Properties of Noncommutative Renyi and Augustin Information

Abstract

Rényi and Augustin information are generalizations of mutual information defined via the Rényi 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 Rényi divergence -- the Petz's, sandwiched, and log-Euclidean versions, that possess meaningful operational interpretation. However, the associated quantum Rényi 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 Rényi and Augustin information from a noncommutative measure-theoretic perspective. Firstly, we prove the uniform equicontinuity for all three quantum versions of Rényi 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 Rényi and Augustin information in the region of $s\in (-1,0)$ 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.

arXiv commentFinal Version

References in corpus (32)

• Rényi Divergence and Kullback-Leibler Divergence
• On quantum Renyi entropies: a new generalization and some properties
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• From Classical to Quantum Shannon Theory
• Quantum Information Processing with Finite Resources -- Mathematical Foundations
• A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks
• Sandwiched Rényi Divergence Satisfies Data Processing Inequality
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• Second-order asymptotics for quantum hypothesis testing
• On the strong converses for the quantum channel capacity theorems
• Fundamental bound on the reliability of quantum information transmission
• Correlation Detection and an Operational Interpretation of the Renyi Mutual Information
• Coding Theorems of Quantum Information Theory
• Strong converse exponent for classical-quantum channel coding
• Lower bounds on the Probability of Error for Classical and Classical-Quantum Channels
• Reliability function of general classical-quantum channel
• Rényi divergences as weighted non-commutative vector valued $L_p$-spaces
• Rényi relative entropies and noncommutative $L_p$-spaces
• Quantum Sphere-Packing Bounds with Polynomial Prefactors
• The Renyi Capacity and Center
• Non-Asymptotic Classical Data Compression with Quantum Side Information
• Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions
• Coding theorems for compound problems via quantum Rényi divergences
• The Augustin Capacity and Center
• The Sphere Packing Bound For Memoryless Channels
• The Sphere Packing Bound via Augustin's Method
• On the Concavity of Auxiliary Function in Classical-Quantum Channels
• Duality between source coding with quantum side information and c-q channel coding
• Conditional Rényi entropy and the relationships between Rényi capacities
• Exact Asymptotics for the Random Coding Error Probability
• The Saddlepoint Approximation: Unified Random Coding Asymptotics for Fixed and Varying Rates
• Minimizing Quantum Renyi Divergences via Mirror Descent with Polyak Step Size