Kupczynski states "Counterfactual expectations E(A_x A_x′), E(B_y B_y′), E(A_x A_x′ B_y B_y′) do not exist and Bell and CHSH inequalities may not be derived". This is misleading. Those expectation values do not exist within his construction of four separate probability models for the four sub-experiments corresponding to each of the four setting pairs. But Kupczynski's conclusion is false. The Bell-CHSH inequalities do hold. The proof is as follows. One constructs a probabilistic coupling of Kupczynski's four models, embedding them in one space, on which the four random variables A_x, A_x', B_y, B_y' are all defined simultaneously, with exactly the same probability distributions as before. The Bell-CHSH inequality therefore does hold for the four correlations of interest to Kupczynski, contrary to his claim that it need not hold.
Many couplings are possible, here is one. Take as sample space a set of tuples "new λ" = (λ_1, λ_2, λ_x, λ_x′, λ_y, λ_y′). This space is just the Cartesian product of the spaces whose existence Kupczynski already hypothesized. Take as probability mass function on this space the product new "new p(λ)" = p(λ_1, λ_2) p_x(λ_x) p_x′(λ_x′) p_y(λ_y) p_y′(λ_y′). Finally, define new measurement functions "new A(λ, x)" = A_x(λ_1, λ_x), "new B(λ, y)" = B_y(λ_2, λ_y) where x can be replaced by x′ and/or y by y′. Now compute E(A_xB_y), also with x replaced by x′ and/or y by y′. It is immediately clear that the four new expectation values of products are identical to those derived in Kupczynski’s framework.
I think this article makes too much of nonlocality and too little of noncommutativity (to say that slightly differently, incompatible measurements). Cirel'son's 1980 paper (this paper's Ref. [92]) and Fine's 1982 papers (this paper's Ref. [18,19]) originated the suggestion that the issue is commutativity, without any need to invoke nonlocality, however I think those three articles are mathematically much less clear than Landau's paper (this paper's Ref. [93]). It might be helpful to mention in future papers that the basic idea that we can prove that there are some pairs of probability measures that do not admit a joint probability measure that has that pair as marginals goes back to Boole in the mid-19th Century, which I learned of from Pitowsky 1995 ([https://doi.org/10.1093/bjps/45.1.95]). Other no-go theorems also depend essentially on whether the measurement algebra is noncommutative, particularly Gleason or Kochen-Specker, without any direct reference needed to nonlocality. To me, nonlocality is closely but only indirectly associated with noncommutativity because of microcausality in QFT, which explicitly requires noncommutativity only at time-like separation (a relationship that is not present in nonrelativistic QM). I suggest that the common worry that nonlocality in classical physics is problematic should not be taken to be too worrying because nonlocal correlations at space-like separation are a commonplace in thermal states of classical field theories (as I argue in a paper in JPhysA 2006, [https://doi.org/10.1088/0305-4470/39/23/018]).
Well, Boole (1854) has a general methodology, one could even say he presents an algorithm, for obtaining necessary and sufficient conditions for existence of couplings, as they are nowadays called by probabilists. With hard work and some creativity one can recognise the original Bell (1964) three-correlations inequalities as part of an exercise for the reader in Boole's very, very fat text-book. Nobody ever wrote an "instructor's manual". It is not clear to me that Pitowsky himself ever worked out the details. So his remark is a nice display of erudition but not much more than that. I suspect I am one of very few people who ever checked that one can indeed go from Boole to Bell in this way. It requires a lot of patience and a lot of careful reading. Nowadays we have convexity theory and linear programming and many tools by which these things can be done with much less sweat and tears.
#1 Hidden variables (λx , λy, λx’, λy’) describe states of measuring devices at the moment of their interaction with the photonic beams in different experimental settings. As we explained on the page 5 of our article : A joint probability distribution of all possible hidden events (λx , λ1, λy, , λ2 , λx’ , λy’, λ2)does not exist because hidden events (λx , λx’ )and ( λy , λy’) may never occur together. This is why one may not prove CHSH assuming the existence of such probability distribution and a non-vanishing E(Ax Ax’ By By’) ,used to prove (2-3,8), does not exist.
