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Some properties of SU(2)-irreducibly covariant quantum channels

Jae Hyeok Shin

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

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.

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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 this thesis, we introduce EPOSIC channels, a class of SU(2) -covariant quantum channels. For each of them, we give a Stinespring representation, a Kraus representation, its Choi matrix, a complementary channel, and its dual map. We show that these channels are the extreme points of all SU(2) -irreducibly covariant channels. As an application of these channels to the theory of quantum information, we study the minimal output entropy of EPOSIC channels, and show that a large class of these channels is a potential example of violating the well-known problem, the additivity problem. We determine the cases where their minimal output entropy is not zero, and obtain some partial results on the fulfillment of their entanglement breaking property. We find a bound of the minimal output entropy of the tensor product of two SU(2) -irreducibly covariant channels. We also get an example of a positive map that is not completely positive.
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.
We investigate information theoretic properties of low rank (less than or equal to 3) quantum channels with [Formula: see text]-symmetry, where we have a complete description. We prove that PPT property coincides with entanglement-breaking property and that degradability seldomly holds in this class. In connection with these results, we will demonstrate how we can compute Holevo and coherent information of those channels. In particular, we exhibit a strong form of additivity violation of coherent information, which resembles the superactivation of coherent information of depolarizing channels.
One of the major achievements of the recently emerged quantum information theory is the introduction and thorough investigation of the notion of a quantum channel which is a basic building block of any data-transmitting or data-processing system. This development resulted in an elaborated structural theory and was accompanied by the discovery of a whole spectrum of entropic quantities, notably the channel capacities, characterizing information-processing performance of the channels. This paper gives a survey of the main properties of quantum channels and of their entropic characterization, with a variety of examples for finite-dimensional quantum systems. We also touch upon the 'continuous-variables' case, which provides an arena for quantum Gaussian systems. Most of the practical realizations of quantum information processing were implemented in such systems, in particular based on principles of quantum optics. Several important entropic quantities are introduced and used to describe the basic channel capacity formulae. The remarkable role of specific quantum correlations-entanglement-as a novel communication resource is stressed.