Positive Maps and Separable Matrices

TL;DR

This paper develops numerical algorithms using semidefinite relaxations to determine positivity of linear maps and separability of matrices in symmetric matrix spaces, providing practical tools for these complex problems.

Contribution

It introduces a hierarchy of semidefinite relaxations to efficiently check positivity and separability, with finite-step detection and certificates for non-separability.

Findings

Positivity can be detected by solving a finite sequence of semidefinite relaxations.

Separable matrices can be decomposed or certified as non-separable through the proposed hierarchy.

Algorithms are based on Lasserre type relaxations for bi-quadratic forms.

Abstract

A linear map between real symmetric matrix spaces is positive if all positive semidefinite matrices are mapped to positive semidefinite ones. A real symmetric matrix is separable if it can be written as a summation of Kronecker products of positive semidefinite matrices. This paper studies how to check if a linear map is positive or not and how to check if a matrix is separable or not. We propose numerical algorithms, based on Lasserre type semidefinite relaxations, for solving such questions. To check the positivity of a linear map, we construct a hierarchy of semidefinite relaxations for minimizing the associated bi-quadratic forms over the unit spheres. We show that the positivity can be detected by solving a finite number of such semidefinite relaxations. To check the separability of a matrix, we construct a hierarchy of semidefinite relaxations. If it is not separable, we can get a mathematical certificate for that; if it is, we can get a decomposition for the separability.