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arXiv:2208.13634 · Quantum

Trade-off relations between measurement dependence and hidden information for factorizable hidden variable models

Ryo Takakura, Kei Morisue, Issei Watanabe, Gen Kimura

Published Mar 14, 2025• 1 citation
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Abstract

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.

DOI
10.22331/q-2025-03-14-1662

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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.
Quantum correlations that violate the Bell inequality cannot be explained by any local-hidden-variable theory that is measurement independent. However, this violation merely signifies the incompatibility of the underlying assumptions of reality, locality, and measurement independence, without providing a quantitative measure of the extent to which each assumption is violated. In contrast, Hall introduced a measure for each of the following assumptions: indeterminism, signaling, and measurement dependence, and Hall also generalized the Bell-Clauser-Horne-Shimony-Holt inequality, which provides a quantitative trade-off relationship between these assumptions. In this paper, we consider the introduction of hidden variables to be an essential assumption of Bell's theorem and introduce a quantification of hidden variables, which we term ``hiddenness.'' We derive a trade-off relation between hiddenness and measurement dependence that applies to any local-hidden-variable theory.
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.
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.
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.
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”.