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arXiv:2107.02808 · Entropy

The Violation of Bell-CHSH Inequalities Leads to Different Conclusions Depending on the Description Used

Aldo F. G. Solis-Labastida, Melina Gastélum Vargas, Jorge G. Hirsch

Published Jul 8, 2021• 3 citations
arXivHTMLPDF

Abstract

Since the experimental observation of the violation of the Bell-CHSH inequalities, much has been said about the non-local and contextual character of the underlying system. However, the hypothesis from which Bell's inequalities are derived differ according to the probability space used to write them. The violation of Bell's inequalities can, alternatively, be explained by assuming that the hidden variables do not exist at all, that they exist but their values cannot be simultaneously assigned, that the values can be assigned but joint probabilities cannot be properly defined, or that averages taken in different contexts cannot be combined. All of the above are valid options, selected by different communities to provide support to their particular research program.

DOI
10.3390/e23070872

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Since the experimental observation of the violation of the Bell-CHSH inequalities, much has been said about the non-local and contextual character of the underlying system. However, the hypothesis from which Bell's inequalities are derived differ according to the probability space used to write them. The violation of Bell's inequalities can, alternatively, be explained by assuming that the hidden variables do not exist at all, that they exist but their values cannot be simultaneously assigned, that the values can be assigned but joint probabilities cannot be properly defined, or that averages taken in different contexts cannot be combined. All of the above are valid options, selected by different communities to provide support to their particular research program.
Bell–CHSH inequalities are commonly interpreted as demonstrating the incompatibility between quantum mechanics and local hidden-variable theories based on pre-existing measurement outcomes. In standard derivations of the CHSH inequality, however, it is implicitly assumed that all possible measurement settings can be assigned simultaneous values within a single global hidden-variable structure. In this paper, we reconsider the Bell–CHSH framework from the viewpoint of the Intrinsic Observability Framework (IOF). We argue that the central issue is not the experimental violation of the inequality itself, but the applicability of the global value assignment assumed in its derivation. In particular, the simultaneous definition of outcomes associated with incompatible measurement settings presupposes a common operational structure shared across all observational configurations. Within IOF, observable outcomes are not treated as globally pre-existing values independent of observational accessibility and decomposition structure. Instead, measurement outcomes are associated with operationally accessible configurations determined through observable–hidden relations. From this perspective, the global assignment required for the CHSH derivation is not operationally justified. We further show that this viewpoint remains fully consistent with the standard quantum-mechanical correlations observed in Bell-type experiments. The present analysis suggests that the Bell–CHSH inequality may be interpreted not as a direct contradiction between quantum mechanics and locality itself, but rather as a limitation of globally defined pre-existing value assignments across incompatible observational configurations.
The violation of Bell inequalities is often interpreted as showing that, if hidden variables exist, they must be contextual and non local. But they can also be explained questioning the probability space employed, or the validity of the Kolmogorov axioms. In this article we explore the additional constrains which can be deduced from two widely used objetive probability theories: frequentism and propensities. One of the strongest objections in the deduction of one version of Bell inequalities goes about the probability space, which assumes the existence of values for the output of the experiment in each run, while only two of the four values can be measured each time, making them counterfactual. It is shown that frequentism rejects the possibility of using counterfactual situations, while long-run propensities allow their use. In this case the introduction of locality and contextuality does not help to explain the violation, and an alternative explanation could point to a failure of the probability. Single case propensities were designed to associate probabilities to single events, but they need to be conditional to the whole universe, and do not have a clear link with the observed relative frequencies. It heavily limits their use.
A violation of Bell-CHSH inequalities does not justify speculations about quantum non-locality, conspiracy and retro-causation. Such speculations are rooted in a belief that setting dependence of hidden variables in a probabilistic model (called a violation of measurement independence (MI)) would mean a violation of experimenters’ freedom of choice. This belief is unfounded because it is based on a questionable use of Bayes Theorem and on incorrect causal interpretation of conditional probabilities. In Bell-local realistic model, hidden variables describe only photonic beams created by a source, thus they cannot depend on randomly chosen experimental settings. However, if hidden variables describing measuring instruments are correctly incorporated into a contextual probabilistic model a violation of inequalities and an apparent violation of no-signaling reported in Bell tests can be explained without evoking quantum non-locality. Therefore, for us, a violation of Bell-CHSH inequalities proves only that hidden variables have to depend on settings confirming contextual character of quantum observables and an active role played by measuring instruments. Bell thought that he had to choose between non-locality and the violation of experimenters’ freedom of choice. From two bad choices he chose non-locality. Today he would probably choose the violation of MI understood as contextuality.