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)].

翻译：利用 [Sason, IEEE Trans. Inf. Th. 59, 7118 (2013)] 和 [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)] 中提出的技术，并基于后者的结果，我们构造了冯·诺依曼熵的一个全局最优连续性界。该界适用于满足吉布斯假设的任意哈密顿量所施加的能量约束下的任何量子态。这完全解决了在该设定下寻找冯·诺依曼熵最优连续性界的问题，而此前已知的结果仅适用于彼此足够接近的态对。我们的主要技术成果——一般能量约束下冯·诺依曼熵的全局最优半连续性界——导出了这一连续性界。为了证明它，我们还推导了具有可数无限字母表和一般约束的随机变量的最优Fano型不等式，以及相同设定下香农熵的最优半连续性界和连续性界。在此过程中，我们改进了 [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)] 中得出的结果。