## Bell-test experiments rule out additivity of expectations

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

### Bell-test experiments rule out additivity of expectations

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This thread is about my claim that the Bell-test experiments do not rule out local realism but Bell's implicit assumption of the additivity of expectation values in the proof of his theorem.

I have explained this claim in considerable detail in one of my preprints on the arXiv: https://arxiv.org/pdf/1704.02876.pdf. Let me quote an essential paragraph from the preprint:

The highlighted part is what was recalled by Peter G. Bergmann, who was Einstein's assistant (i.e., his postdoc), and reported by Abner Shimony (my Ph.D. mentor) in one of his papers.

Thus the flaw in Bell's theorem was pointed out by Einstein in 1938 when Bell was hardly ten years old. It is so basic that it is mindboggling why it has not been recognized before.

The flaw has to do with the assumption of the additivity of expectation values in any derivation of a Bell-type inequality. That assumption is illegal for hidden variable theories of any kind, as pointed out by Einstein in the mid-1930s, and independently by Bell himself just before 1964, in the context of von Neumann's theorem against hidden variables. Just because the additivity of expectation values follows by following mathematics does not mean that it is meaningful physically. Einstein saw that clearly, and so did Bell in the above-mentioned context. Consequently, what is ruled out by the Bell-test experiments is not local realism but additivity of expectation values --- which is not permissible for hidden variable theories to begin with.

To understand Einstein's point, let r and s be the eigenvalues of the QM operators R and S, respectively. Then the eigenvalue of the operator R + S is not r + s if R and S do not commute.

Let us say the eigenvalue of the operator R + S is t when R and S do not commute, where t is not equal to r + s.

Then the hidden variable counterpart of the QM equation < R > + < S > = < R + S > is not < r > + < s > = < r + s >. It is < r > + < s > = < t >.

To not recognize this within his theorem is Bell's mistake, as I have explained in considerable detail in my paper: https://arxiv.org/pdf/1704.02876.pdf.

If you do not have time to read my paper, then here is the upshot of it. Bell's derivation of his inequalities involves the additivity of expectation values:

< r > + < s > = < r + s >.

This is an assumption. Whether it is a justified assumption or not is irrelevant. What matters is that without this assumption the derivation of Bell inequalities does not go through.

Now we do the experiments and discover that the bounds of +/-2 on the Bell-CHSH inequalities are exceeded. The only rational conclusion from that is to conclude that the assumption of the additivity of expectation values is ruled out by the experiments. Anything else is pure speculation. That is my argument in a nutshell.

Here is a nice picture of Peter G. Bergmann (the man in the black suit), taken at an old gathering at Osgood Hill, North Andover, Massachusetts. I am kneeling down, second from your left.

Since everything I say is subject to challenge and doubt among some proponents of Bell's theorem, here is proof that I did collaborate with Abner Shimony, my Ph.D. mentor, and Peter G. Bergmann did collaborate with Albert Einstein, his post-doctoral mentor: https://mathscinet.ams.org/mathscinet/f ... ?version=2.

And here is a nice picture of John S. Bell and Abner Shimony. It was taken by me during a historic conference on the foundations of quantum physics, held on Mount Erice, Sicily, 1989.

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Joy Christian
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### Re: Bell-test experiments rule out additivity of expectation

Nice picture! You told the place but not the year. It would be interesting to know the year.
I do not respond to your claim on Bell's mistake because we already discussed it in another thread and we both made our points clear.
I can only say that if your claim is correct it is a big issue, at least for the field of quantum foundations and I do not know what it would mean for the field of quantum computation.
Correct or not, I think your paper is publishable. On the other hand, I have seen obviously erroneous papers on the Bell theorem published by authentic peer-review journals. It seems that if for some reason the correct sympathetic reviewers are assigned to you, you get your manuscript published.
For example, I now a case of researchers that in 1972 claimed the Bell theorem is incorrect because it implies incompatible experiments then, in 2020, they published another article issuing the same claim but in a different form and I am not talking about open access.
Juso

### Re: Bell-test experiments rule out additivity of expectation

Juso wrote:
Nice picture! You told the place but not the year. It would be interesting to know the year.

