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Lobachevskii Journal of Mathematics

The Role of Covariancy in Calculating Capacity of Quantum Channels Generated by Projective Unitary Representations of Non-Abelian Groups

Grigori Gennadievich Amosov, L. A. Ryskin

Published Jun 1, 2024• 1 citation
Publisher

Abstract

Abstract We study quantum channels generated by irreducible projective unitary representations of finite groups. It is shown that to calculate a classical capacity of channel it is not necessary to claim that it is covariant with respect to a representation. It helps to consider non-Abelian groups because in this case such channels are not covariant respect to the generating representation.

DOI
10.1134/s1995080224603102

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Abstract We study quantum channels generated by irreducible projective unitary representations of finite groups. It is shown that to calculate a classical capacity of channel it is not necessary to claim that it is covariant with respect to a representation. It helps to consider non-Abelian groups because in this case such channels are not covariant respect to the generating representation.
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⊗Uc is simply reducible (with Uc 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.
In matrix algebras, we introduce a class of linear maps that are irreducibly covariant with respect to the finite group generated by the Weyl operators. In particular, we analyze the irreducibly covariant quantum channels, that is, the completely positive and trace-preserving linear maps. Interestingly, imposing additional symmetries leads to the so-called generalized Pauli channels, which were recently considered in the context of the non-Markovian quantum evolution. Finally, we provide examples of irreducibly covariant positive but not necessarily completely positive maps.
We study four well-known capacities of a two-parameter family of qubit Pauli channels. These are the channels that are covariant under the SO(2)\text{SO}(2)SO(2) group and contain the depolarizing channel as a special case. We find exact expressions for the classical capacity and entanglement-assisted capacities, and analytically determine the regions where the quantum capacity of the channel vanishes. We then use a flag extension to find upper bound for the quantum capacity and private capacity of these channels in the entire region of parameter space and also obtain the lower bound for the quantum capacity by calculating the single-shot quantum capacity numerically. In conjunction with previous results on depolarizing channels, our result is one step forward for determining the capacities of the full Pauli channel.
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We study mixed unitary channels generated by finite subgroups of the group of all unitary operators in a Hilbert space. Based on the majorization theory we introduce techniques allowing to calculate different characteristics of output states of channels. A class of channels has been allocated for which the use of entangled states doesn't give any advantage under taking supremum and infimum for output characteristics of channels. In particular, lpl_plp​-norms are multiplicative and the minimal entropy is additive with respect to taking tensor products of channels. As an important application of the obtained results the classical capacity of channel is calculated in the evident form. We compare our techniques with the informational characteristics of Boson quantum channels.