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Physics Letters A

Hidden variable model for the Bell-Clauser-Horne-Shimony-Holt test based on the Bell-Kochen-Specker model for a qubit

F. De Zela

Published Jul 17, 2026• 0 citations
Publisher

Abstract

The validity of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality for hidden-variable models of the Bell type and the quantum prediction that said inequality can be violated are two indisputable assertions. However, one can prove wrong the assertion that no hidden-variable model is able to reproduce quantum predictions about the Bell-CHSH test. Here, we reproduce all quantum predictions of Bell-CHSH tests by means of the Bell-Kochen-Specker hidden-variable model for a qubit plus Einstein-Podolsky-Rosen “criterion of physical reality”.

DOI
10.1016/j.physleta.2026.132006

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The validity of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality for hidden-variable models of the Bell type and the quantum prediction that said inequality can be violated are two indisputable assertions. However, one can prove wrong the assertion that no hidden-variable model is able to reproduce quantum predictions about the Bell-CHSH test. Here, we reproduce all quantum predictions of Bell-CHSH tests by means of the Bell-Kochen-Specker hidden-variable model for a qubit plus Einstein-Podolsky-Rosen “criterion of physical reality”.
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ABSTRACT A hidden variables (HVs) model is reported, which reproduces quantum predictions for Bell‐Clauser‐Horne‐Shimony‐Holt (Bell‐CHSH) tests. The existence of such a model poses some limitations to quantum certifications that rely on Bell‐CHSH inequality violations. The reported model does not prove wrong Bell's theorem. The latter assumes the factorability of the probability density p AB , which rules the stochastic behavior of the HVs. The reported HVs model is based on an extended form of p AB , which is suggested by Lebesgue's decomposition theorem for bounded functions. The considered p AB complies with locality and realism, and also with measurement independence, parameter independence and outcome independence.
In this work we aim to analyze the Clauser-Horne-Shimony-Holt CHSH inequality strictly in the context of probability theory. In the course of assembling inequality we have to take care not to produce assumptions a priori, that is, physically or intuitively accepted suppositions. Of course, this does not mean that after these considerations, we put the adequate physical conditions suitable and generally accepted in these contexts. This allows us to clearly visualize the possibility of finding a greater inequality than that of CHSH in which it is included. This inequality does not contradict the CHSH inequality. This result is suported by using a robust computational simulation, showing the possibility for obtaining an inequality for quantum mechanics that is not violated and allows random hidden variables without any conflict with Bell's inequality.
We emphasize the role of the precise correlations loophole in attempting to connect the CHSH type inequalities with the EPR-argument. The possibility to test theories with hidden variables experimentally by using such inequalities is questioned. The role of the original Bell inequality is highlighted. The interpretation of the CHSH inequality in the spirit of Bohr, as a new test of incompatibility, is presented. The positions of Bohr, Einstein, Podolsky, Rosen, Bell, Clauser, Horne, Shimony, Holt, De Broglie, Hertz, and Boltzmann on interrelation of theory and experiment are enlightened.
Abstract This chapter presents a detailed discussion of Bell inequalities and their experimental tests. Starting from the Aharonov-Bohm version of the Einstein-Podolsky-Rosen (EPR) paradox, we review EPR’s criteria for realism and completeness and Bell’s locality condition, before deriving the Clauser-Horne-Shimony-Holt (CHSH) inequality. This is followed by an exposition on Bell’s original inequality and other Bell inequalities by Wigner, as well as Clauser and Horne. We then discuss the milestones in the experimental violation of Bell inequalities, from the experiment of Freedman and Clauser in the 1970s, to the efforts of Aspect et al. in the 1980s, the experiments by the teams of Zeilinger and Gisin in the 1990s, and culminating the loophole-free tests in 2015. Finally we turn to a brief discussion of the interpretations of quantum mechanics, focusing on contrasting the realist views held by, e.g., Einstein and Bell, and the informational views with proponents such as Brukner and Zeilinger
Many future directions of scientific endeavors depend on quantum theory and the precise interpretation and significance of the entanglement of quantum-particles. This interpretation depends in turn on the physical meaning of so called Bell-tests that are mostly performed using entangled photons and randomly switched polarizers to measure their polarization at distant locations. This paper presents a detailed critique of the well known theory of Bell tests given by Clauser-Horne-Shimony-Holt (CHSH). It is demonstrated that several important steps of the CHSH derivations contain serious inaccuracies of the underlying physics and probability theory and even a calculus error. As a consequence, the Bell-CHSH theory cannot be used to demonstrate extreme and opposite interpretations of entanglement such as super-luminal influences or alternatively super-determinism that cast aspersions on Einstein’s concepts of locality and separability.