Extending the Fisher metric to density matrices

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.