# Kupczynski’s Contextual Locally Causal Probabilistic Models Are Constrained by Bell’s Theorem

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## Abstract

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## 1. Introduction

Next, MK explained what was for him the heart of the matter:Bell–CHSH inequalities [named for the Clauser, Horne Shimony, Holt generalization of Bell’s original inequality] are trivial algebraic properties satisfied by each line of an $N\times 4$ spreadsheet containing ±1 entries, and, thus, it is surprising that their violation in some experiments allows us to speculate about the existence of non-local influences in nature and casts doubt on the existence of the objective external physical reality. Such speculations are rooted in incorrect interpretations of quantum mechanics and in a failure of local realistic hidden variable models to reproduce quantum predictions for spin polarization correlation experiments (SPCEs). In these models, one uses a counterfactual joint probability distribution of only pairwise measurable random variables $(A,{A}^{\prime},B,{B}^{\prime})$ to prove Bell–CHSH inequalities.

Finally, he presented his own metaphysical conclusions regarding quantum interpretations and the foundations of quantum mechanics:In SPCE, Alice and Bob, using 4 incompatible pairs of experimental settings, estimate imperfect correlations between clicks registered by their detectors. Clicks announce the detection of photons and are coded by ±1. Expectations of corresponding random variables – $E\left(AB\right)$, $E\left(A{B}^{\prime}\right)$, $E\left({A}^{\prime}B\right)$, and $E\left({A}^{\prime}{B}^{\prime}\right)$ – are estimated and compared with quantum predictions. These estimates significantly violate CHSH inequalities. Since variables $(A,{A}^{\prime})$ and $(B,{B}^{\prime})$ cannot be measured jointly, neither $N\times 4$ spreadsheets nor a joint probability distribution of $(A,{A}^{\prime},B,{B}^{\prime})$ exist, thus Bell–CHSH inequalities may not be derived. Nevertheless, imperfect correlations between clicks in SPCE may be explained in a locally causal way, if contextual setting-dependent parameters describing measuring instruments are correctly included in the description.

The Bell–CHSH inequalities certainly can be seen as trivial algebraic properties of some very elementary mathematical structures. However, what MK calls speculations are actually arguments which depart from physical assumptions. MK’s general objections were already answered by John Bell himself in Bell (1975) [2], and the assumptions behind the Bell–CHSH inequalities have been clearly stated by Bell (1981) [3]. We come back to these matters later.The violation of Bell–CHSH inequalities may not therefore justify the existence of a spooky action at the distance, super-determinism, or speculations that an electron can be both here and a meter away at the same time. In this paper we review and rephrase several arguments proving that such conclusions are unfounded. Entangled photon pairs cannot be described as pairs of socks nor as pairs of fair dice producing in each trial perfectly correlated outcomes. Thus, the violation of inequalities confirms only that the measurement outcomes and ‘the fate of photons’ are not predetermined before the experiment is done. It does not allow for doubt regarding the objective existence of atoms, electrons, and other invisible elementary particles which are the building blocks of the visible world around us.

## 2. A Fallacious Argument

Indeed, according to quantum mechanics, noncommuting observables cannot be measured simultaneously. However, this fact is irrelevant. The fact that there is no logical impediment to the eventual existence of a joint probability distribution, even though measurements $(A,{A}^{\prime})$ and $(B,{B}^{\prime})$ are physically impossible, can be proved with a counterexample. Use of Fine’s theorem [13] makes this job easy. In [14], a counterexample is constructed where we have four incompatible random experiments à la Bell satisfying the CHSH inequalities. Hence, according to Fine’s theorem, they admit a joint distribution with appropriate marginals reproducing the “actual” experimental probability distributions.Since variables $(A,{A}^{\prime})$ and $(B,{B}^{\prime})$ cannot be measured jointly, neither $N\times 4$ spreadsheets nor a joint probability distribution of $(A,{A}^{\prime},B,{B}^{\prime})$ exist, ……

He then continues explaining thatWe have 4 incompatible experiments, labeled by $(x,y)$, and only 2 outcomes are outputted in each trial, thus JP [a joint probability distribution] of 4 random variables ${A}_{1},{A}_{-1},{B}_{1},{B}_{-1}$ does not exist.