Thanks!

The first picture was taken in May 1986, during a conference on the history of General Relativity, held at a Boston University conference center, located at Osgood Hill, North Andover, Massachusetts. Notable attendees were Abhay Ashtekar, Peter Bergmann, John Stachel, John Earman, John Norton, and Peter Havas (among many others).

The second picture, of Bell and Shimony, was taken by me during a historic conference on the foundations of quantum physics, held at Mount Erice, Sicily, in August 1989.
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Joy Christian
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### Re: Bell-test experiments rule out additivity of expectation

Lovely pictures indeed! However, Einstein's argument has some small print which Joy has overlooked.

"Einstein then said that there is no reason why this premise should hold in a state not acknowledged by quantum mechanics if R, S etc. are not simultaneously measurable".

Einstein implicitly confirms that according to quantum mechanics, it does hold under a quantum mechanical state. It holds when we prepare a state in a state rho, one of the states acknowledged by quantum mechanics!

The Bell-test experiments confirm quantum mechanics and moreover generate results which cannot be explained by local realism. They certainly do not rule out additivity of expectation values within quantum mechanics. On the contrary the fact that they confirm quantum mechanics confirms the additivity of expectation values in quantum mechanics even of non-commuting observables. Hence any local realistic model which can mimic QM predictions must also reproduce the constraints of linearity *coming from quantum mechanics*.

It is certainly worth writing a paper on all this, perhaps together?

PS Joy writes "everything I say is subject to challenge and doubt among some proponents of Bell's theorem...". But not everything he says is challenged or doubted! Only the statements which are obviously untrue are challenged. Which is as it should be. It does not entail disrespect.
gill1109
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### Re: Bell-test experiments rule out additivity of expectation

gill1109 wrote:
Einstein implicitly confirms that according to quantum mechanics, it does hold under a quantum mechanical state.

It is universally agreed, by all parties involved, past and present, that the additivity of expectation values holds within quantum mechanics. But it does not hold for dispersion-free states:

John S. Bell wrote:
[von Neumann's] essential assumption is: Any real linear combination of expectation values ... is the expectation value of the combination. This is true for quantum mechanical states; it is required by von Neumann of the hypothetical dispersion free states also. ... But for a dispersion free state (which has no statistical character) the expectation value of an observable must equal one of its eigenvalues. ... The essential assumption [of von Neumann] can be criticized as follows. ... (from Section 3 of Chapter 1 of Bell's book).

Evidently, you have not understood the Einstein-Bell argument against the additivity of expectation values within hidden variable theories. I also doubt that you have read my linked paper.
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Joy Christian
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### Re: Bell-test experiments rule out additivity of expectation

Joy Christian wrote:
gill1109 wrote:
Einstein implicitly confirms that according to quantum mechanics, it does hold under a quantum mechanical state.

It is universally agreed, by all parties involved, past and present, that the additivity of expectation values holds within quantum mechanics. But it does not hold for dispersion-free states:

John S. Bell wrote:
[von Neumann's] essential assumption is: Any real linear combination of expectation values ... is the expectation value of the combination. This is true for quantum mechanical states; it is required by von Neumann of the hypothetical dispersion free states also. ... But for a dispersion free state (which has no statistical character) the expectation value of an observable must equal one of its eigenvalues. ... The essential assumption [of von Neumann] can be criticized as follows. ... (from Section 3 of Chapter 1 of Bell's book).

Evidently, you have not understood the Einstein-Bell argument against the additivity of expectation values within hidden variable theories. I also doubt that you have read my linked paper.
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I have read all your papers, Joy. They are a gold mine.