A construction proposed in the comment is artificial and is devoid of any physical justification. On paper one may define many different probabilistic couplings. A simplest intuitive probabilistic coupling was implicitly defined by Bell and used to prove his inequalities. Such coupling was inconsistent with contextual character quantum observables represented by non-commuting operators and with the experimental data in Bell Tests. In fact the data in Bell Tests proved that such coupling was impossible. Since many random variables describing these data are inconsistently connected they should be discussed using Contextuality by Default framework . More details my be found in: Kupczynski, M.: Contextuality-by-Default Description of Bell Tests: Contextuality as the Rule and Not as an Exception. Entropy 2021, 23(9), 1104; https://doi.org/10.3390/e23091104: Kupczynski, M.: Comment on Causal Networks and Freedom of Choice in Bell’s Theorem. International Journal of Quantum Foundations, Volume 8, Issue 2, pages 117-124 (2022) , https://doi.org https://doi.org /10.48550/arXiv.2201.08483; Kupczynski M:A comment on Bell's Theorem Logical Consistency, arXiv:2202.09639 [quant-ph]: https://doi.org/10.48550/arXiv.2202.09639
#2 I did not realize that I talked too much about nonlocality. Perhaps you are right, but reading what is being claimed on blogs and in books addressed to a general public I tried to give convincing arguments that all speculations about quantum nonlocality based on EPRB experiment and on the violation of Bell inequalities are unfounded. Such speculations are are based on incorrect interpretation of QM or on incorrect mental images of quantum phenomena. See also: Khrennikov, A. Two Faced Janus of Quantum Nonlocality , Entropy 2020, 22(3), 303; https://doi.org/10.3390/e22030303
I agree with you that one should not worry about nonlocality neither in classical physics nor in quantum physics. QFT is explicitly local. Moreover the apparent violation of non-signaling in Bell Tests does not violate Einsteinian non-signaling: Kupczynski M. Is Einsteinian no-signaling violated in Bell tests? Open Physics, 2017, 15 , 739-753, DOI: https://doi.org/10.1515/phys-2017-0087,2017.
You say "A construction proposed in the comment is artificial and is devoid of any physical justification. On paper one may define many different probabilistic couplings. A simplest intuitive probabilistic coupling was implicitly defined by Bell and used to prove his inequalities. Such coupling was inconsistent with contextual character quantum observables represented by non-commuting operators and with the experimental data in Bell Tests. In fact the data in Bell Tests proved that such coupling was impossible. Since many random variables describing these data are inconsistently connected they should be discussed using Contextuality by Default framework".
A mathematician works on paper. Many couplings are mathematically possible. Whether or not they make physical sense is irrelevant: the paper we are discussing here makes clear mathematical claims which are wrong.
It is very important to get the reasoning clear and correct, and the mathematics transparent. The data on Bell tests does not prove that couplings are impossible in general. It might show that some experiments have worrisome imperfections. It is important to criticise the experiments in order to goad the experimenters to do better. In order to do that, it is important to use clear and correct reasoning and mathematics. They are smart people. They are not convinced by circular arguments.
It is fine that people explore the possibility to model these phenomena using probabilistic local causality (local hidden variables, local realism). But you cannot prove that probabilistic local causality is the correct description of nature by assuming that it is true.
Please take account of the saying "Das Wissen von heute ist der Irrtum von morgen". Keep an open mind.
You write "such coupling was inconsistent with contextual character quantum observables represented by non-commuting operators and with the experimental data in Bell Tests". Yes! That's exactly what Bell, and the experimental data, are telling us! It is strange that you are fighting Bell's theorem while exactly agreeing with what it tells us.
