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IEEE Transactions on Information Theory

Concavity of the auxiliary function appearing in quantum reliability function

Jun Ichi Fujii, Ritsuo Nakamoto, Kenjiro Yanagi

Published Jul 1, 2006• 7 citations
Publisher

Abstract

Reliability functions characterize the asymptotic behavior of the error probability for transmission of data on a channel. Holevo introduced the quantum channel, and gave an expression for a random-coding lower bound involving an auxiliary function. Holevo, Ogawa, and Nagaoka conjectured that this auxiliary function is concave. Here we give a proof of this conjecture.

DOI
10.1109/tit.2006.876248

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Reliability functions characterize the asymptotic behavior of the error probability for transmission of data on a channel. Holevo introduced the quantum channel, and gave an expression for a random-coding lower bound involving an auxiliary function. Holevo, Ogawa, and Nagaoka conjectured that this auxiliary function is concave. Here we give a proof of this conjecture.
The auxiliary function of a classical channel appears in two fundamental quantities, the random coding exponent and the sphere-packing exponent, which yield upper and lower bounds on the error probability of decoding, respectively. A crucial property of the auxiliary function is its concavity, and this property consequently leads to several important results in finite blocklength analysis. In this paper, we prove that the auxiliary function of a classical-quantum channel also enjoys the same concavity property, extending an earlier partial result to its full generality. We also prove that the auxiliary function satisfies the data-processing inequality, among various other important properties. Furthermore, we show that the concavity property of the auxiliary function enables a geometric interpretation of the random coding exponent and the sphere-packing exponent of a classical-quantum channel. The key component in our proof is an important result from the theory of matrix geometric means.
© 2016 IEEE. The auxiliary function of a classical channel appears in two fundamental quantities, the random coding exponent and the sphere-packing exponent, which yield upper and lower bounds on the error probability of decoding, respectively. A crucial property of the auxiliary function is its concavity, and this property consequently leads to several important results in finite blocklength analysis. In this paper, we prove that the auxiliary function of a classical-quantum channel also enjoys the same concavity property, extending an earlier partial result to its full generality. We also prove that the auxiliary function satisfies the data-processing inequality, among various other important properties. Furthermore, we show that the concavity property of the auxiliary function enables a geometric interpretation of the random coding exponent and the sphere-packing exponent of a classical-quantum channel. The key component in our proof is an important result from the theory of matrix geometric means.