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SU(2)-Irreducibly Covariant Quantum Channels and Some Applications

Muneerah Al Nuwairan

Published Jan 1, 2015• 3 citations
Publisher

Abstract

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.

DOI
10.20381/ruor-2813

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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.
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.
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.
Quantum channels can be described via a unitary coupling of system and environment, followed by a trace over the environment state space. Taking the trace instead over the system state space produces a different mapping which we call the conjugate channel. We explore the properties of conjugate channels and describe several different methods of construction. In general, conjugate channels map M_d to M_d' with d < d', and different constructions may differ by conjugation with a partial isometry. We show that a channel and its conjugate have the same minimal output entropy and maximal output p-norm. It then follows that the additivity and multiplicativity conjectures for these measures of optimal output purity hold for a product of channels if and only if they also hold for the product of their conjugates. This allows us to reduce these conjectures to the special case of maps taking M_d to M_d' with a minimal representation of dimension at most d. We find explicit expressions for the conjugates for a number of well-known examples, including entanglement-breaking channels, unital qubit channels, the depolarizing channel, and a subclass of random unitary channels. For the entanglement-breaking channels, channels this yields a new class of channels for which additivity and multiplicativity of optimal output purity can be established. For random unitary channels using the generalized Pauli matrices, we obtain a new formulation of the multiplicativity conjecture. The conjugate of the completely noisy channel plays a special role and suggests a mechanism for using noise to transmit information.