We prove a variety of new and refined uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality and the notion of maximal coupling of random variables. We then employ this classical result to derive the first tight energy-constrained continuity bound for the von Neumann entropy of states of infinite-dimensional quantum systems, when the Hamiltonian is the number operator, which is arguably the most relevant Hamiltonian in the study of infinite-dimensional quantum systems in the context of quantum information theory. The above scheme works only for Shannon- and von Neumann entropies. Hence, to deal with more general entropies, e.g. $\alpha$-R\'enyi and $\alpha$-Tsallis entropies, with $\alpha \in (0,1)$, for which continuity bounds are known only for finite-dimensional systems, we develop a novel approximation scheme which relies on recent results on operator H\"older continuous functions and the equivalence of all Schatten norms in special spectral subspaces of the Hamiltonian. This approach is, as we show, motivated by continuity bounds for $\alpha$-R\'enyi and $\alpha$-Tsallis entropies of random variables that follow from the H\"older continuity of the entropy functionals. Bounds for $\alpha>1$ are provided, too. Finally, we settle an open problem on related approximation questions posed in the recent works by Shirokov on the so-called Finite-dimensional Approximation (FA) property.

翻译：我们证明了一系列关于无穷状态空间上经典随机变量和无穷维系统量子态熵的新的和改进的一致连续性界。我们首次获得了具有可数无穷字母表的随机变量香农熵的紧致连续性估计，该证明依赖于一个新的均值约束Fano型不等式和随机变量的最大耦合概念。随后，我们利用这一经典结果推导了当哈密顿量为数算符时无穷维量子系统冯·诺依曼熵的首个紧致能量约束连续性界——数算符在量子信息论中无穷维量子系统的研究中堪称最相关的哈密顿量。上述方案仅适用于香农熵和冯·诺依曼熵。为处理更一般的熵（例如$\alpha \in (0,1)$时的$\alpha$-Rényi熵和$\alpha$-Tsallis熵，其连续性界此前仅对有限维系统已知），我们开发了一种新的近似方案，该方案基于算子Hölder连续函数的最新结果以及哈密顿量特定谱子空间中所有Schatten范数的等价性。我们指出，这种方法的动机源于遵循熵泛函Hölder连续性的随机变量$\alpha$-Rényi熵和$\alpha$-Tsallis熵的连续性界。本文也提供了$\alpha>1$时的界。最后，我们解决了Shirokov近期关于所谓有限维近似（FA）性质的研究中提出的相关近似问题的开放性疑问。