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A minimal classical model illustrating superdeterminism and the violation of Bell inequalities

Renzo Mosetti

Published Nov 27, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.17740274

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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.
Bell's inequality was originally derived under the assumption that experimenters are free to select detector settings independently of any local ``hidden variables'' that might affect the outcomes of measurements on entangled particles. This assumption has come to be known as ``measurement independence'' (also referred to as ``freedom of choice'' or ``settings independence''). For a two-setting, two-outcome Bell test, we derive modified Bell inequalities that relax measurement independence, for either or both observers, while remaining locally causal. We describe the loss of measurement independence for each observer using the parameters M1{M}_{1}M1​ and M2{M}_{2}M2​, as defined by Hall in 2010, and also by a more complete description that adds two new parameters, which we call $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{1}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{2}}$, deriving a modified Bell inequality for each description. These ``relaxed'' inequalities subsume those considered in previous work as special cases, and quantify how much the assumption of measurement independence needs to be relaxed in order for a locally causal model to produce a given violation of the standard Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. We show that both relaxed Bell inequalities are tight bounds on the CHSH parameter by constructing locally causal models that saturate them. For any given Bell inequality violation, the new two-parameter and four-parameter models each require significantly less mutual information between the hidden variables and measurement settings than previous models. We conjecture that the new models, with optimal parameters, require the minimum possible mutual information for a given Bell violation. We further argue that, contrary to various claims in the literature, relaxing freedom of choice need not imply superdeterminism.
A simple classical non-local dynamical system with random initial conditions and an output projecting the state variable on selected axes has been defined to mimic a two-channel quantum coincidence experiment. Non-locality is introduced by a parameter connecting the initial conditions to the selection of the projection axes. The statistics of the results shows violations up to 100% of the Bell's inequality, in the form of Clauser-Horne- Shimony-Holt (CHSH), strongly depending on the non-locality parameter. Discussions on the parallelism with Bohmian mechanics are given.
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.
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”.
Quantum theory predicts violations of Bell’s inequality. This implies a contradiction with at least one of three statistical conditions, known as “Realism” (joint distribution of relevant observables), “Locality” (conditional separability of pairwise observations, given a set of hidden variables), and “Free Will” (statistical independence between hidden variables and measurement settings). Here we describe a classical system with “clockwork” properties that is unquestionably local. It produces maximal violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality. It contradicts every one of the three listed formal conditions. Nonetheless, it expresses the qualitative markers of ontological Realism, ontological Locality, and ontological Free Will, all at the same time. In other words, the three listed conditions do not correspond to the three ontological concepts with the same names. Therefore, there is no conflict between quantum theory and Local Realism.
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.