Together with Justo Pastor Lambare from Paraguay, we wrote up a critique of the Kupczynski "Frontiers in Physics" paper and have submitted it to the same journal. Now Marian can start up a PubPeer discussion of our paper, if he likes! Let the debate continue!
https://arxiv.org/abs/2208.09930
Kupczynski's contextual setting-dependent parameters offer no escape from Bell-CHSH
Richard D. Gill, Justo Pastor Lambare
In a sequence of papers, Marian Kupczynski has argued that Bell's theorem can be circumvented if one takes correct account of contextual setting-dependent parameters describing measuring instruments. We show that this is not true. Taking account of such parameters in the way he suggests, the Bell-CHSH inequality can still be derived. Violation thereof by quantum mechanics cannot be easily explained away: quantum mechanics and local realism (including Kupczynksi's expanded concept of local realism) are not compatible with one another. Further inspection shows that Kupczynski is actually falling back on the detection loophole. Since 2015, numerous loophole-free experiments have been performed, in which the Bell-CHSH inequality is violated, so despite any other possible imperfections of such experiments, Kupczynski's escape route for local realism is not available.
The arXiv preprint https://arxiv.org/abs/2208.09930 by Gill and Lambare has been submitted to Frontiers in Physics. That is why it is formatted using the LaTeX style files of that journal, as required for submission. Submission to the journal does not entail that a paper may not be placed on a preprint server or personal home page.
I asked for open review, ie, referee reports are also published. I hope the referees will agree to this too.
Bell in 1964 assumed perfect correlations at equal settings. He later dropped this assumption, and explicitly allowed hidden variables to be located anywhere in the past light cones of the points in space-time where settings are entered into the two measurement devices. This weaker assumption together with the assumption that the probability distribution of the hidden variables does not depend on the settings is exactly the assumption of probabilistic causality and no-conspiracy made by Kupsczynski. MK stated that “Bell-CHSH inequalities may not be derived” for his set-up, but we did derive them for his set-up. Therefore, their violation cannot be explained by his assumptions. The rest of his paper makes clear that he explains possible violation of CHSH by exploiting the detection loophole.
I too recommend readers of this PubPeer thread to read MK’s paper in its entirety before reading our preprint. Further background material would be the book of collected papers by John Bell, “Speakable and unspeakable in quantum mechanics”.
I agree with Kupczynski when he says that it is surprising that the empirical violations of the Bell inequalities lead to "speculate about the existence of non-local influences in nature and casts doubt on the existence of the objective external physical reality." It is fair to point out that John Bell never claimed that. What Bell claimed about his inequalities is that they prove a local hidden variable model, under reasonable assumptions, cannot reproduce the quantum correlations. Metaphysical speculations about realism, objective reality, etc. are not Bell's fault. The problem with Kupczynski's claims is that he goes far beyond that also claiming that the Bell inequality is a meaningless mathematical result describing impossible to perform experiments. When somebody makes such a blatant claim implying the incompetence of a large part of the scientific community and respected experimental physicists that purportedly wasted their careers trying to falsify a meaningless "impossible to perform" experiment, we must be suspicious of such claims.
Thank you for your remark. I never claimed that Bell's inequality is meaningless. I find John Bell's contribution to the discussions on the foundations of QM and beautiful experiments performed to test various inequalities very important. I met and discussed with him couple of times in 1976 and I respected him a lot . Probably he respected me also , because in 1982 he sent me a copy of his handwritten letter to Pitovsky. The only two other he sent this letter were Stapp and d'Espagnat
I would like to cite below few paragraphs from my recent paper:Kupczynski M:A comment on Bell's Theorem Logical Consistency, arXiv:2202.09639 [quant-ph]: https://doi.org/10.48550/arXiv.2202.09639 ; " Local predetermination of outcomes of experiments, by some ontic properties of signals, is called usually: local realism, classicality or counterfactual definiteness (CFD). Since different authors attach a different meaning to the notion of realism, thus CFD understood as local predetermination of outcomes is less ambiguous.
