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arXiv:1409.2536 · IEEE Transactions on Information Theory

Coding theorem and strong converse for quantum channels

Andreas Winter

Published Jan 1, 1999• 450 citations
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Abstract

We present a new proof of Holevo's (1973, 1977) coding theorem for transmitting classical information through quantum channels, and its strong converse. The technique is largely inspired by Wolfwitz's (1964) combinatorial approach using types of sequences. As a byproduct of our approach which is independent of previous ones, both in the coding theorem and the converse, we can give a new proof of Holevo's information bound.

DOI
10.1109/18.796385

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We present a new proof of Holevo's (1973, 1977) coding theorem for transmitting classical information through quantum channels, and its strong converse. The technique is largely inspired by Wolfwitz's (1964) combinatorial approach using types of sequences. As a byproduct of our approach which is independent of previous ones, both in the coding theorem and the converse, we can give a new proof of Holevo's information bound.
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.
In this correspondence, we give an alternative proof of the direct part of the classical-quantum channel coding theorem (the Holevo-Schumacher-Westmoreland (HSW) theorem), using ideas of quantum hypothesis testing. In order to show the existence of good codes, we invoke a limit theorem, relevant to the quantum Stein's lemma, in quantum hypothesis testing as the law of large numbers used in the classical case. We also apply a greedy construction of good codes using a packing procedure of noncommutative operators. Consequently we derive an upper bound on the coding error probability, which is used to give an alternative proof of the HSW theorem. This approach elucidates how the Holevo information applies to the classical-quantum channel coding problems
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