Structure of irreducibly covariant quantum channels for finite 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.

43 pages, 3 figures

References in corpus (6)

• Converse bounds for private communication over quantum channels
• Matrix Product States: Symmetries and Two-Body Hamiltonians
• Remarks on the classical capacity of quantum channel
• Additivity of minimal entropy output for a class of covariant channels
• Conditions for optimal input states for discrimination of quantum channels
• A direct proof of dimerization in a family of SU(n)-invariant quantum spin chains

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