All the Bell inequalities

Abstract

Bell inequalities are derived for any number of observers, any number of alternative setups for each one of them, and any number of distinct outcomes for each experiment. It is shown that if a physical system consists of several distant subsystems, and if the results of tests performed on the latter are determined by local variables with objective values, then the joint probabilities for triggering any given set of distant detectors are convex combinations of a finite number of Boolean arrays, whose components are either 0 or 1 according to a simple rule. This convexity property is both necessary and sufficient for the existence of local objective variables. It leads to a simple graphical method which produces a large number of generalized Clauser- Horne inequalities corresponding to the faces of a convex polytope. It is plausible that quantum systems whose density matrix has a positive partial transposition satisfy all these inequalities, and therefore are compatible with local objective variables, even if their quantum properties are essentially nonlocal.