Chain Rules for Smooth Min- and Max-Entropies

TL;DR

This paper investigates the chain rule properties of smooth min- and max-entropy, establishing inequalities that generalize the classical chain rule for von Neumann entropy to one-shot information theory.

Contribution

It derives a set of inequalities for the chain rule of smooth min- and max-entropy, extending classical entropy relations to the one-shot setting.

Findings

Chain rule for smooth min- and max-entropy expressed as inequalities.

Reduces to von Neumann entropy chain rule in the i.i.d. case.

Provides foundational tools for one-shot information theory.

Abstract

The chain rule for the Shannon and von Neumann entropy, which relates the total entropy of a system to the entropies of its parts, is of central importance to information theory. Here we consider the chain rule for the more general smooth min- and max-entropy, used in one-shot information theory. For these entropy measures, the chain rule no longer holds as an equality, but manifests itself as a set of inequalities that reduce to the chain rule for the von Neumann entropy in the i.i.d. case.