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International Journal of Mathematics

The extreme points of SU(2)-irreducibly covariant channels

Muneerah Al Nuwairan

Published Apr 21, 2014• 19 citations
Publisher

Abstract

In this paper, we introduce EPOSIC channels, a class of SU(2)-covariant quantum channels. For each of them, we give a Kraus representation, its Choi matrix, a complementary channel, and its dual map. We show that they are the extreme points of all SU(2)-irreducibly covariant channels. As an application of these channels, we get an example of a positive map that is not completely positive.

DOI
10.1142/s0129167x14500487

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In this paper, we introduce EPOSIC channels, a class of SU(2)-covariant quantum channels. For each of them, we give a Kraus representation, its Choi matrix, a complementary channel, and its dual map. We show that they are the extreme points of all SU(2)-irreducibly covariant channels. As an application of these channels, we get an example of a positive map that is not completely positive.
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In quantum information theory, there are few quantum channels whose specific formulas and properties are known.In this thesis, we introduce EPOSIC channels, a class of SU(2)-irreducibly covariant quantum channels, and compute their χ-capacity and entropy in some cases.We review the basics of group representation theory and quantum information theory, and we construct EPOSIC channels, a new class of quantum channels, using representation theory.We also study specific formulas and properties of EPOSIC channels.
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 present a complete characterization of quantum channels covariant under the action of SU(2), a symmetry that arises naturally across a wide range of physical settings. By exploiting the constraints imposed by rotational symmetry, we derive the spectral decomposition of the associated Choi–Jamiołkowski operators and obtain explicit closed-form expressions for the corresponding Kraus operators. This unified framework provides a systematic approach to analyzing the structure and dynamics of SU(2)-covariant noise, with direct applications to quantum communication, information processing, and symmetry-constrained quantum protocols.
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