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International Journal of Quantum Information

THE HSW CHANNEL CAPACITY FOR THE DIAGONAL UNITAL QUDIT CHANNELS

Yu-Chun Wu, Zheng-Wei Zhou, Guang‐Can Guo

Published Dec 1, 2004• 0 citations
Publisher

Abstract

The Holevo–Schumacher–Westermoreland (HSW) channel capacity for the diagonal unital qudit channels is considered. In Phys. Rev.A69, 022302, the HSW channel capacity for the diagonal unital qudit channels Φ is given as χ(Φ)=1- min ρH(Φ(ρ)), where minimization is over the input states of the channel. In this paper, using the concavity of von Neumann entropy, we show that using only pure states we can work out its HSW channel capacity. Hence, our result simplifies the computation.

DOI
10.1142/s021974990400047x

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The Holevo–Schumacher–Westermoreland (HSW) channel capacity for the diagonal unital qudit channels is considered. In Phys. Rev.A69, 022302, the HSW channel capacity for the diagonal unital qudit channels Φ is given as χ(Φ)=1- min ρH(Φ(ρ)), where minimization is over the input states of the channel. In this paper, using the concavity of von Neumann entropy, we show that using only pure states we can work out its HSW channel capacity. Hence, our result simplifies the computation.
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.
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.
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.
This chapter introduces the notion of noisy quantum channels , and the different types of “quantum noise” that affect qubit messages passed through such channels. The main types of noisy channel reviewed here are the depolarizing , bit-flip , phase-flip , and bit-phase-flip channels. Then the quantum channel capacity χ is defined through the Holevo–Schumacher–Westmoreland (HSW) theorem. Such a theorem can conceptually be viewed as the elegant quantum counterpart of Shannon's (noisy) channel coding theorem , which was described in Chapter 13. Here, I shall not venture into the complex proof of the HSW theorem but only provide a background illustrating the similarity with its classical counterpart. The resemblance with the channel capacity χ and the Holevo bound , as described in Chapter 21, and with the classical mutual information H ( X ; Y ), as described in Chapter 5, are both discussed. For advanced reference, a hint is provided as to the meaning of the still not fully explored concept of quantum coherent information . Several examples of quantum channel capacity, derived from direct applications of the HSW theorem, along with the solution of the maximization problem, are provided. Noisy quantum channels The notion of “noisiness” in a classical communication channel was first introduced in Chapter 12, when describing channel entropy . Such a channel can be viewed schematically as a probabilistic relation between two random sources, X for the originator, and Y for the recipient.