On random classical marginal problems with applications to quantum information theory

TL;DR

This paper investigates the probability of the existence of joint distributions in random classical marginal problems modeled by graphs, with applications to quantum information theory, particularly in Bell scenarios.

Contribution

It introduces a graph-based encoding of the classical marginal problem and provides probabilistic estimates, connecting to quantum mechanics through Bell scenarios.

Findings

Estimated probabilities for joint distribution existence in random graph models.

Analyzed volume ratios between local and non-signaling polytopes in Bell scenarios.

Connected classical marginal problems to quantum mechanical concepts like Fine's theorem.

Abstract

In this paper, we study random instances of the classical marginal problem. We encode the problem in a graph, where the vertices have assigned fixed binary probability distributions, and edges have assigned random bivariate distributions having the incident vertex distributions as marginals. We provide estimates on the probability that a joint distribution on the graph exists, having the bivariate edge distributions as marginals. Our study is motivated by Fine's theorem in quantum mechanics. We study in great detail the graphs corresponding to CHSH and Bell-Wigner scenarios providing rations of volumes between the local and non-signaling polytopes.