Although Kupczynski’s former statements reduce to a denial of Fine’s celebrated theorem, it is baffling that he accepts Fine’s theorem by explaining at the beginning of the second paragraph of section 5 in [12]Lambare and Franco incorrectly conclude: “according to Fine’s theorem, a joint probability distribution of ${A}_{1},{A}_{-1},{B}_{1},{B}_{-1}$ exists, although the experiments are incompatible.”

Since, in our counterexample, the CHSH inequalities hold, denying the existence of a JP for the probabilities in such an experiment amounts to rejecting Fine’s theorem. Kupczynski’s attempt to invalidate the counterexample is contradictory: one cannot reject the counterexample and accept Fine’s theorem.Fine demonstrated, that CHSH are necessary and sufficient conditions for the existence of JP of 4 only pair-wise measurable random variables.

## 3. Kupczynski’s Framework

## 4. The Mathematical Details

## 5. An Alternative Approach

## 6. What John Bell Already Said about This

The objection of de la Peña, Cetto, and Brody [17] is based on a misinterpretation of the demonstration of the theorem. In the course of it reference is made to $A({a}^{\prime},\mu ),B({b}^{\prime},\mu )$ as well as to $A(a,\mu ),B(b,\mu )$. These authors say “Clearly, since $A,{A}^{\prime},B,{B}^{\prime}$ are all evaluated for the same $\mu $, they must refer to four measurements carried out on the same electron–positron pair. We can suppose, for instance, that ${A}^{\prime}$ is obtained after A, and ${B}^{\prime}$ after B″. But by no means. We are not at all concerned with sequences of measurements on a given particle, or of pairs of measurements on a given pair of particles. We are concerned with experiments in which for each pair the ‘spin’ of each particle is measured once only. The quantities $A({a}^{\prime},\mu ),B({b}^{\prime},\mu )$ are just the same functions $A(a,\mu ),B(b,\mu )$ with different arguments.

Earlier in the same paper he writesIt is notable that in this argument nothing is said about the locality, or even localizability, of the variable $\lambda $. These variables could well include, for example, quantum mechanical state vectors, which have no particular localization in ordinary space time.

Bell goes on to derive the CHSH inequalities, where now, his functions A and B would be the conditional expectations of the outcome value at each measurement location, conditional on the local setting and the sources of correlation. Thus, Bell’s $A(a,\lambda )$ would be Kupczynski’s ${\sum}_{{\lambda}_{x}}{A}_{x}({\lambda}_{1},{\lambda}_{x}){p}_{x}\left({\lambda}_{x}\right)$ with $\lambda \equiv ({\lambda}_{1},{\lambda}_{2})$ and $a\equiv x$.It seems reasonable to expect that if sufficiently many such causal factors can be identified and held fixed [together, they form $\lambda $] … $\lambda $ denotes any number of other variables that might be relevant the residual fluctuations will be independent. … That is to say we suppose that there are variables $\lambda $, which, if only we knew them, would allow decoupling of the fluctuations.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Some Logical Issues

In [5], the above Equations (11)–(14) correspond to MK’s contextual model while (19)–(22) describe Bell’s model. Kupczynski explicitly states that the calculated expectations of Bell and his contextual model have the same values. The obvious conclusion is that, even if we assume that MK is correct and Bell’s formulae are “impossible to implement”, they give the same result as his “contextual model”, namely, MK’s model describing actual experiments, and, hence, these also satisfy the Bell inequality.Although the expectations calculated using the Equations (11)–(14) and (19)–(22) have the same values, the two sets of formulae describe different experiments.

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**Figure 1.**Spatio-temporal disposition of one trial of a Bell experiment. (Figure 7 from Bell (1981) [3], “Bertlmann’s socks and the nature of reality”).

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**MDPI and ACS Style**

Gill, R.D.; Lambare, J.P.
Kupczynski’s Contextual Locally Causal Probabilistic Models Are Constrained by Bell’s Theorem. *Quantum Rep.* **2023**, *5*, 481-495.
https://doi.org/10.3390/quantum5020032

**AMA Style**

Gill RD, Lambare JP.
Kupczynski’s Contextual Locally Causal Probabilistic Models Are Constrained by Bell’s Theorem. *Quantum Reports*. 2023; 5(2):481-495.
https://doi.org/10.3390/quantum5020032

**Chicago/Turabian Style**

Gill, Richard D., and Justo Pastor Lambare.
2023. "Kupczynski’s Contextual Locally Causal Probabilistic Models Are Constrained by Bell’s Theorem" *Quantum Reports* 5, no. 2: 481-495.
https://doi.org/10.3390/quantum5020032