I understand Einstein’s argument. You still have not answered mine. There is no contradiction. My argument used only the additivity within quantum mechanics. Which Einstein agreed with. Please think carefully about the argument I gave; don’t jump to conclusions.
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### Re: Bell-test experiments rule out additivity of expectation

gill1109 wrote:
I understand Einstein’s argument. You still have not answered mine. There is no contradiction. My argument used only the additivity within quantum mechanics. Which Einstein agreed with. Please think carefully about the argument I gave; don’t jump to conclusions.

Ok. Let me respond to Gill's argument in more detail. Here is his argument, which I reproduce from the other thread:

gill1109 wrote:
Consider three observables A, B and C such that C = A + B. Suppose none commute with one another. There are three separate experiments which allow one to experimentally determine <A>, <B>, and <C>, each by averaging many measurements of A, B or C on systems prepared in the same state rho. One will discover that <C> = <A> + <B>.

This is because <A> = trace(rho A), <B> = trace(rho B), <C> = trace(rho C), “trace” is linear.

A hidden variable model for these experiments is a classical probability space with three random variables X, Y and Z defined on it, such that the probability distributions of the three random variables exactly reproduce the probability distributions of the outcomes of measurements of the three observables. In particular, their mean values are the same. Hence E(X) = <A>, E(Y) = <B>, E(Z) = <C>. Hence E(X) + E(Y) = E(Z).

The additivity follows from the linearity of the trace operator. The hidden variable model by definition mimics observable features of the quantum model (probabilities, expectation values...).

The mistake in the above argument is obvious, and, as I noted in the other thread, it is precisely the one that was pointed out by Einstein in the mid-1930s, and later by Bell and others.

The mistake is that, while C = A + B by construction, Z is not equal to X + Y, if X, Y, and Z are the eigenvalues of A, B, and C, respectively, as they must be in any hidden variable theory:

John S. Bell wrote:
[von Neumann's] essential assumption is: Any real linear combination of expectation values ... is the expectation value of the combination. This is true for quantum mechanical states; it is required by von Neumann of the hypothetical dispersion free states also. ... But for a dispersion free state (which has no statistical character) the expectation value of an observable must equal one of its eigenvalues. ... The essential assumption [of von Neumann] can be criticized as follows. ... (from Section 3 of Chapter 1 of Bell's book).
Joy Christian
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### Re: Bell-test experiments rule out additivity of expectation

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Joy Christian
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### Re: Bell-test experiments rule out additivity of expectation

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I have received a decision email from a journal concerning my paper I have liked above: https://arxiv.org/pdf/1704.02876.pdf.

The paper has been rejected based on two reviewer reports. The first reviewer recommends outright rejection based on an argument identical to the one Justo has put forward in this forum. I do not agree with that argument and might consider fighting back.

The second reviewer has enthusiastically recommended the paper for publication with very positive comments: "The manuscript by Joy Christian is a virtuoso performance and tells us much about the possible analogies and failures of von Neuman's and Bell's no-go proofs. I find Christian's logic convincing and definitely recommend that this work be published." The report then goes on to make some minor criticisms and concludes: "Besides all these minor criticisms, Christian's paper is excellent and definitely should be published."

The paper is nevertheless rejected on the grounds that the journal's "current publication program is not well suited for it, and must regretfully decline your offer to let us publish it."
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Joy Christian
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### Re: Bell-test experiments rule out additivity of expectation

Joy Christian wrote:.
I have received a decision email from a journal concerning my paper I have liked above: https://arxiv.org/pdf/1704.02876.pdf.

The paper has been rejected based on two reviewer reports. The first reviewer recommends outright rejection based on an argument identical to the one Justo has put forward in this forum. I do not agree with that argument and might consider fighting back.

The second reviewer has enthusiastically recommended the paper for publication with very positive comments: "The manuscript by Joy Christian is a virtuoso performance and tells us much about the possible analogies and failures of von Neuman's and Bell's no-go proofs. I find Christian's logic convincing and definitely recommend that this work be published." The report then goes on to make some minor criticisms and concludes: "Besides all these minor criticisms, Christian's paper is excellent and definitely should be published."

