Computational tools for solving a marginal problem with applications in Bell non-locality and causal modeling

TL;DR

This paper reviews and introduces computational tools, including a new geometric algorithm, for solving marginal problems with applications in Bell non-locality and causal modeling, enabling the discovery of Bell inequalities and analysis of quantum nonlocality.

Contribution

It presents a new geometric algorithm for polyhedral projection and applies it to derive Bell inequalities and analyze quantum nonlocality in complex networks.

Findings

Discovered many tight entropic Bell inequalities in tripartite scenarios

Applied algorithms to complex causal networks

Identified quantum states that violate derived inequalities

Abstract

Marginal problems naturally arise in a variety of different fields: basically, the question is whether some marginal/partial information is compatible with a joint probability distribution. To this aim, the characterization of marginal sets via quantifier elimination and polyhedral projection algorithms is of primal importance. In this work, before considering specific problems, we review polyhedral projection algorithms with focus on applications in information theory, and, alongside known algorithms, we also present a newly developed geometric algorithm which walks along the face lattice of the polyhedron in the projection space. One important application of this is in the field of quantum non-locality, where marginal problems arise in the computation of Bell inequalities. We apply the discussed algorithms to discover many tight entropic Bell inequalities of the tripartite Bell scenario as well as more complex networks arising in the field of causal inference. Finally, we analyze the usefulness of these inequalities as nonlocality witnesses by searching for violating quantum states.