Monotone Riemannian metrics on density matrices with non-monotone scalar curvature

TL;DR

This paper investigates the scalar curvature of monotone Riemannian metrics on quantum state spaces, revealing that not all such metrics behave as expected with respect to the mixedness of states, and provides conditions for different behaviors.

Contribution

It introduces a mathematical condition for monotone metrics to have a local minimum at the maximally mixed state, challenging previous assumptions about scalar curvature behavior.

Findings

Some monotone metrics do not have maximum scalar curvature at the maximally mixed state.

A mathematical condition for local minima of scalar curvature at the maximally mixed state is established.

Examples of metrics with different curvature behaviors are provided.

Abstract

The theory of monotone Riemannian metrics on the state space of a quantum system was established by Denes Petz in 1996. In a recent paper he argued that the scalar curvature of a statistically relevant - monotone - metric can be interpreted as an average statistical uncertainty. The present paper contributes to this subject. It is reasonable to expect that states which are more mixed are less distinguishable than those which are less mixed. The manifestation of this behavior could be that for such a metric the scalar curvature has a maximum at the maximally mixed state. We show that not every monotone metric fulfils this expectation, some of them behave in a very different way. A mathematical condition is given for monotone Riemannian metrics to have a local minimum at the maximally mixed state and examples are given for such metrics.