On the convex characterisation of the set of unital quantum channels

TL;DR

This paper explores the convex structure of unital quantum channels, providing parametrizations, classifying extreme points by Kraus rank, and introducing new families of qutrit channels with explicit examples.

Contribution

It offers a partial characterization of unital quantum channels, introduces a new family of channels with Kraus rank four, and classifies extreme points based on Kraus rank.

Findings

Parametrization of unital quantum channels.

Classification of extreme points by Kraus rank.

Explicit examples of channels with Kraus rank three and four.

Abstract

In this paper, we consider the convex set of $d$ dimensional unital quantum channels. In particular, we parametrise a family of maps and through this parametrisation we provide a partial characterisation of the set of unital quantum maps with respect to this family of channels. For the case of qutrit channels, we consider the extreme points of the set and their classification with respect to the Kraus rank. In this setting, we see that the parametrised family of maps corresponds to maps with Kraus rank three. Furthermore, we introduce a novel family of qutrit unital quantum channels with Kraus rank four to consider the extreme points of the set over all possible Kraus ranks. We construct explicit examples of these two families of channels and we consider the question of whether these channels correspond to extreme points of the set of quantum unital channels. Finally, we demonstrate how well-known channels relate to the examples presented.