Positively Factorizable Maps

TL;DR

This paper introduces and characterizes a new class of linear maps on complex matrices that factor through tracial von Neumann algebras with positive entries, connecting to quantum information and convex geometry.

Contribution

It defines positively factorizable maps, characterizes those with Choi rank 2, and explores their applications in identifying non-CPSD doubly nonnegative matrices.

Findings

Choi matrix of maps factoring through abelian algebras is CP.

Fully characterized positively factorizable maps with Choi rank 2.

Identified non-CPSD doubly nonnegative matrices using these maps.

Abstract

We initiate a study of linear maps on $M_n(\mathbb{C})$ that have the property that they factor through a tracial von Neumann algebra $(\mathcal{A,\tau})$ via operators $Z\in M_n(\mathcal{A})$ whose entries consist of positive elements from the von-Neumann algebra. These maps often arise in the context of non-local games, especially in the synchronous case. We establish a connection with the convex sets in $\mathbb{R}^n$ containing self-dual cones and the existence of these maps. The Choi matrix of a map of this kind which factors through an abelian von-Neumann algebra turns out to be a completely positive (CP) matrix. We fully characterize positively factorizable maps whose Choi rank is 2. We also provide some applications of this analysis in finding doubly nonnegative matrices which are not CPSD. A special class of these examples is found from the concept of Unextendible Product Bases in quantum information theory.