Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki-Fannes-Winter technique

TL;DR

This paper develops a quasi-classical adaptation of the Alicki-Fannes-Winter technique to derive continuity bounds for quantum entropies and related measures, with applications to energy constraints and classical-quantum systems.

Contribution

It introduces a quasi-classical version of the Alicki-Fannes-Winter technique and applies it to obtain near-optimal continuity bounds for quantum entropies under various constraints.

Findings

Universal continuity bound for von Neumann entropy under energy constraints

Semi-continuity bounds for quantum conditional entropy and entanglement of formation

Continuity bounds for classical entropic characteristics in multi-mode systems

Abstract

We consider a quasi-classical version of the Alicki-Fannes-Winter technique widely used for quantitative continuity analysis of characteristics of quantum systems and channels. This version allows us to obtain continuity bounds under constraints of different types for quantum states belonging to subsets of a special form that can be called "quasi-classical". Several applications of the proposed method are described. Among others, we obtain the universal continuity bound for the von Neumann entropy under the energy-type constraint which in the case of one-mode quantum oscillator is close to the specialized optimal continuity bound presented recently by Becker, Datta and Jabbour. We obtain semi-continuity bounds for the quantum conditional entropy of quantum-classical states and for the entanglement of formation in bipartite quantum systems with the rank/energy constraint imposed only on one state. Semi-continuity bounds for entropic characteristics of classical random variables and classical states of a multi-mode quantum oscillator are also obtained.