On the characterization of probabilities: A generalization of Bell’s inequalities

Abstract

A partial solution to the problem of generalizing Bell’s inequalities to arbitrary numbers of physical properties is proposed. It is first assumed that the considered sets of probabilities correspond to events which satisfy a postulate ensuring that they form an orthomodular partially ordered set admitting a full set of states. In this framework a theorem generalizing Bell’s inequalities to an arbitrary finite number of events is proven. An interpretation of these results in Hilbert space is indicated. Conditions characterizing the classicality of so-called correlation probabilities are then found, and a method for verifying inequalities involving measurable correlations is discussed.