Quantum marginal problem and representations of the symmetric group

TL;DR

This paper addresses the quantum marginal problem by providing a complete set of linear inequalities for spectra and exploring the problem through symmetric group representation theory, with extensive tables for up to 4 qubits.

Contribution

It offers a comprehensive solution to the quantum marginal problem using linear inequalities and connects it to symmetric group representations, including detailed tables for small systems.

Findings

Complete set of linear inequalities for quantum spectra

Extensive tables of marginal inequalities for up to 4 qubits

Reduction of the problem to symmetric group representation theory

Abstract

We discuss existence of mixed state of multicomponent system with given spectrum and given reduced density matrices. We give a complete solution of the problem in terms of linear inequalities on the spectra, accompanied with extensive tables of marginal inequalities, including arrays up to 4 qubits. In the second part of the paper we pursue another approach based on reduction of the problem to representation theory of the symmetric group.