Strong converse theorems using Rényi entropies

Abstract

We use a Rényi entropy method to prove strong converse theorems for certain information-theoretic tasks which involve local operations and quantum or classical communication between two parties. These include state redistribution, coherent state merging, quantum state splitting, measurement compression with quantum side information, randomness extraction against quantum side information, and data compression with quantum side information. The method we employ in proving these results extends ideas developed by Sharma [arXiv:1404.5940], which he used to give a new proof of the strong converse theorem for state merging. For state redistribution, we prove the strong converse property for the boundary of the entire achievable rate region in the $(e,q)$-plane, where $e$ and $q$ denote the entanglement cost and quantum communication cost, respectively. In the case of measurement compression with quantum side information, we prove a strong converse theorem for the classical communication cost, which is a new result extending the previously known weak converse. For the remaining tasks, we provide new proofs for strong converse theorems previously established using smooth entropies. For each task, we obtain the strong converse theorem from explicit bounds on the figure of merit of the task in terms of a Rényi generalization of the optimal rate. Hence, we identify candidates for the strong converse exponents for each task discussed in this paper. To prove our results, we establish various new entropic inequalities, which might be of independent interest. These involve conditional entropies and mutual information derived from the sandwiched Rényi divergence. In particular, we obtain novel bounds relating these quantities, as well as the Rényi conditional mutual information, to the fidelity of two quantum states.

40 pages, 5 figures; v4: Accepted for publication in Journal of Mathematical Physics

References in corpus (8)

• Coding Theorem and Strong Converse for Quantum Channels
• Leftover Hashing Against Quantum Side Information
• The mother of all protocols: Restructuring quantum information's family tree
• The information-theoretic costs of simulating quantum measurements
• Renyi generalizations of the conditional quantum mutual information
• Duality of privacy amplification against quantum adversaries and data compression with quantum side information
• State redistribution as merging: introducing the coherent relay
• A strong converse for the quantum state merging protocol

Cited by in corpus (18)

• Quantum Resource Theories
• Resource theory of asymmetric distinguishability
• Renyi relative entropies of quantum Gaussian states
• Non-Asymptotic Classical Data Compression with Quantum Side Information
• Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification
• Computable Rényi mutual information: Area laws and correlations
• Convex-split and hypothesis testing approach to one-shot quantum measurement compression and randomness extraction
• A one-shot achievability result for quantum state redistribution
• Relative entropy for von Neumann subalgebras
• Thermodynamic Constraints on Quantum Information Gain and Error Correction: A Triple Trade-Off
• Communication Complexity of One-Shot Remote State Preparation
• Relating relative Rényi entropies and Wigner-Yanase-Dyson skew information to generalized multiple quantum coherences
• One-shot quantum state redistribution and quantum Markov chains
• Operator-valued Schatten spaces and quantum entropies
• Upper Bounds on the Distillable Randomness of Bipartite Quantum States
• A simple coin for a $2d$ entangled walk
• Generalized Entropic Quantum Speed Limits
• Uniform continuity bound for sandwiched Rényi conditional entropy