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 building on results from the latter, we construct a globally optimal continuity bound for the von Neumann entropy under energy constraints imposed by arbitrary Hamiltonians, satisfying the Gibbs hypothesis. In particular, this provides a precise expression for the modulus of continuity of the von Neumann entropy over the set of states with bounded energy for infinite-dimensional quantum systems. Thus, it completely solves the problem of finding an optimal continuity bound for the von Neumann entropy in this setting, which was previously known only for pairs of states which were sufficiently close to each other. This continuity bound follows from a globally optimal semicontinuity bound for the von Neumann entropy under general energy constraints, which is our main technical result.

翻译：利用[Sason, IEEE Trans. Inf. Th. 59, 7118 (2013)]与[Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)]提出的技术，并基于后者的研究成果，我们为满足吉布斯假设的任意哈密顿量所施加能量约束下的冯·诺依曼熵，构建了一个全局最优的连续性界。特别地，这为无限维量子系统中能量有界状态集合上冯·诺依曼熵的连续性模提供了精确表达式。因此，该结果完全解决了在此设定下寻找冯·诺依曼熵最优连续性界的问题，而此前仅对彼此充分接近的状态对已知相关结果。该连续性界源自我们主要的技术成果——一般能量约束下冯·诺依曼熵的全局最优半连续性界。