Smooth Max-Information as One-Shot Generalization for Mutual Information

TL;DR

This paper explores the properties of smooth max-information, establishing its equivalence across definitions and deriving new chain rules, thereby extending the smooth entropy framework to quantum mutual information.

Contribution

It introduces equivalence relations among different smoothed max-information definitions and derives new chain rules, advancing the understanding of one-shot quantum information measures.

Findings

Different smoothed max-information definitions are essentially equivalent.

New chain rules for max-information are derived in terms of min- and max-entropies.

The results extend smooth entropy formalism to quantum mutual information.

Abstract

We study formal properties of smooth max-information, a generalization of von Neumann mutual information derived from the max-relative entropy. Recent work suggests that it is a useful quantity in one-shot channel coding, quantum rate distortion theory and the physics of quantum many-body systems. Max-information can be defined in multiple ways. We demonstrate that different smoothed definitions are essentially equivalent (up to logarithmic terms in the smoothing parameters). These equivalence relations allow us to derive new chain rules for the max-information in terms of min- and max-entropies, thus extending the smooth entropy formalism to mutual information.