On Faces of the set of Quantum Channels

TL;DR

This paper investigates the geometric structure of the set of quantum channels, specifically analyzing the dimensions of its faces using Choi matrices, which enhances understanding of the convex set's face structure.

Contribution

It computes the maximum dimension of proper faces of the set of quantum channels and characterizes face dimensions generated by channels with Choi matrix rank 2.

Findings

Maximum dimension of a proper face of quantum channels set

Possible face dimensions for channels with Choi matrix rank 2

Enhanced understanding of the convex structure of quantum channels

Abstract

A linear map $L$ from ${\mathbb C}^{n \times n}$ into ${\mathbb C}^{n \times n}$ is called a quantum channel if it is completely positive and trace preserving. The set ${\cal L}_n$ of all such quantum channels is known to be a compact convex set. While the extreme points of ${\cal L}_n$ can be characterized, not much is known about the structure of its higher dimensional faces. Using the so called Choi matrix $Z(L)$ associated with the quantum channel $L$, we compute the maximum dimension of a proper face of ${\cal L}_n$, and in addition the possible dimensions of faces generated by $L$ when $rank \ Z(L)=2 $.