Quantum Relative-alpha-Entropies: A Structural and Geometric Perspective

TL;DR

This paper introduces a new quantum relative-alpha-entropy that extends Umegaki's entropy, highlighting its geometric properties and advantages over classical divergence frameworks.

Contribution

It presents a novel quantum divergence outside the f-divergence class, with unique convexity, invariance, and geometric features, and establishes its classical correspondence.

Findings

The divergence exhibits nonlinear convexity for alpha > 1.

It is additive under tensor products and invariant under unitaries.

It corresponds exactly to classical relative-alpha-entropy via Nussbaum-Szkola distributions.

Abstract

Most quantum divergences derive their structure from classical f-divergences or Renyi-type constructions, a dependence that obscures several quantum geometric effects. We introduce a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. The proposed divergence exhibits a nonlinear convexity property, which yields a generalized convexity result for the Petz-Renyi divergence for alpha greater than one, complementing the known convexity for alpha less than one. It is additive under tensor products, invariant under unitary transformations, and depends only on the relative geometry of quantum states rather than their absolute magnitudes. Using Nussbaum-Szkola-type distributions, we also establish an exact correspondence of this divergence with classical relative-alpha-entropy. This reveals relative-alpha-entropy as a fundamentally geometric notion of quantum distinguishability not captured by existing divergence frameworks.