Multipartite Bell-type inequalities for arbitrary numbers of settings and outcomes per site

Abstract

For a multipartite correlation experiment with an arbitrary number of settings and any spectral type of outcomes at each site, we introduce a single general representation incorporating in a unique manner all Bell-type inequalities for either correlation functions or joint probabilities that have been introduced or will be introduced in the literature. Specifying this general representation for correlation functions, we prove that the form of any correlation Bell-type inequality does not depend on a spectral type of observed outcomes, in particular, on their numbers at different sites, and is determined only by extremal values of outcomes at each site. We also specify the general form of bounds in Bell-type inequalities on joint probabilities. Our approach to the derivation of Bell-type inequalities is universal, concise and can be applied to a multipartite correlation experiment with outcomes of any spectral type, discrete or continuous. We, in particular, prove that, for an N-partite quantum state, possibly, infinite dimensional, admitting the 2x...x2-setting LHV description, the Mermin-Klyshko inequality holds for any two bounded quantum observables per site, not necessarily dichotomic.