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

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.