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Annalen der Physik

Possible Vulnerability of Bell‐Clauser‐Horne‐Shimony‐Holt Tests Used for Quantum Certification

F. De Zela

Published Dec 29, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1002/andp.202500437

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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.
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 pABp_{AB}pAB​, which rules the stochastic behavior of the HVs. The reported HVs model is based on an extended form of pABp_{AB}pAB​, which is suggested by Lebesgue's decomposition theorem for bounded functions. The considered pABp_{AB}pAB​ complies with locality and realism, and also with measurement independence, parameter independence and outcome independence.
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”.
We present an extremely simple linear dynamical model which, through initial conditions modified as a function of the measurement settings, produces violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality. The model is not a classi- cal counterexample to Bell’s theorem, since it explicitly breaks the assumption of independence between hidden variables and measurement settings. Instead, it pro- vides a clear example of superdeterminism and can be used as an educational tool to illustrate the crucial role of the assumptions of locality, realism, and freedom of choice in Bell’s theorem.
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.
Bell's theorem shows that no local hidden-variable model can explain the measurement statistics of a quantum system shared between two parties, thus ruling out a classical (local) understanding of nature. In this paper we demonstrate that by relaxing the positivity restriction in the hidden-variable probability distribution it is possible to derive quasiprobabilistic Bell inequalities whose sharp upper bound is written in terms of a negativity witness of said distribution. This provides an analytic solution for the amount of negativity necessary to violate the Clauser-Horne-Shimony-Holt inequality by an arbitrary amount, therefore revealing the amount of negativity required to emulate the quantum statistics in a Bell test.
The Bell theorem is explored in terms of a trade-off relation between underlying assumptions within the hidden variable model framework. In this paper, recognizing the incorporation of hidden variables as one of the fundamental assumptions, we propose a measure termed `hidden information' taking account of their distribution. This measure quantifies the number of hidden variables that essentially contribute to the empirical statistics. For factorizable models, hidden variable models that satisfy `locality' without adhering to the measurement independence criterion, we derive novel relaxed Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequalities. These inequalities elucidate trade-off relations between measurement dependence and hidden information in the CHSH scenario. It is also revealed that the relation gives a necessary and sufficient condition for the measures to be realized by a factorizable model.