On the monotonicity of scalar curvature in classical and quantum information geometry

TL;DR

This paper investigates the monotonicity properties of scalar curvature in classical and quantum information geometry, proposing conjectures and linking them to existing conjectures like Petz's, supported by numerical data.

Contribution

It introduces new conjectures about quantum alpha-geometries and their scalar curvature monotonicity, connecting these to the Petz conjecture.

Findings

Numerical data supports conjectures about quantum alpha-geometries.

Proposes that the scalar curvature monotonicity implies Petz's conjecture.

Links classical and quantum information geometry through scalar curvature properties.

Abstract

We study the statistical monotonicity of the scalar curvature for the alpha-geometries on the simplex of probability vectors. From the results obtained and from numerical data we are led to some conjectures about quantum alpha-geometries and Wigner-Yanase-Dyson information. Finally we show that this last conjecture implies the truth of the Petz conjecture about the monotonicity of the scalar curvature of the Bogoliubov-Kubo-Mori monotone metric.