Continuity bounds for quantum entropies arising from a fundamental entropic inequality

Abstract

We establish a tight upper bound for the difference in von Neumann entropies between two quantum states, ${\rho }_1$ and ${\rho }_2$. This bound is expressed in terms of the von Neumann entropies of the mutually orthogonal states derived from the Jordan-Hahn decomposition of the difference operator $({\rho }_1-{\rho }_2)$. This yields a novel entropic inequality that implies the well-known Audenaert-Fannes (AF) inequality. In fact, it also leads to a refinement of the AF inequality. We employ this inequality to obtain a uniform continuity bound for the quantum conditional entropy of two states whose marginals on the conditioning system coincide. We additionally use it to derive a continuity bound for the quantum relative entropy in both variables. Interestingly, the fundamental entropic inequality is also valid in infinite dimensions.

v4: Corresponds to published version. v3: Strengthened the main result, and significantly simplified the proof. v2: Extended results in multiple directions. 19 pages, 1 figure

References in corpus (27)

• Min- and Max- Relative Entropies and a New Entanglement Monotone
• Quantum Information Processing with Finite Resources -- Mathematical Foundations
• A Sharp Fannes-type Inequality for the von Neumann Entropy
• Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints
• Continuity of quantum conditional information
• On the quantum Renyi relative entropies and related capacity formulas
• Continuity bounds on the quantum relative entropy - II
• On asymptotic continuity of functions of quantum states
• Tight bound on relative entropy by entropy difference
• Entropy and relative entropy from information-theoretic principles
• Optimal Extensions of Resource Measures and their Applications
• Uniform continuity bounds for information characteristics of quantum channels depending on input dimension and on input energy
• Continuity bounds for information characteristics of quantum channels depending on input dimension
• Continuity of quantum entropic quantities via almost convexity
• Adaptation of the Alicki-Fannes-Winter method for the set of states with bounded energy and its use
• Improved Quantum Algorithms for Fidelity Estimation
• Optimal uniform continuity bound for conditional entropy of classical--quantum states
• Advanced Alicki-Fannes-Winter method for energy-constrained quantum systems and its use
• Upper continuity bounds on the relative $q$-entropy for $q>1$
• Continuity of characteristics of composite quantum systems
• Modulus of convexity for operator convex functions
• General Continuity Bounds for Quantum Relative Entropies
• Bounds on the Concavity of Quantum Entropy
• A Tight Uniform Continuity Bound for Equivocation
• Continuity of entropies via integral representations
• Unified framework for continuity of sandwiched Rényi divergences
• Quantum relative entropy: general convergence criterion and preservation of convergence under completely positive linear maps

Cited by in corpus (1)

• Additivity of quantum capacities in simple non-degradable quantum channels