Structure of irreducibly covariant quantum channels for finite groups

TL;DR

This paper characterizes quantum channels covariant under irreducible group representations, providing spectral decompositions and conditions for channels to be valid and entanglement-breaking, with explicit examples for specific groups.

Contribution

It offers an explicit spectral decomposition of covariant quantum channels and necessary conditions for their validity, advancing understanding of symmetry in quantum information.

Findings

Spectral decomposition of covariant channels expressed via group representations

Necessary and sufficient eigenvalue conditions for channels to be completely positive

Characterization of entanglement-breaking covariant channels for specific groups

Abstract

We obtain an explicit characterization of linear maps, in particular, quantum channels, which are covariant with respect to an irreducible representation ($U$) of a finite group ($G$), whenever $U \otimes U^c$ is simply reducible (with $U^c$ being the contragradient representation). Using the theory of group representations, we obtain the spectral decomposition of any such linear map. The eigenvalues and orthogonal projections arising in this decomposition are expressed entirely in terms of representation characteristics of the group $G$. This in turn yields necessary and sufficient conditions on the eigenvalues of any such linear map for it to be a quantum channel. We also obtain a wide class of quantum channels which are irreducibly covariant by construction. For two-dimensional irrreducible representations of the symmetric group $S(3)$, and the quaternion group $Q$, we also characterize quantum channels which are both irreducibly covariant and entanglement breaking.