Generalized Entropies

Abstract

We study an entropy measure for quantum systems that generalizes the von Neumann entropy as well as its classical counterpart, the Gibbs or Shannon entropy. The entropy measure is based on hypothesis testing and has an elegant formulation as a semidefinite program, a type of convex optimization. After establishing a few basic properties, we prove upper and lower bounds in terms of the smooth entropies, a family of entropy measures that is used to characterize a wide range of operational quantities. From the formulation as a semidefinite program, we also prove a result on decomposition of hypothesis tests, which leads to a chain rule for the entropy.

21 pages

References in corpus (9)

• Quantum information can be negative
• Quantum cryptography with finite resources: unconditional security bound for discrete-variable protocols with one-way post-processing
• Leftover Hashing Against Quantum Side Information
• A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks
• Finite blocklength converse bounds for quantum channels
• The decoupling approach to quantum information theory
• Smooth Renyi Entropies and the Quantum Information Spectrum
• Sine distance for quantum states
• Semidefinite programs for completely bounded norms

Cited by in corpus (37)

• Quantum Resource Theories
• Quantum conditional mutual information and approximate Markov chains
• Quantum fidelity measures for mixed states
• alpha-z-relative Renyi entropies
• One-Shot Operational Quantum Resource Theory
• Convex geometry of quantum resource quantification
• Beyond heat baths: Generalized resource theories for small-scale thermodynamics
• Renyi generalizations of the conditional quantum mutual information
• Quantum Relative Lorenz Curves
• Second-Order Asymptotics for the Classical Capacity of Image-Additive Quantum Channels
• Introducing one-shot work into fluctuation relations
• Thermodynamic Capacity of Quantum Processes
• Beyond heat baths II: Framework for generalized thermodynamic resource theories
• Coherence and entanglement measures based on Rényi relative entropies
• Fundamental Finite Key Limits for One-Way Information Reconciliation in Quantum Key Distribution
• One-shot lossy quantum data compression
• Relative Thermalization
• Sequential decoding of a general classical-quantum channel
• Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification
• Resource theory of quantum uncomplexity
• A one-shot achievability result for quantum state redistribution
• Quantum Markov chains, sufficiency of quantum channels, and Renyi information measures
• Macroscopic Thermodynamic Reversibility in Quantum Many-Body Systems
• Duality of channels and codes
• Bounding the forward classical capacity of bipartite quantum channels
• Fidelity-Based Smooth Min-Relative Entropy: Properties and Applications
• Thermodynamic Implementations of Quantum Processes
• The refined quantum extremal surface prescription from the asymptotic equipartition property
• Communication Complexity of One-Shot Remote State Preparation
• On privacy amplification, lossy compression, and their duality to channel coding
• Lower Bounds for Quantum Parameter Estimation
• Smooth entropy in axiomatic thermodynamics
• Fixed Error Asymptotics For Erasure and List Decoding
• Towards Optimal Quantum Ranging -- Hypothesis Testing for an Unknown Return Signal
• On distinguishability distillation and dilution exponents
• Local Invariance of Divergence-based Quantum Information Measures
• Tight relations and equivalences between smooth relative entropies