On the Curvature of Monotone Metrics and a Conjecture Concerning the Kubo-Mori Metric

TL;DR

This paper investigates the curvature properties of monotone metrics on quantum state spaces, focusing on the Kubo-Mori metric, and addresses Petz's conjecture about scalar curvature behavior with state mixing.

Contribution

It provides a detailed analysis of curvature for monotone metrics, offers a formal proof of a key concavity conjecture related to the Kubo-Mori metric, and explores implications for quantum information geometry.

Findings

Scalar curvature of the Kubo-Mori metric increases with state mixing.

Confirmed the concavity of a specific function related to the metric.

Provided insights into the geometric structure of quantum state spaces.

Abstract

It is the aim of this article to determine curvature quantities of an arbitrary Riemannian monotone metric on the space of positive matrices resp. nonsingular density matrices. Special interest is focused on the scalar curvature due to its expected quantum statistical meaning. The scalar curvature is explained in more detail for three examples, the Bures metric, the largest monotone metric and the Kubo-Mori metric. In particular, we show an important conjecture of Petz concerning the Kubo-Mori metric up to a formal proof of the concavity of a certain function on R_+^3. This concavity seems to be numerically evident. The conjecture of Petz asserts that the scalar curvature of the Kubo-Mori metric increases if one goes to more mixed states.