Tight relations and equivalences between smooth relative entropies

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 R\'enyi divergences.