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arXiv:1711.10823 · Journal of Mathematical Physics

Quantum channels irreducibly covariant with respect to the finite group generated by the Weyl operators

Katarzyna Siudzińska, Dariusz Chruściński

Published Mar 1, 2018• 28 citations
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Abstract

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.

DOI
10.1063/1.5013604

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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.
Quantum channels define key objects in quantum information theory. They are represented by completely positive trace-preserving linear maps in matrix algebras. We analyze a family of quantum channels defined through the use of the Weyl operators. Such channels provide generalization of the celebrated qubit Pauli channels. Moreover, they are covariant with respective to the finite group generated by Weyl operators. In what follows, we study self-adjoint Weyl channels by providing a special Hermitian representation. For a prime dimension of the corresponding Hilbert space, the self-adjoint Weyl channels contain well-known generalized Pauli channels as a special case. We propose multipartite generalization of Weyl channels. In particular, we analyze the power of prime dimensions using finite fields and study the covariance properties of these objects.
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.
A generalization of the Choi-Jamiolkowski isomorphism for completely positive maps between operator algebras is introduced. Particular emphasis is placed on the case of normal unital completely positive maps defined between von Neumann algebras. This generalization is applied especially to the study of maps which are covariant under actions of a symmetry group. We highlight with the example of, e.g., phase-shift-covariant quantum channels the ease of this method in particular in the case of a compact symmetry group. We also discuss the case of channels which are covariant under actions of the Euclidean group of rigid motions in 3 dimensions.
Quantum channels, a subset of quantum maps, describe the unitary and nonunitary evolution of quantum systems. We study a generalization of the concept of Pauli maps to the case of multipartite high-dimensional quantum systems through the use of the Weyl operators. The condition for such maps to be valid quantum channels, i.e., complete positivity, is derived in terms of Fourier transform matrices. From these conditions, we find the extreme points of this set of channels and identify an elegant algebraic structure nested within them. In turn, this allows us to expand upon the concept of ``component erasing channels'' introduced in earlier work by the authors. We show that these channels are completely characterized by elements drawn of finite cyclic groups. An algorithmic construction for such channels is presented and the smallest subsets of erasing channels which generate the whole set are determined.
Abstract In this paper, we discuss positive maps induced by (irreducibly) covariant linear operators for finite groups. The application of group theory methods allows deriving some new results of a different kind. In particular, a family of necessary conditions for positivity, for such objects is derived, stemming either from the definition of a positive map or the novel method inspired by the inverse reduction map. In the low-dimensional cases, for the permutation group S(3) and the quaternion group Q, the necessary and sufficient conditions are given, together with the discussion on their decomposability. In higher dimensions, we present positive maps induced by a three-dimensional irreducible representation of the permutation group S(4) and d-dimensional representation of the monomial unitary group MU(d). In the latter case, we deliver if and only if condition for the positivity and compare the results with the method inspired by the inverse reduction. We show that the generalised Choi map can be obtained by considered covariant maps induced by the monomial unitary group. As an additional result, a novel interpretation of the Fujiwara–Algolet conditions for positivity and complete positivity is presented. Finally, in the end, a new form of an irreducible representation of the symmetric group S(n) is constructed, allowing us to simplify the form of certain Choi–Jamiołkowski images derived for irreducible representations of the symmetric group.