  Chentsov studied Riemannian metrics on the set of probability measures from
the point of view of decision theory. He proved that up to a constant factor
the Fisher information is the only metric which is monotone under stochastic
transformations. The present paper deals with monotone metrics on the space of
finite density matrices on the basis motivated by quantum mechanics. A
characterization of those metrics is given in terms of operator monotone
functions. Several concrete metrics are constructed and analyzed, in
particular, instead of the uniqueness in the probabilistic case, there is a
large class of monotone metrics. Some of those appeared already in the physics
literature a long time ago. A limiting procedure to pure states is discussed as
well.
