Tight relations and equivalences between smooth relative entropies

• URL: http://arxiv.org/abs/2501.12447v2
• Date: Sun, 02 Feb 2025 14:54:36 GMT
• Title: Tight relations and equivalences between smooth relative entropies
• Authors: Bartosz Regula, Ludovico Lami, Nilanjana Datta,
• Abstract summary: 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. We also introduce a modified proof technique based on matrix geometric means and a tightened gentle measurement lemma.
• Score: 12.699007098398805
• License:
• Abstract: 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 smoothed 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 smoothed 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, sharpening also bounds that connect the max-relative entropy with R\'enyi divergences.

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