The paper is nevertheless rejected on the grounds that the journal's "current publication program is not well suited for it, and must regretfully decline your offer to let us publish it."
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I regret to hear that. According to what you said, I would agree with the first reviewer but I think that the fair thing to do in these cases is a third review. However, editors have the right to make decisions according to their judgment.

If your paper is published, I am willing to write a comment on it explaining why I believe is incorrect. But it would not be for personal reasons, it would be just to discuss ideas.
Justo

### Re: Bell-test experiments rule out additivity of expectation

Justo wrote:
If your paper is published, I am willing to write a comment on it explaining why I believe is incorrect. But it would not be for personal reasons, it would be just to discuss ideas.

Sure. But regardless of whether my paper is published, Bell's theorem has long been rendered irrelevant for physics, because any claim that no local-realistic theory can reproduce all of the predictions of quantum mechanics is demonstrably wrong. I have presented a strictly local-realistic model of all quantum correlations in my Royal Society paper, supported by three other papers that specifically discuss the singlet correlations. Here are the references to all four of my papers in case anyone is not familiar with my work on quantum correlations:

(1) Quantum correlations are weaved by the spinors of the Euclidean primitives, Royal Society Open Science, https://royalsocietypublishing.org/doi/ ... sos.180526 (2018),

(2) Macroscopic observability of spinorial sign changes under 2pi rotations, International Journal of Theoretical Physics, https://link.springer.com/article/10.10 ... 014-2412-2 (2015),

(3) Bell's theorem versus local realism in a quaternionic model of physical space, IEEE Access, https://ieeexplore.ieee.org/document/8836453 (2019),

(4) Dr. Bertlmann's socks in the quaternionic world of ambidextral reality, IEEE Access, https://ieeexplore.ieee.org/document/9226414 (2020).

Note also that claimed criticisms of my model exist, especially by Gill. But I do not take Gill's criticisms seriously. Although I do have to respond to some of them for sociological reasons.
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Joy Christian
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### Re: Bell-test experiments rule out additivity of expectation

Joy Christian wrote:.
I have received a decision email from a journal concerning my paper I have liked above: https://arxiv.org/pdf/1704.02876.pdf.

The paper has been rejected based on two reviewer reports. The first reviewer recommends outright rejection based on an argument identical to the one Justo has put forward in this forum. I do not agree with that argument and might consider fighting back.

The second reviewer has enthusiastically recommended the paper for publication with very positive comments: "The manuscript by Joy Christian is a virtuoso performance and tells us much about the possible analogies and failures of von Neuman's and Bell's no-go proofs. I find Christian's logic convincing and definitely recommend that this work be published." The report then goes on to make some minor criticisms and concludes: "Besides all these minor criticisms, Christian's paper is excellent and definitely should be published."

The paper is nevertheless rejected on the grounds that the journal's "current publication program is not well suited for it, and must regretfully decline your offer to let us publish it."
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Why did they send it out for review if it wasn't well suited for the Journal. This is highly unusual. The editor should reject it outright if that is the case. Are the review reports confidential or can you post it here for us to shred?
minkwe

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### Re: Bell-test experiments rule out additivity of expectation

minkwe wrote:
Joy Christian wrote:.
I have received a decision email from a journal concerning my paper I have liked above: https://arxiv.org/pdf/1704.02876.pdf.

The paper has been rejected based on two reviewer reports. The first reviewer recommends outright rejection based on an argument identical to the one Justo has put forward in this forum. I do not agree with that argument and might consider fighting back.

The second reviewer has enthusiastically recommended the paper for publication with very positive comments: "The manuscript by Joy Christian is a virtuoso performance and tells us much about the possible analogies and failures of von Neuman's and Bell's no-go proofs. I find Christian's logic convincing and definitely recommend that this work be published." The report then goes on to make some minor criticisms and concludes: "Besides all these minor criticisms, Christian's paper is excellent and definitely should be published."

