Bell’s Inequality for Trichotomic Observables

Aleksandr V. Belinskii

Peter Nyman, Irina Basieva

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.

Angelo Baracca, Silvio Bergia, Roberto Livi +1 more

Aleksandr V. Belinskii

Bell's paradoxes, due to the fundamental properties of light and the nature of the photon, are discussed within a single framework with a view to checking the hypothesis that a stationary, non-negative, joint probability distribution function exists. This hypothesis, related to the local theory of hidden parameters as a possible interpretation of quantum theory, enables experimentally verifiable Bell's inequalities to be formulated. The dependence of these inequalities on the number of observers V is considered. Quantum theory predicts the breakdown of Bell's inequalities in optical experiments. It is shown that as V increases, the requirements on the quantum effectiveness of the detector, η, are reduced from η > 2/3 at V = 2 to η > 1/2 for V → ∞. Examples of joint probability distribution functions are given for illustrative purposes, and a way to resolve the Greenberg–Horne–Zeilinger (GHZ) paradox is suggested.