between the notations and . I mean: you can use either, it is a matter of taste. Do you want to think of A as a function of two variables, or an indexed family of functions of one variable? You can think either way, it comes down to the same thing.

[minkwe]

Needless to say, your statement is obviously and demonstrably false. Do you want me to derive Bell's inequalities from that expression or are you ready to withdraw your statement.

No need to waste your time. The trivial derivation can be found in Eqs. (11) to (14) of this paper: http://arxiv.org/abs/1501.03393.

Thanks Joy, I was hoping Heine will flip a couple of pages to equations (13) to (15) of Bell's paper and verify that plays no role in the derivation of the inequality anyone can can follow along from Bells equation (13):



, Since
and after factorizing out , remembering that we get


The second term on the right is .
Do I need to continue? We now see that we get Bell's inequality without any physics whatsoever. All we need is theory-agnostic outcomes and the expectation value of the product of the paired outcomes , or as Bell would say . The inequality is therefore theory-agnostic and has nothing whatsoever to do with locality.

[minkwe]

Minkwe is right, yet again!

Thanks!