Properties of Noncommutative Rényi and Augustin Information
Journal
Communications in Mathematical Physics
Date Issued
2022
Author(s)
Abstract
R?nyi and Augustin information are generalizations of mutual information defined via the R?nyi divergence, playing a significant role in evaluating the performance of information processing tasks by virtue of its connection to the error exponent analysis. In quantum information theory, there are three generalizations of the classical R?nyi divergence—the Petz’s, sandwiched, and log-Euclidean versions, that possess meaningful operational interpretation. However, the associated quantum R?nyi and Augustin information are much less explored compared with their classical counterpart, and lacking crucial properties hinders applications of these quantities to error exponent analysis in the quantum regime. The goal of this paper is to analyze fundamental properties of the R?nyi and Augustin information from a noncommutative measure-theoretic perspective. Firstly, we prove the uniform equicontinuity for all three quantum versions of R?nyi and Augustin information, and it hence yields the joint continuity of these quantities in order and prior input distributions. Secondly, we establish the concavity of the scaled R?nyi and Augustin information in the region of s? (- 1 , 0) for both Petz’s and the sandwiched versions. This completes the open questions raised by Holevo (IEEE Trans?Inf?Theory 46(6):2256–2261, 2000), and Mosonyi and Ogawa (Commun?Math?Phys 355(1):373–426, 2017). For the applications, we show that the strong converse exponent in classical-quantum channel coding satisfies a minimax identity, which means that the strong converse exponent can be attained by the best constant composition code. The established concavity is further employed to prove an entropic duality between classical data compression with quantum side information and classical-quantum channel coding, and a Fenchel duality in joint source-channel coding with quantum side information. ? 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
Type
journal article