Factorizations of Quantum Channels

TL;DR

This paper explores the relationship between quantum channel factorizations and low-rank solutions to linear matrix equations, providing new characterizations of extreme points in the set of factorizable channels.

Contribution

It establishes a connection between quantum channel factorizations and low-rank matrix solutions, and characterizes certain extreme points in the set of factorizable channels.

Findings

Factorization of quantum channels relates to low-rank linear matrix solutions.

Channels factorized by a direct integral lie in the convex hull of channels factorized by the components.

Identifies non-trivial extreme points in the set of factorizable quantum channels.

Abstract

We find a connection between the existence of a factorization of a quantum channel and the existence of low-rank solutions to certain linear matrix equations. Using this, we show that if a quantum channel is factorized by a direct integral of factors, it must lie in the convex hull of quantum channels which are factorized respectively by the factors in the direct integral. We use this to characterize some non-trivial extreme points in the set of factorizable quantum channels and give an example.