Talk:Bell's theorem

Bad ending to the section on Bell's 1964 paper.

The section on Bell's paper now ends with

• In other words, Bell's 1964 paper uses the EPR criterion to argue that local hidden variables exist, and then demonstrates that local hidden variables are incompatible with quantum mechanics.

This is not good. Bell is our hero in this article. Here he argues "that local hidden variables exist". That is not what Bell says. If we claim that it is what he said, we need a secondary reference for that claim. I tried a simple fix but @XOR'easter reverted it.

Also what is a " probability-1 prediction "? The reference says nothing like that. Johnjbarton (talk) 01:50, 15 February 2025 (UTC)

It's just paraphrasing the beginning of Bell's section 2: Since we can predict in advance the result of measuring any chosen component of ${\vec {\sigma }}_{2}$, by previously measuring the same component of ${\vec {\sigma }}_{1}$, it follows that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state. Perfect correlation, predicting with probability 1, it's all in there. (The EPR criterion takes "certainty" and "probability equal to unity" as synonymous.) Bell argues from the EPR criterion, a form of locality, and the perfect (anti)correlations of the spin singlet state that a local hidden variable $\lambda$ would have to exist, and then he shows that the predictions based on the assumption of $\lambda$ existing are inconsistent with quantum physics. XOR'easter (talk) 02:59, 15 February 2025 (UTC)

The section of Bell you quote starts out: "the EPR argument is the following". So it is EPR that argues that local hidden variables exist, not Bell. Bell demonstrated this is inconsistent with QM. Thus EPR was wrong. That is the take-away we should leave the reader. Johnjbarton (talk) 03:41, 15 February 2025 (UTC)

I changed "probability-1" to "completely certain" (I know I've read both in the literature, but the latter might be less jargon-y) and have made various other phrasing adjustments. XOR'easter (talk) 05:14, 15 February 2025 (UTC)

Thank you for adding the refs and changes. Let me see if I can convince you to go further.

The paragraph uses the term "EPR criterion" three times. This indirect phrasing obscures the content. Bell's paper uses it zero times. The first paragraph of the section of Homes/Selleri on Bell's paper describes the EPR paper in terms of hidden variables and reality across distant measurements. Similarly Clauser/Shimony describe hidden variables and locality. Our description of Bell's paper using the indirect phrasing "EPR criterion" to describe Bell's work means we need to make this equivalence clear.

Assuming you don't want to reduce or seperate the "EPR criterion" from that last paragraph, I made a change in the wording of the sentence to first establish the equivalence before giving Bells QM result. Johnjbarton (talk) 00:00, 16 February 2025 (UTC)

He then demonstrates that this cannot hold true in general.

The first two sentences of "Bell (1964)" says:

• Bell's 1964 paper points out that under restricted conditions, local hidden-variable models can reproduce the predictions of quantum mechanics. He then demonstrates that this cannot hold true in general.

In my opinion a reader will parse this as "Bell theorem cannot hold in general". It's too hard to sort out what "this" means in the second sentence.

More important this is inappropriately sourced to Bell's paper. Nothing in Bell's paper makes his case in this way. These two sentences are unsourced analysis. Johnjbarton (talk) 00:12, 16 February 2025 (UTC)

I don't see why a reader would be likely to make that mistake. What else could "this" refer to, except "local hidden-variable models [reproducing] the predictions of quantum mechanics." I'm open to rephrasing the line, but I don't see why it would be necessary, strictly speaking.

See Bell's section 3. Firstly, there is no difficulty in giving a hidden variable account of spin measurements on a single particle. [...] So in this simple case there is no difficulty in the view that the result of every measurement is determined by the value of an extra variable, and that the statistical features of quantum mechanics arise because the value of this variable is unknown in individual instances. And then: Secondly, there is no difficulty in reproducing [...] the only features of [the quantum correlations] commonly used in verbal discussions of this problem. He says, in considerable detail, how to build a hidden-variable model for a single two-level system, and how to build a local hidden-variable model for the cases where the two detectors are oriented exactly parallel, exactly anti-parallel, or exactly orthogonal. And he does all this before his main result, the theorem he proves in section 4. And this isn't forgotten history or anything like that. For example, Mermin's review article "Hidden variables and the two theorems of John Bell" gives a detailed treatment of Bell's LHV model for a single qubit at the end of section 3. Max Jammer's Philosophy of Quantum Mechanics (1974) notes that Bell's doubts about the relevance—not mathematical correctness!—of von Neumann's proof for the problem of hidden variables in quantum mechanics were raised by the very fact that he succeeded in constructing a consistent hidden variable theory for particles of spin-1/2 without translational motion (pp. 303–304). Jammer then spends the better part of a page going into detail. The LHV model for a two-level or spin-1/2 system was important enough for Bell that he used it in both the 1964 paper and the one that got delayed to 1966. XOR'easter (talk) 00:29, 16 February 2025 (UTC)

Thank you. As usual Mermin is perfectly clear. I added Mermin's paper as ref and rephrased the sentence in a way that I think is clearer. Johnjbarton (talk) 01:53, 16 February 2025 (UTC)