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arXiv:0808.1007 · Physical Review A

Quantum capacity of a class of compound channels

Igor Bjelaković, Holger Boche, Janis Nötzel

Published Oct 28, 2008• 27 citations
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Abstract

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.

DOI
10.1103/physreva.78.042331

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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.
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.
This work contains results concerning transmission of entanglement and subspaces as well as generation of entanglement in the limit of arbitrary many uses of compound- and arbitrarily varying quantum channels (CQC, AVQC). In both cases, the channel is described by a set of memoryless channels. Only forward communication between one sender and one receiver is allowed. A code is said to be „good“ only, if it is „good“ for every channel out of the set. Both settings describe a scenario, in which sender and receiver have only limited channel knowledge. For different amounts of information about the channel available to sender or receiver, coding theorems are proven for the CQC. For the AVQC, both deterministic and randomised coding schemes are considered. Coding theorems are proven, as well as a quantum analogue of the Ahlswede-dichotomy. The connection to zero-error capacities of stationary memoryless quantum channels is investigated. The notion of symmetrisability is defined and used for both classes of channels.
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 onlya prioriinformation 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 a universal classical approximation of the quantum relative entropy which in turn relies on a universal hypothesis testing result.
We determine the capacity of compound classicalquantum 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.
If a sender and a receiver lack precise knowledge about the communication line that connects them, designing a scheme to reliably transmit information becomes more challenging. This has been studied in classical and quantum information theory in the context of compound channel models and arbitrarily varying channel models. However, a fully quantum version of system uncertainty allows for an even more challenging coding scenario with entangled channel uses. This type of model has previously been investigated for classical and quantum capacity. Here, we address the problem of entanglement-assisted capacity in the presence of such system uncertainty. We find that, under the assumption of a finite environment dimension, it is equal to a corresponding compound capacity. Intriguingly, our results imply that in certain fully quantum arbitrarily varying channel models, the entanglement-assisted capacity can be positive while the classical capacity is equal to zero, a phenomenon that does not occur in regular single-channel coding.
We study universal quantum codes for entanglement-assisted quantum communication over compound quantum channels. In this setting, sender and receiver do not know the specific channel that will be used for communication, but only know the set that the channel is selected from. We investigate different variations of the problem: uninformed users, informed receiver, informed sender, and feedback assistance. We derive single-letter formulas for all corresponding channel capacities. Our proofs are based on one-shot decoupling bounds and properties of smooth entropies.