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

TL;DR

This paper introduces a universal framework for Bell-type inequalities applicable to any number of settings and outcomes in multipartite quantum experiments, unifying previous inequalities and extending their applicability.

Contribution

It provides a general representation of Bell-type inequalities that is independent of outcome spectral types and applies to both correlation functions and joint probabilities.

Findings

The form of correlation Bell inequalities depends only on extremal outcomes, not outcome spectral types.

The approach unifies and extends Bell inequalities to arbitrary outcomes, including continuous spectra.

The Mermin-Klyshko inequality applies to any two bounded observables per site, regardless of outcome type.

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.