Rényi divergence inequalities via interpolation, with applications to generalised entropic uncertainty relations

Abstract

We investigate quantum Rényi entropic quantities, specifically those derived from 'sandwiched' divergence. This divergence is one of several proposed Rényi generalisations of the quantum relative entropy. We may define Rényi generalisations of the quantum conditional entropy and mutual information in terms of this divergence, from which they inherit many desirable properties. However, these quantities lack some of the convenient structure of their Shannon and von Neumann counterparts. We attempt to bridge this gap by establishing divergence inequalities for valid combinations of Rényi order which replicate the chain and decomposition rules of Shannon and von Neumann entropies. Although weaker in general, these inequalities recover equivalence when the Rényi parameters tend to one. To this end we present Rényi mutual information decomposition rules, a new approach to the Rényi conditional entropy tripartite chain rules and a more general bipartite comparison. The derivation of these results relies on a novel complex interpolation approach for general spaces of linear operators. These new comparisons allow us to employ techniques that until now were only available for Shannon and von Neumann entropies. We can therefore directly apply them to the derivation of Rényi entropic uncertainty relations. Accordingly, we establish a family of Rényi information exclusion relations and provide further generalisations and improvements to this and other known relations, including the Rényi bipartite uncertainty relations.

References in corpus (24)

• The Uncertainty Principle in the Presence of Quantum Memory
• Entropic Uncertainty Relations and their Applications
• Entanglement-Assisted Classical Capacity of Noisy Quantum Channels
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• Sandwiched Rényi Divergence Satisfies Data Processing Inequality
• Quantum Reverse Shannon Theorem
• Improved entropic uncertainty relations and information exclusion relations
• Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies
• On the quantum Renyi relative entropies and related capacity formulas
• An area law for entanglement from exponential decay of correlations
• Chain Rules for Smooth Min- and Max-Entropies
• Correlation Detection and an Operational Interpretation of the Renyi Mutual Information
• The decoupling approach to quantum information theory
• Strong converse exponent for classical-quantum channel coding
• Relating different quantum generalizations of the conditional Renyi entropy
• Renyi generalizations of the conditional quantum mutual information
• Operational Interpretation of Renyi Information Measures via Composite Hypothesis Testing Against Product and Markov Distributions
• Smooth Max-Information as One-Shot Generalization for Mutual Information
• Generalized statistics: yet another generalization
• The world according to Renyi: thermodynamics of fractal systems
• Convexity and Operational Interpretation of the Quantum Information Bottleneck Function
• Impossibility of Local State Transformation via Hypercontractivity
• Chain rules for quantum Rényi entropies
• Conjectured strong complementary-correlations tradeoff