Optimal uniform continuity bound for conditional entropy of classical--quantum states

Abstract

In this short note, I show how a recent result of Alhejji and Smith [arXiv:1909.00787] regarding an optimal uniform continuity bound for classical conditional entropy leads to an optimal uniform continuity bound for quantum conditional entropy of classical--quantum states. The bound is optimal in the sense that there always exists a pair of classical--quantum states saturating the bound, and so no further improvements are possible. An immediate application is a uniform continuity bound for entanglement of formation that improves upon the one previously given by Winter in [arXiv:1507.07775]. Two intriguing open questions are raised regarding other possible uniform continuity bounds for conditional entropy, one about quantum--classical states and another about fully quantum bipartite states.

arXiv commentv2: 11 pages, accepted for publication in Quantum Information Processing

References in corpus (15)

• Mixed State Entanglement and Quantum Error Correction
• 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
• Approaches for approximate additivity of the Holevo information of quantum channels
• Continuity of quantum channel capacities
• Approximate Degradable Quantum Channels
• Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels
• Tight continuity bounds for the quantum conditional mutual information, for the Holevo quantity and for capacities of quantum channels
• Quantum and private capacities of low-noise channels
• Quantum conditional entropy for infinite-dimensional systems
• 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
• Adaptation of the Alicki-Fannes-Winter method for the set of states with bounded energy and its use
• A Tight Uniform Continuity Bound for Equivocation

Cited by in corpus (10)

• Advanced Alicki-Fannes-Winter method for energy-constrained quantum systems and its use
• Bounding quantum capacities via partial orders and complementarity
• Continuity of characteristics of composite quantum systems
• A tight uniform continuity bound for the Arimoto-Rényi conditional entropy and its extension to classical-quantum states
• Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki-Fannes-Winter technique
• Continuity of entropies via integral representations
• Entanglement cost for infinite-dimensional physical systems
• Continuity bounds for quantum entropies arising from a fundamental entropic inequality
• Fundamental limitations to key distillation from Gaussian states with Gaussian operations
• Lipschitz continuity of quantum-classical conditional entropies with respect to angular distance and related properties