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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Qubit channels that achieve capacity with two states

Dominic William Berry

Published Aug 18, 2005• 0 citations
Publisher

Abstract

Four signalling states are sufficient to achieve the Holevo capacity for qubit channels, but in many cases are not necessary. There are examples known where the capacity is achieved for two orthogonal input states, two nonorthogonal states, and examples where three states are necessary and sufficient. Many previous results were obtained for a class of channel for which three states are sufficient. In this study, a systematic theory for this class of channel was developed. Simple criteria are presented that, when satisfied, mean that two states are sufficient for the ensemble. When these criteria are satisfied, there is a simple method to determine whether the states in the ensemble should be orthogonal or nonorthogonal. When the criteria are not satisfied, it is still possible that two states are sufficient, though it is possible that three states are necessary. In the case where three states are necessary, the form of the optimal ensemble is predicted. These results provide an efficient method for calculating the Holevo capacity for all channels in this class.

DOI
10.1117/12.616222

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Four signalling states are sufficient to achieve the Holevo capacity for qubit channels, but in many cases are not necessary. There are examples known where the capacity is achieved for two orthogonal input states, two nonorthogonal states, and examples where three states are necessary and sufficient. Many previous results were obtained for a class of channel for which three states are sufficient. In this study, a systematic theory for this class of channel was developed. Simple criteria are presented that, when satisfied, mean that two states are sufficient for the ensemble. When these criteria are satisfied, there is a simple method to determine whether the states in the ensemble should be orthogonal or nonorthogonal. When the criteria are not satisfied, it is still possible that two states are sufficient, though it is possible that three states are necessary. In the case where three states are necessary, the form of the optimal ensemble is predicted. These results provide an efficient method for calculating the Holevo capacity for all channels in this class.
This paper considers a class of qubit channels for which three states are always sufficient to achieve the Holevo capacity. For these channels, it is known that there are cases where two orthogonal states are sufficient, two nonorthogonal states are required, or three states are necessary. Here a systematic theory is given which provides criteria to distinguish cases where two states are sufficient, and determine whether these two states should be orthogonal or nonorthogonal. In addition, we prove a theorem on the form of the optimal ensemble when three states are required, and present efficient methods of calculating the Holevo capacity.
The notion of the Holevo capacity for arbitrarily constrained infinite dimensional quantum channels is introduced. It is shown that despite nonexistence of an optimal ensemble in this case it is possible to define the notion of the output optimal average state for such a channel. The characterization of the output optimal average state and a minimax expression for the Holevo capacity are obtained. This makes it possible to prove equivalence of several additivity properties for infinite dimensional quantum channels. The notion of the χ\chiχ-function for an infinite dimensional channel is considered, its strong concavity and lower semicontinuity are shown. The problem of continuity of the Holevo capacity is also discussed. It is shown that the Holevo capacity is continuous function of a channel in the finite dimensional case while in general it is only lower semicontinuous. This conclusion is confirmed by the example. The main result of this note is the statement that additivity of the Holevo capacity for all finite dimensional channels implies additivity of the Holevo capacity for all infinite dimensional channels with arbitrary constraints. The subadditivity of the χ\chiχ-function for two infinite dimensional channels with one of them noiseless or entanglement breaking is also proved.
Holevo capacity is the maximum rate at which a quantum channel can reliably transmit classical information without entanglement. However, calculating the Holevo capacity of arbitrary quantum channels is a nontrivial and computationally expensive task since it requires the numerical optimization over all possible input quantum states. In this paper, we consider discrete Weyl channels (DWCs) and exploit their symmetry properties to model DWC as a classical symmetric channel. We characterize lower and upper bounds on the Holevo capacity of DWCs using simple computational formulae. Then, we provide a sufficient and necessary condition where the upper and lower bounds coincide. The framework in this paper enables us to characterize the exact Holevo capacity for most of the known special cases of DWCs.
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.
An example is given of a qubit quantum channel which requires four inputs to maximize the Holevo capacity. The example is one of a family of channels which are related to 3-state channels. The capacity of the product channel is studied and numerical evidence presented which strongly suggests additivity. The numerical evidence also supports a conjecture about the concavity of output entropy as a function of entanglement parameters. However, an example is presented which shows that for some channels this conjecture does not hold for all input states. A numerical algorithm for finding the capacity and optimal inputs is presented and its relation to a relative entropy optimization discussed.