The Alicki-Fannes-Winter technique in the quasi-classical settings: advanced version and its applications
M. E. Shirokov
TL;DR
This work advances the Alicki-Fannes-Winter technique by introducing a refined, quasi-classical framework whose auxiliary lemmas optimize the core bounds. The authors derive universal, faithful semicontinuity and local lower bounds for central quantum information quantities (e.g., von Neumann entropy, quantum relative entropy, quantum conditional entropy, mutual information, and entanglement measures) under rank and energy constraints, with strong emphasis on commuting states. The resulting bounds are sharper and more broadly applicable, including infinite-dimensional settings and extensions to discrete random variables and oscillator-classical states, offering a unified approach that improves both accuracy and technical simplicity. The framework promises significant impact on quantitative continuity analyses in quantum information theory and related probabilistic settings.
Abstract
We describe an advanced version of the AFW-technique proposed in [Lett. Math. Phys., 113, 121 (2023)],[Lobachevskii J. Math., 44(6), 2169 (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).