Bell's theorem is a no-go theorem stating that quantum mechanics cannot be reproduced by a physical theory based on realism, freedom to choose experimental settings and two locality conditions: setting (SI) and outcome (OI) independence. We provide a novel analysis of what it takes to violate Bell's inequality within the framework in which both realism and freedom of choice are assumed, by showing that it is impossible to model a violation without having information in one laboratory about both the setting and the outcome at the distant one. While it is possible that outcome information can be revealed from shared hidden variables, the assumed experimenter's freedom to choose the settings ensures that the setting information must be non-locally transferred even when the SI condition is obeyed. The amount of transmitted information about the setting that is sufficient to violate the CHSH inequality up to its quantum mechanical maximum is 0.736 bits.
Bell's theorem is a no-go theorem stating that quantum mechanics cannot be reproduced by a physical theory based on realism, freedom to choose experimental settings and two locality conditions: setting (SI) and outcome (OI) independence. We provide a novel analysis of what it takes to violate Bell's inequality within the framework in which both realism and freedom of choice are assumed, by showing that it is impossible to model a violation without having information in one laboratory about both the setting and the outcome at the distant one. While it is possible that outcome information can be revealed from shared hidden variables, the assumed experimenter's freedom to choose the settings ensures that the setting information must be non-locally transferred even when the SI condition is obeyed. The amount of transmitted information about the setting that is sufficient to violate the CHSH inequality up to its quantum mechanical maximum is 0.736 bits.
Bell's theorem shows that local realistic theories place strong restrictions on observable correlations between different systems, giving rise to Bell's inequality which can be violated in experiments using entangled quantum states. Bell's theorem is based on the assumptions of realism, locality, and the freedom to choose between measurement settings. In experimental tests, "loopholes" arise which allow observed violations to still be explained by local realistic theories. Violating Bell's inequality while simultaneously closing all such loopholes is one of the most significant still open challenges in fundamental physics today. In this paper, we present an experiment that violates Bell's inequality while simultaneously closing the locality loophole and addressing the freedom-of-choice loophole, also closing the latter within a reasonable set of assumptions. We also explain that the locality and freedom-of-choice loopholes can be closed only within nondeterminism, i.e., in the context of stochastic local realism.
The Bell inequality constrains the outcomes of measurements on pairs of distant entangled particles. The Bell contradiction states that the Bell inequality is inconsistent with the calculated outcomes of these quantum experiments. This contradiction led many to question the underlying assumptions, viz. so-called realism and locality. The probability model underlying the Bell inequality is generally left implicit. We propose an explicit probability model for the CHSH version of the Bell experiment. This model has only two simultaneously observable detector settings per measurement, and therefore does not assume realism. The quantum expectation now becomes a conditional expectation, given the two detector settings. This probability model is in full agreement with both quantum mechanics and experiments. As a result, the model satisfies the Bell inequality; there are no so-called violations. We extend this model to include a hidden variable. This extended model is not Bell-separable. This non-separability implies that the model is non-deterministic or non-local (or both).
Loophole-free violations of Bell inequalities imply that at least one of the assumptions behind local hidden-variable theories must fail. Here, we show that, if only one fails, then it has to fail completely, therefore excluding models that partially constrain freedom of choice or allow for partial retrocausal influences, or allow partial instantaneous actions at a distance. Specifically, we show that (i) any hidden-variable theory with outcome independence (OI) and arbitrary joint relaxation of measurement independence (MI) and parameter independence (PI) can be experimentally excluded in a Bell-like experiment with many settings on high-dimensional entangled states, and (ii) any hidden-variable theory with MI, PI and arbitrary relaxation of OI can be excluded in a Bell-like experiment with many settings on qubit-qubit entangled states. Loophole-free violations of Bell inequalities imply that at least one of the assumptions formalising local hidden-variable theories must fail. Here, the authors show that, if only one is false, then it has to fail completely, therefore excluding models that allow partial instantaneous actions at a distance, partially constrain freedom of choice or allow for partial retrocausal influences.
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.
Bell's theorem states that no description of a Bell experiment can be simultaneously local, realistic in the sense of counterfactual definiteness, and free of conspiracy between settings and hidden state. The recent generation of experiments has confirmed the predicted violation of the CHSH inequality, so one of the assumptions must be abandoned. Which one, and how one reconstructs a coherent worldview after doing so, is a question on which many authors disagree. This paper is written by three such authors. All three reject both counterfactual definiteness and conspiratorial violation of statistical independence of setting choices and state. After a joint exposition of the classical half of Bell's theorem in the language of Pearl-style causal graphs, a joint summary of the loophole-free experiments, and a joint survey of the recent literature, each author states where they have presently arrived. Gill accepts irreducible and non-local quantum randomness and finds the choice between locality and realism a false dichotomy. In his later works, Bell derives counterfactual definiteness from classical local causality, and that is what has to go. The metaphysical concepts "realism", "locality", "causality" need to be reconsidered. Helland reconstructs the Hilbert-space formalism from a theory of accessible variables, and from this theory he concludes that every observer must be limited in a specific sense. Jongejan proposes a geometric hidden-variable construction in which the degree of violation of the CHSH inequality depends on the number of dimensions of space, Tsirelson's bound corresponding to three dimensions. The authors conclude with a discussion.
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.