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arXiv:2505.00882 · Lobachevskii Journal of Mathematics

The Alicki–Fannes–Winter Technique in the Quasi-classical Settings: Advanced Version and Its Applications

Максим Евгеньевич Широков

Published Jun 1, 2025• 0 citations
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Abstract

Abstract We describe an advanced version of the AFW-technique proposed in [M. E. Shirokov, ‘‘Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki–Fannes–Winter technique’’, Lett. Math. Phys. 113, 121 (2023); M. E. Shirokov, ‘‘Local lower bounds on characteristics of quantum and classical systems’’, Lobachevskii J. Math. 44, 2169–2191 (2023)] which allows us to obtain lower semicontinuity bounds, continuity bounds and local lower bounds for characteristics of quantum systems and discrete random variables. We consider applications of the new version of the AFW-technique to several basic characteristics of quantum systems (the von Neumann entropy, the energy-type functionals, the quantum relative entropy, the conditional entropy and the entanglement of formation).

DOI
10.1134/s1995080225607891

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Abstract We describe an advanced version of the AFW-technique proposed in [M. E. Shirokov, ‘‘Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki–Fannes–Winter technique’’, Lett. Math. Phys. 113, 121 (2023); M. E. Shirokov, ‘‘Local lower bounds on characteristics of quantum and classical systems’’, Lobachevskii J. Math. 44, 2169–2191 (2023)] which allows us to obtain lower semicontinuity bounds, continuity bounds and local lower bounds for characteristics of quantum systems and discrete random variables. We consider applications of the new version of the AFW-technique to several basic characteristics of quantum systems (the von Neumann entropy, the energy-type functionals, the quantum relative entropy, the conditional entropy and the entanglement of formation).
We consider methods for obtaining local lower bounds on characteristics of quantum (resp. classical) systems, i.e., lower bounds valid in the trace norm ϵ\epsilonϵ -neighborhood of a given state (resp. probability distribution). Our basic tool is the quasi-classical modification of the Alicki–Fannes–Winter technique which gives faithful one-side continuity bounds with the rank/energy constraint imposed on only one of the two quantum states (resp. probability distributions). We also propose a new universal method that allows us to obtain easy computable faithful local lower bounds on many important characteristics of quantum (resp. classical) systems. The main attention is paid to infinite-dimensional systems.
Abstract We describe the most general form of the Alicki–Fannes–Winter method for obtaining uniform continuity bounds for characteristics of composite quantum systems and consider several modifications of this method, which make it applicable to a wide class of characteristics in both finite-dimensional and infinite-dimensional cases. We present uniform continuity bounds for the most important characteristics of composite quantum systems. Along with the results obtained earlier by various authors, we describe a number of new results proved by the proposed methods. In particular, we obtain new continuity bounds for the quantum discord, one-way classical correlation and its regularization in finite-dimensional and infinite-dimensional bipartite quantum systems.
Using techniques proposed in [Sason, IEEE Trans. Inf. Th. 59, 7118 (2013)] and [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)], and based on the results from the latter, we construct a globally optimal continuity bound for the von Neumann entropy. This bound applies to any state under energy constraints imposed by arbitrary Hamiltonians that satisfy the Gibbs hypothesis. This completely solves the problem of finding an optimal continuity bound for the von Neumann entropy in this setting, previously known only for pairs of states that are sufficiently close to each other. Our main technical result, a globally optimal semicontinuity bound for the von Neumann entropy under general energy constraints, leads to this continuity bound. To prove it, we also derive an optimal Fano-type inequality for random variables with a countably infinite alphabet and a general constraint, as well as optimal semicontinuity and continuity bounds for the Shannon entropy in the same setting. In doing so, we improve the results derived in [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)].
In this short note, I show how a recent result of Alhejji and Smith (A tight uniform continuity bound for equivocation, 2019. arXiv:1909.00787v1) 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 the entanglement of formation that improves upon the one previously given by Winter (Commun Math Phys 347(1):291–313, 2016. 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.