Quantum channels with quantum group symmetry

TL;DR

This paper explores how compact quantum groups serve as symmetry groups for quantum channels, revealing their structure and extreme points, and extending classical symmetry concepts to quantum group settings.

Contribution

It introduces the concept of covariant channels under quantum group symmetry and characterizes their extreme points, generalizing previous results to non-commutative symmetries.

Findings

Quantum group symmetry leads to covariant channels with a well-structured convex set.

Extreme points of covariant channels are identified under multiplicity-free fusion rules.

Examples include quantum permutation groups and $SU_q(2)$, highlighting differences from classical groups.

Abstract

In this paper we will demonstrate that any compact quantum group can be used as symmetry groups for quantum channels, which leads us to the concept of covariant channels. We, then, unearth the structure of the convex set of covariant channels by identifying all extreme points under the assumption of multiplicity-free condition for the associated fusion rule, which provides a wide generalization of some recent results. The presence of quantum group symmetry contrast to the group symmetry will be highlighted in the examples of quantum permutation groups and $SU_q(2)$. In the latter example, we will see the necessity of the Heisenberg picture coming from the non-Kac type condition. This paper ends with the covariance with respect to projective representations, which leads us back to Weyl covariant channels and its fermionic analogue.