  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.
