# A minimax approach to one-shot entropy inequalities

## TL;DR

This paper introduces a minimax approach to derive tighter inequalities among one-shot entropic quantities in quantum information theory, simplifying complex quantum problems to commutative cases and potentially advancing quantum Shannon theory.

## Contribution

It presents a novel minimax method that simplifies quantum entropy inequalities to commutative cases, leading to tighter bounds and new insights.

## Key findings

- Derived tighter entropy inequalities using the minimax approach
- Simplified quantum problems to commutative cases for easier analysis
- Applied method to a joint smoothing problem in quantum Shannon theory

## Abstract

One-shot information theory entertains a plethora of entropic quantities, such as the smooth max-divergence, hypothesis testing divergence and information spectrum divergence, that characterize various operational tasks and are used to prove the asymptotic behavior of various tasks in quantum information theory. Tight inequalities between these quantities are thus of immediate interest. In this note we use a minimax approach (appearing previously for example in the proofs of the quantum substate theorem), to simplify the quantum problem to a commutative one, which allows us to derive such inequalities. Our derivations are conceptually different from previous arguments and in some cases lead to tighter relations. We hope that the approach discussed here can lead to progress in open problems in quantum Shannon theory, and exemplify this by applying it to a simple case of the joint smoothing problem.