A minimax approach to one-shot entropy inequalities

Abstract

One-shot information theory entertains a plethora of entropic quantities, such as the smooth max-divergence, hypothesis testing divergence and information spectrum divergence, that characterize various operational tasks and are used to prove the asymptotic behavior of various tasks in quantum information theory. Tight inequalities between these quantities are thus of immediate interest. In this note we use a minimax approach (appearing previously for example in the proofs of the quantum substate theorem), to simplify the quantum problem to a commutative one, which allows us to derive such inequalities. Our derivations are conceptually different from previous arguments and in some cases lead to tighter relations. We hope that the approach discussed here can lead to progress in open problems in quantum Shannon theory, and exemplify this by applying it to a simple case of the joint smoothing problem.

References in corpus (9)

• 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
• Min- and Max- Relative Entropies and a New Entanglement Monotone
• A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks
• Sandwiched Rényi Divergence Satisfies Data Processing Inequality
• Duality Between Smooth Min- and Max-Entropies
• Monotonicity of a relative Rényi entropy
• A largely self-contained and complete security proof for quantum key distribution
• On simultaneous min-entropy smoothing

Cited by in corpus (13)

• Entropy of a quantum channel
• On a gap in the proof of the generalised quantum Stein's lemma and its consequences for the reversibility of quantum resources
• Fundamental Limits on Correlated Catalytic State Transformations
• An information-theoretic treatment of quantum dichotomies
• One-Shot Yield-Cost Relations in General Quantum Resource Theories
• A solution of the generalised quantum Stein's lemma
• Moderate deviation expansion for fully quantum tasks
• Fidelity-Based Smooth Min-Relative Entropy: Properties and Applications
• Thermodynamic Implementations of Quantum Processes
• Parallelization of Adaptive Quantum Channel Discrimination in the Non-Asymptotic Regime
• Asymptotic quantification of entanglement with a single copy
• Asymptotic Equipartition Theorems in von Neumann algebras
• Tight relations and equivalences between smooth relative entropies