Sets of Marginals and Pearson-Correlation-based CHSH Inequalities for a Two-Qubit System

TL;DR

This paper explores the properties of quantum mass functions and their marginals, introducing a generalized CHSH inequality based on Pearson correlation to characterize jointly classicable variables in quantum graphical models.

Contribution

It characterizes the set of marginals for jointly classicable variables and generalizes the CHSH inequality to include Pearson correlations, proving a prior conjecture.

Findings

Characterization of marginals for jointly classicable variables

Generalization of CHSH inequality using Pearson correlation

Proof of a conjecture by Pozsgay et al.

Abstract

Quantum mass functions (QMFs), which are tightly related to decoherence functionals, were introduced by Loeliger and Vontobel [IEEE Trans. Inf. Theory, 2017, 2020] as a generalization of probability mass functions toward modeling quantum information processing setups in terms of factor graphs. Simple quantum mass functions (SQMFs) are a special class of QMFs that do not explicitly model classical random variables. Nevertheless, classical random variables appear implicitly in an SQMF if some marginals of the SQMF satisfy some conditions; variables of the SQMF corresponding to these "emerging" random variables are called classicable variables. Of particular interest are jointly classicable variables. In this paper we initiate the characterization of the set of marginals given by the collection of jointly classicable variables of a graphical model and compare them with other concepts associated with graphical models like the sets of realizable marginals and the local marginal polytope. In order to further characterize this set of marginals given by the collection of jointly classicable variables, we generalize the CHSH inequality based on the Pearson correlation coefficients, and thereby prove a conjecture proposed by Pozsgay et al. A crucial feature of this inequality is its nonlinearity, which poses difficulties in the proof.