A sufficient family of necessary inequalities for the compatibility of quantum marginals

TL;DR

This paper introduces a countable family of inequalities that precisely characterizes the compatibility of quantum marginals, providing both necessary and sufficient conditions for quantum state compatibility.

Contribution

It presents a novel, complete set of inequalities that fully determine when quantum marginals are compatible, advancing the understanding of the quantum marginal problem.

Findings

The family of inequalities is necessary for all compatible quantum marginals.

The inequalities are sufficient: incompatible marginals violate at least one inequality.

This work offers a complete characterization of quantum marginal compatibility.

Abstract

The quantum marginal problem is concerned with characterizing which collections of quantum states on different subsystems are compatible in the sense that they are the marginals of some multipartite quantum state. Presented here is a countable family of inequalities, each of which is necessarily satisfied by any compatible collection of quantum states. Additionally, this family of inequalities is shown to be sufficient: every incompatible collection of quantum states will violate at least one inequality belonging to the family.