The precise one-shot characterisation of operational tasks in classical and quantum information theory relies on different forms of smooth entropic quantities. A particularly important connection is between the hypothesis testing relative entropy and the smooth max-relative entropy, which together govern many operational settings. We first strengthen this connection into a type of equivalence: we show that the hypothesis testing relative entropy is equivalent to a variant of the smooth max-relative entropy based on the information spectrum divergence, which can be alternatively understood as a measured smooth max-relative entropy. Furthermore, we improve a fundamental lemma due to Datta and Renner that connects the different variants of the smooth max-relative entropy, introducing a modified proof technique based on matrix geometric means and a tightened gentle measurement lemma. We use the unveiled connections and tools to strictly improve on previously known one-shot bounds and duality relations between the smooth max-relative entropy and the hypothesis testing relative entropy, establishing provably tight bounds between them. The results then allow us to refine other divergence inequalities, in particular sharpening bounds that connect the max-relative entropy with Rényi divergences.

翻译：在经典与量子信息理论中，操作任务的精确一次性刻画依赖于不同形式的平滑熵量。一个特别重要的联系是假设检验相对熵与平滑最大相对熵之间的关系，这两个量共同支配着许多操作场景。我们首先将这一联系强化为一种等价性类型：证明假设检验相对熵等价于基于信息谱散度的平滑最大相对熵的变体，该变体也可理解为测量平滑最大相对熵。此外，我们改进了由Datta和Renner提出的连接不同平滑最大相对熵变体的基本引理，引入了一种基于矩阵几何均值的修正证明技巧以及收紧的温和测量引理。利用所揭示的联系和工具，我们严格改进了先前已知的关于平滑最大相对熵与假设检验相对熵之间的一次性界限和对偶关系，建立了它们之间可证明的紧致界限。这些结果随后使我们能够精炼其他散度不等式，特别是强化了连接最大相对熵与Rényi散度的界限。