Properties of Noncommutative Renyi and Augustin Information
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The scaled Rényi information plays 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 Rényi divergence---the Petz's, sandwiched, and log-Euclidean versions, that possess meaningful operational interpretation. However, these scaled noncommutative Rényi informations are much less explored compared with their classical counterpart, and lacking crucial properties hinders applications of these quantities to refined performance analysis. The goal of this paper is thus to analyze fundamental properties of scaled Rényi information from a noncommutative measure-theoretic perspective. Firstly, we prove the uniform equicontinuity for all three quantum versions of Rényi information, hence it yields the joint continuity of these quantities in the orders and priors. Secondly, we establish the concavity in the region of s∈(-1,0) for both Petz's and the sandwiched versions. This completes the open questions raised by Holevo [https://ieeexplore.ieee.org/document/868501/IEEE Trans. Inf. Theory, 46(6):2256--2261, 2000], Mosonyi and Ogawa [https://doi.org/10.1007/s00220-017-2928-4/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. 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 in the forthcoming papers.
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