  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.
