On quantum Renyi entropies: a new generalization and some properties

TL;DR

This paper introduces a new quantum generalization of Renyi entropies that unifies various existing quantum entropies and satisfies key properties like data-processing inequalities and duality relations.

Contribution

A novel quantum Renyi entropy definition that encompasses multiple existing entropies and satisfies important theoretical properties.

Findings

The new entropy includes von Neumann, min-, max-, and collision entropies as special cases.

It satisfies data-processing inequalities and duality relations.

It provides a unified framework for quantum entropies.

Abstract

The Renyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of Renyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Renyi entropies that contains the von Neumann entropy, min-entropy, collision entropy and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.