On the curvature of monotone metrics and a conjecture concerning the Kubo-Mori metric

Abstract

The manifold of trace one positive complex n×n-matrices represents the space of faithful mixed states of a finite dimensional quantum system. Riemannian monotone metrics on these manifolds are in one-to-one correspondence with operator monotone functions. These metrics generalize the classical Fisher metric of statistical distinguishability of probability distributions to the quantum case and are of interest in quantum statistics and information theory. Thus, it is natural to ask for curvature quantities of these Riemannian metrics. It is known that the classical Fisher metric is of positive constant curvature. The quantum case considered here is much more complicated. Most informations are available for the Bures metric, a special distinguished monotone metric. Partial curvature results are also known for the Kubo-Mori metric. It is the aim of this article to determine curvature quantities of an arbitrary Riemannian monotone metric. The resulting scalar curvature is explained in more detail for three examples. In particular, we consider an important conjecture of Petz asserting that the scalar curvature of the Kubo-Mori metric increases if one goes to more mixed states. This assertion is shown up to a formal proof of the concavity of a certain function on R + 3 . However, this concavity seems to be numerically evident and we give certain function plots supporting this concavity.