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Remark on the HSW Channel Capacityfor the Qubit Unital Channel

Guang-Can Guo

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

The Holevo-Schumacher-Westermoreland (HSW) channel capacity for the qudit unital channels was considered. A simpler formula for computing HSW channel capacity was given. And the HSW channel capacity can be determined by using only pure states. This result simplifies the computation of HSW channel capacity for the qudit unital channel.

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The relative entropy description of Holevo-Schumacher-Westmoreland (HSW) classical channel capacity is applied to single qubit channels. A simple formula for the relative entropy of qubit density matrices in the Bloch sphere representation is derived. This formula is combined with the King-Ruskai-Szarek-Werner qubit channel ellipsoid picture to analyze several unital and non-unital qubit channels in detail. An alternate proof that the optimal HSW signalling states for single qubit unital channels are those states with the minimum channel output entropy is presented. The derivation is based on the symmetries of the qubit relative entropy formula and the King-Ruskai-Szarek-Werner qubit channel ellipsoid picture. A proof is given that the average output density matrix of any set of optimal HSW signalling states for a (qubit or non-qubit) quantum channel is unique.
Using the unique nature of the average output state of an optimal signalling ensemble, we prove that for a special class of qudit unital channels, the Holevo-Schumacher-Westmoreland channel capacity is $\mathcal{C}={\mathrm{log}}_{2}(d)\ensuremath{-}{\mathrm{min}}_{\ensuremath{\rho}}\mathcal{S}(\mathcal{E}(\ensuremath{\rho})),$ where d is the dimension of the qudit. The result is extended to products of the same class of unital qudit channels. Thus, the connection between the minimum von Neumann entropy at the channel output and the transmission rate for classical information over quantum channels extends beyond the qubit domain.
This thesis studies classical communication over quantum channels. Chapter 1 describes an algebraic technique which extends several previously known qubit channel capacity results to the qudit quantum channel case. Chapter 2 derives a formula for the relative entropy function of two qubit density matrices in terms of their Bloch vectors. The application of the Bloch vector relative entropy formula to the determination of Holevo-Schumacher-Westmoreland (HSW) capacities for qubit quantum channels is discussed. Chapter 3 outlines several numerical simulation results which support theoretical conclusions and conjectures discussed in Chapters 1 and 2. Chapter 4 closes the thesis with comments, examples and discussion on the additivity of Holevo Chi and the HSW channel capacity.
Additivity of the Holevo capacity is proved for product channels, under the condition that one of the channels is a unital qubit channel, with the other completely arbitrary. As a byproduct this proves that the Holevo bound is the classical information capacity of such qubit channels, and provides an explicit formula for this classical capacity. Additivity of minimal entropy and multiplicativity of p-norms are also proved under the same assumptions. The proof relies on a new bound for the p-norm of an output state from the half-noisy phase-damping channel.