The paper is nevertheless rejected on the grounds that the journal's "current publication program is not well suited for it, and must regretfully decline your offer to let us publish it."
.

Why did they send it out for review if it wasn't well suited for the Journal. This is highly unusual. The editor should reject it outright if that is the case.

The thought occurred to me as well. They also say that the basis for their decision is the referee reports. So it is all a bit confusing. Unfortunately, journal editors have a lot of powers.

minkwe wrote:
Are the review reports confidential or can you post it here for us to shred?

Review reports are always confidential. By posting the information that I have posted is already a violation of some unwritten rules of publication ethics. Moreover, I have decided to appeal the decision because I do not agree with the comments by the first reviewer. It is worth addressing them if only to refute the argument head-on. So, for now, I cannot post the reports here.
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Joy Christian
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### Re: Bell-test experiments rule out additivity of expectation

Joy Christian wrote:Review reports are always confidential. By posting the information that I have posted is already a violation of some unwritten rules of publication ethics. Moreover, I have decided to appeal the decision because I do not agree with the comments by the first reviewer. It is worth addressing them if only to refute the argument head-on. So, for now, I cannot post the reports here.

There is a movement to have review reports published: https://www.nature.com/articles/d41586-020-00309-9. The process is badly in need of more transparency.
minkwe

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### Re: Bell-test experiments rule out additivity of expectation

Joy Christian wrote:
gill1109 wrote:
I understand Einstein’s argument. You still have not answered mine. There is no contradiction. My argument used only the additivity within quantum mechanics. Which Einstein agreed with. Please think carefully about the argument I gave; don’t jump to conclusions.

Ok. Let me respond to Gill's argument in more detail. Here is his argument, which I reproduce from the other thread:

gill1109 wrote:
Consider three observables A, B and C such that C = A + B. Suppose none commute with one another. There are three separate experiments which allow one to experimentally determine <A>, <B>, and <C>, each by averaging many measurements of A, B or C on systems prepared in the same state rho. One will discover that <C> = <A> + <B>.

This is because <A> = trace(rho A), <B> = trace(rho B), <C> = trace(rho C), “trace” is linear.

A hidden variable model for these experiments is a classical probability space with three random variables X, Y and Z defined on it, such that the probability distributions of the three random variables exactly reproduce the probability distributions of the outcomes of measurements of the three observables. In particular, their mean values are the same. Hence E(X) = <A>, E(Y) = <B>, E(Z) = <C>. Hence E(X) + E(Y) = E(Z).

The additivity follows from the linearity of the trace operator. The hidden variable model by definition mimics observable features of the quantum model (probabilities, expectation values...).

The mistake in the above argument is obvious, and, as I noted in the other thread, it is precisely the one that was pointed out by Einstein in the mid-1930s, and later by Bell and others.

The mistake is that, while C = A + B by construction, Z is not equal to X + Y, if X, Y, and Z are the eigenvalues of A, B, and C, respectively, as they must be in any hidden variable theory:

John S. Bell wrote:
[von Neumann's] essential assumption is: Any real linear combination of expectation values ... is the expectation value of the combination. This is true for quantum mechanical states; it is required by von Neumann of the hypothetical dispersion free states also. ... But for a dispersion free state (which has no statistical character) the expectation value of an observable must equal one of its eigenvalues. ... The essential assumption [of von Neumann] can be criticized as follows. ... (from Section 3 of Chapter 1 of Bell's book).

There is no reason for X + Y = Z to hold in the presumed underlying dispersion-free states in a possibly contextual hidden variables theory. The Kochen-Specker theorem on non-contextual hidden variables models is less interesting, more subtle, and its proof more complex, than Bell’s theorem. Bell’s theorem is about locally-contextual hidden variables.

Over the years I have found Joy’s work a great inspiration! It has taught me a lot and led to discoveries of which I am very proud. I only hope that some time we will come to a sufficient alignment concerning understanding of some basic mathematical facts that we could write a paper together. And Fred and Michel can join in too.

