On the monotonicity conjecture for the curvature of the Kubo-Mori metric

TL;DR

This paper investigates Petz's conjecture that the scalar curvature of the quantum state space with the Kubo-Mori metric increases with state mixing, providing partial proofs and numerical evidence for the conjecture.

Contribution

The authors group the summands of the scalar curvature and prove their monotonicity in some cases, offering numerical evidence for the remaining terms, and relate the conjecture's validity between real and complex density matrices.

Findings

Proved monotonicity for some summands of the scalar curvature

Provided numerical evidence for the monotonicity of remaining summands

Established that Petz's conjecture for complex matrices implies it for real matrices

Abstract

The canonical correlation or Kubo-Mori scalar product on the state space of a finite quantum system is a natural generalization of the classical Fisher metric. This metric is induced by the von Neumann entropy or the relative entropy of the quantum mechanical states. An important conjecture of Petz that the scalar curvature of the state space with Kubo-Mori scalar product as Riemannian metric is monotone with respect to the majorisation relation of states: the scalar curvature is increases if one goes to more mixed states. We give an appropriate grouping for the summands in the expression for the scalar curvature. The conjecture will follows from the monotonicity of the summands. We prove the monotonicity for some of these summands and we give numerical evidences that the remaining terms are monotone too. Note that the real density matrices form a submanifold of the complex density matrices. We prove that if Petz's conjecture true for complex density matrices then it is true for real density matrices too.