Such assumption was proven incorrect, but it was not stupid. Reinhold Bertlmann remembers, what his friend John said to him:” I’m a realist…I think that in actual daily practice all scientists are realists, they believe that the world is really there, that it is not a creation of their mind. They feel that there are things there to be discovered, not a world to be invented but a world to be discovered. So I think that realism is a natural position for a scientist and in this debate about the meaning of quantum mechanics I do not know any good arguments against realism.”[38].
Local realism understood as CFD, automatically implies MI and the existence of JP. Bell Theorem and its implications are now well understood and nobody questions Bell’s Theorem logical consistency. Bell inequalities are violated in various Bell Tests, what only proves that LTHVM and SHVM provide an incorrect and an oversimplified description of these experiments"
"The violation of Bell inequalities neither proves completeness of QM nor impossibility of a local and causal description of experimental data. It only proves, that hidden variables depend on settings confirming contextual character of quantum observables and an active role played by measuring instruments.
It is high time to stop speculating about nonlocality , freedom of choice, retro-causality etc. Bell was a realist, thus he thought that he had to choose between nonlocality and superdeterminism. From two bad choices he chose nonlocality. Today probably he would choose contextuality"
Well, our paper proves you are wrong on this score. Or more precisely: your arguments are faulty, your mathematical claims are false. Yes, I agree that there is a contextual character of quantum observables and that measuring instruments play an active role. The violation of Bell inequalities in well conducted experiments does prove the impossibility of a local and causal description of experimental data according to your own definition of these terms. Whether or not well enough conducted experiments have already been done is a matter for debate. The completeness or incompleteness of QM is also another matter. We have pointed out by explicit counterexample that your paper starts off by making a false mathematical claim. It is interesting that you don’t recognise this fact. It has nothing to do with physics. Purely to do with elementary probability theory. Maybe there is some confusion of language here. One has to distinguish between probability distributions and expectation values, for instance. One necessarily, when building a coupling, creates a new probability space where random variables are defined having some of the same distributional properties as other variables on different probability spaces. To understand this, and to debate it, requires use of the precise language of probability theory and knowledge of the precise meanings of its technical terms.
Dear Marian, above, you cited your comments in another paper,
Kupczynski M:A comment on Bell's Theorem Logical Consistency, arXiv:2202.09639 [quant-ph]: https://doi.org/10.48550/arXiv.2202.09639
"Local realism understood as CFD, automatically implies MI and the existence of JP. Bell Theorem and its implications are now well understood and nobody questions Bell’s Theorem logical consistency. Bell inequalities are violated in various Bell Tests, what only proves that LTHVM and SHVM provide an incorrect and an oversimplified description of these experiments."
"The violation of Bell inequalities neither proves completeness of QM nor impossibility of a local and causal description of experimental data. It only proves, that hidden variables depend on settings confirming contextual character of quantum observables and an active role played by measuring instruments."
"It is high time to stop speculating about nonlocality , freedom of choice, retro-causality etc. Bell was a realist, thus he thought that he had to choose between nonlocality and superdeterminism. From two bad choices he chose nonlocality. Today probably he would choose contextuality."
Yet, in the paper by yourself which we are discussing here, you reproduce, in a section called "CONTEXTUAL DESCRIPTION OF SPIN POLARIZATION CORRELATION EXPERIMENTS", predictions of a model of your own, which you proposed in your 2017 papers
[59] Kupczynski M. Can we close the Bohr-Einstein quantum debate? Phil Trans R Soc A. (2017) 375:20160392. doi: 10.1098/rsta.2016.0392
[60] Kupczynski M. Is Einsteinian no-signalling violated in Bell tests? Open Physics. (2017) 15:739–753. doi: 10.1515/phys-2017-0087.
You then go on to state " Bell and CHSH inequalities may not be derived". Yet we have derived the Bell and CHSH inequalities for this situation. I am not aware of any moral injunction preventing a mathematician from writing down a mathematically correct proof of a true mathematical theorem.
Do you agree with my claim? If so, don't you agree that some correction to your paper (and perhaps also to your two 2017 papers) is needed?
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