The key word is “dispersion-free”. Bell means, to use statistical language, conditional on the values of the underlying hidden variables.

In general, a hidden variables theory only has to reproduce observable, statistical predictions of QM. Anything more is an inessential luxury. Von Neuman’s “mistake” was to impose more. He could have had physical reasons to want that (and this has been argued by some experts in the field). But there are is no mathematical necessity.
gill1109
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### Re: Bell-test experiments rule out additivity of expectation

gill1109 wrote:
There is no reason for X + Y = Z to hold in the presumed underlying dispersion-free states in a possibly contextual hidden variables theory.

There is no reason to assume Z = X + Y in a local contextual hidden variable theory. And yet, Bell explicitly assumes Z = X + Y in such a theory. Indeed, if he does not assume Z = X + Y so that

E(X) + E(Y) =/= E(X + Y),

then the proof of his theorem does not go through. That is the argument in my paper. Consequently, what is ruled out by Bell-test experiments is the assumption E(X) + E(Y) = E(X + Y). QED.
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### Re: Bell-test experiments rule out additivity of expectation

Some people seem to be hung up on the idea that Joy used QM in his paper which Bell did not but let me ask this then:
Is Bell's theorem about a demonstration that when you substitute QM expectation values into Bell's formula you get a result which exceeds the limits Bell claims should exist? Doesn't this completely defeat the suggestion that no QM is involved in Bell's analysis? Otherwise could anyone who believes that explain to us how you can demonstrate that QM "violates" Bell's inequality without using QM.
Let us start from the CHSH:

$S_{LR}=E(a,b)-E\left(a,b'\right)+E\left(a',b\right)+E\left(a',b'\right)$

Now image that point at which Bell has substituted the expectations from QM into the CHSH and we now have

$S_{QM}=E_{QM}(a,b)-E_{QM}\left(a,b'\right)+E_{QM}\left(a',b\right)+E_{QM}\left(a',b'\right)$

What is this quantity $S_{QM}$ that is claimed to violate $S_{LR}$?

Bell's criticism of Von Neumann was summarized by Bell as follows:
It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made.

In the context of the EPR-B experiment, the relationship $S_{LR}$ is an impossible relation between results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made. Whereas, $S_{QM}$ is an objectively measurable prediction of quantum mechanics. To borrow from the CFD thread, in the context of the coin-reading machine, it was not the objectively measurable outcomes of the coin-reading machine $P(H)_1 + P(T)_2 = 1.5$ that violated anything. it was an impossible relation $P(H)_1 + P(T)_1 = 1$ between results of incompatible measurements either of which might be made on a given occasion but only one of which can, in fact, be made.

Therefore Joy is absolutely right. Bell's criticism of Von Neumann applies to his own theorem also.
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### Re: Bell-test experiments rule out additivity of expectation

gill1109 wrote:The key word is “dispersion-free”. Bell means, to use statistical language, conditional on the values of the underlying hidden variables.

By dispersion-free, Bell simply means deterministic.
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### Re: Bell-test experiments rule out additivity of expectation

minkwe wrote:
gill1109 wrote:The key word is “dispersion-free”. Bell means, to use statistical language, conditional on the values of the underlying hidden variables.

By dispersion-free, Bell simply means deterministic.

He means more than deterministic. He means that after you have conditioned on the values taken by the hidden variables in any particular case, everything else becomes deterministic.

In probability theory, a random variable X is modelled by a deterministic function X(omega) of an underlying sample point omega in a sample space Omega of "elementary outcomes". Physicists prefer the name lambda to the usual Kolmogorovian name omega.
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### Re: Bell-test experiments rule out additivity of expectation

gill1109 wrote:He means more than deterministic. He means that after you have conditioned on the values taken by the hidden variables in any particular case, everything else becomes deterministic.

That's what deterministic means.
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