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IEEE International Symposium on Information Theory (ISIT)

Classical capacity of averaged quantum channels

Igor Bjelaković, Holger Boche

Published Jul 1, 2008• 2 citations
Publisher

Abstract

In this paper we extend recent coding results by Datta and Dorlas on classical capacity of averaged quantum channels with finitely many memoryless branches to arbitrary number of branches. Only assumption in our approach is that the channel satisfies some weak measurability properties. Our approach to the direct coding theorem is based on our previous work on compound classical-quantum channels. The weak converse requires an alternative characterization of the essential infimum and the remaining proof proceeds via application of Holevo’s bound and Fano’s inequality.

DOI
10.1109/isit.2008.4594953

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In this paper we extend recent coding results by Datta and Dorlas on classical capacity of averaged quantum channels with finitely many memoryless branches to arbitrary number of branches. Only assumption in our approach is that the channel satisfies some weak measurability properties. Our approach to the direct coding theorem is based on our previous work on compound classical-quantum channels. The weak converse requires an alternative characterization of the essential infimum and the remaining proof proceeds via application of Holevo’s bound and Fano’s inequality.
In this paper we address the issue of universal or robust communication over quantum channels. Specifically, we consider a memoryless communication scenario with channel uncertainty which is an analog of the compound channel in classical information theory. We determine the quantum capacity of finite compound channels and arbitrary compound channels with an informed decoder. Our approach in the finite case is based on the observation that perfect channel knowledge at the decoder does not increase the capacity of finite quantum compound channels. As a consequence, we obtain a coding theorem for finite quantum averaged channels, the simplest class of channels with long-term memory. The extension of these results to quantum compound channels with uninformed encoder and decoder and infinitely many constituents remains an open problem.
Coding theorems and (strong) converses for memoryless quantum communication channels and quantum sources are proved: for the quantum source the coding theorem is reviewed, and the strong converse proven. For classical information transmission via quantum channels we give a new proof of the coding theorem, and prove the strong converse, even under the extended model of nonstationary channels. As a by-product we obtain a new proof of the famous Holevo bound. Then multi-user systems are investigated, and the capacity region for the quantum multiple access channel is determined. The last chapter contains a preliminary discussion of some models of compression of correlated quantum sources, and a proposal for a program to obtain operational meaning for quantum conditional entropy. An appendix features the introduction of a notation and calculus of entropy in quantum systems.
We determine the capacity of compound classical-quantum channels. As a consequence we obtain the capacity formula for the averaged classical-quantum channels. The capacity result for compound channels demonstrates, as in the classical setting, the existence of reliable universal classical-quantum codes in scenarios where the only a priori information about the channel used for the transmission of information is that it belongs to a given set of memoryless classical-quantum channels. Our approach is based on the universal classical approximation of the quantum relative entropy which in turn relies on the universal hypothesis testing results.
The capacity of a classical-quantum channel (or, in other words, the classical capacity of a quantum channel) is considered in the most general setting, where no structural assumptions such as the stationary memoryless property are made on a channel. A capacity formula as well as a characterization of the strong converse property is given just in parallel with the corresponding classical results of Verdu-Han (1994) which are based on the so-called information-spectrum method. The general results are applied to the stationary memoryless case with or without cost constraint on inputs, whereby a deep relation between the channel coding theory and the hypothesis testing for two quantum states is elucidated.