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IEEE Information Theory Workshop (ITW)

Classical capacities of compound quantum channels

Igor Bjelaković, Holger Boche

Published May 1, 2008• 2 citations
Publisher

Abstract

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.

DOI
10.1109/itw.2008.4578689

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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.
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.
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 consider the task of entanglement assisted message transmission under presence of a compound memoryless quantum channel. In this model, the completely positive and trace preserving map governing the channel statistics is, instead of being perfectly known, only revealed as member of a certain set of channels. Therefore, coding schemes have to be used which are simultaneously reliable for each member of this set. Utilizing universal codes for classical-quantum channels, we introduce optimal universal coding schemes for entanglement assisted message transmission of compound quantum channels. The resulting coding theorem together with a corresponding converse statement leads us to a single-letter expression for the entanglement assisted message transmission capacity of compound quantum channels.
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 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.
The one-shot classical capacity of a quantum channel quantifies the amount of classical information that can be transmitted through a single use of the channel such that the error probability is below a certain threshold. In this work, we show that this capacity is well approximated by a relative-entropy-type measure defined via hypothesis testing. Combined with a quantum version of Stein's lemma, our results give a conceptually simple proof of the well-known Holevo-Schumacher-Westmoreland theorem for the capacity of memoryless channels. More generally, we obtain tight capacity formulas for arbitrary (not necessarily memoryless) channels.