  The theory of monotone Riemannian metrics on the state space of a quantum
system was established by Denes Petz in 1996. In a recent paper he argued that
the scalar curvature of a statistically relevant - monotone - metric can be
interpreted as an average statistical uncertainty. The present paper
contributes to this subject. It is reasonable to expect that states which are
more mixed are less distinguishable than those which are less mixed. The
manifestation of this behavior could be that for such a metric the scalar
curvature has a maximum at the maximally mixed state. We show that not every
monotone metric fulfils this expectation, some of them behave in a very
different way. A mathematical condition is given for monotone Riemannian
metrics to have a local minimum at the maximally mixed state and examples are
given for such metrics.
