Bell’s theory with no locality assumption

Gen Kimura, Yugo Susuki, Kei Morisue

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.

Aleksandr V. Belinskii

Willy De Baere, A. Mann, M. Revzen

Liberato De, Augusto Garuccio

A general approach to Bell-type inequalities for trichotomic observables is presented. With this formalism, which explicitly includes no-counts events, we show that the limits of all Bell-type inequalities are functions of the supplementary assumptions. This approach enables us to include in the causal and local theory the same assumptions used in the quantum-mechanical analysis of an experiment, and to deduce the inequality that must be satisfied by any local realistic model incorporating these assumptions. We prove that, using the quantum-mechanical assumption of ``random nondetection'' in the causal and local approach, a generalized Bell-type inequality is deduced whose upper limit can be violated by the quantum-mechanical predictions for symmetrical experimental setups involving detectors with quantum efficiency greater than 0.811. \textcopyright{} 1996 The American Physical Society.

Karl Raimund Popper

Márton Gömöri, Carl Hoefer

Abstract A widespread view among physicists is that Bell’s theorem rests on an implicit assumption of “classicality,” in addition to locality. According to this understanding, the violation of Bell’s inequalities poses no challenge to locality, but simply reinforces the fact that quantum mechanics is not classical. The paper provides a critical analysis of this view. First we characterize the notion of classicality in probabilistic terms. We argue that classicality thus construed has nothing to do with the validity of classical physics, nor of classical probability theory, contrary to what many believe. At the same time, we show that the probabilistic notion of classicality is not an additional premise of Bell’s theorem, but a mathematical corollary of locality in conjunction with the standard auxiliary assumptions of Bell. Accordingly, any theory that claims to get around the derivation of Bell’s inequalities by giving up classicality, in fact has to give up one of those standard assumptions. As an illustration of this, we look at two recent interpretations of quantum mechanics, Reinhard Werner’s operational quantum mechanics and Robert Griffiths’ consistent histories approach, that are claimed to be local and non-classical, and identify which of the standard assumptions of Bell’s theorem each of them is forced to give up. We claim that while in operational quantum mechanics the Common Cause Principle is violated, the consistent histories approach is conspiratorial. Finally, we relate these two options to the idea of realism, a notion that is also often identified as an implicit assumption of Bell’s theorem.