Denes Petz, Csaba Sudar
Geometries of Quantum States
The paper is published: J. Math. Phys. 37, 6 (1996) 2662-2673

MSC:

60K40 Other physical applications of random processes

46L60 Applications of selfadjoint operator algebras to physics, See also {46N50, 46N55, 47D45, 81T05, 82B10, 82C10}

54E70 Probabilistic metric spaces

81S25 Quantum stochastic calculus

82B10 Quantum equilibrium statistical mechanics (general)

Abstract: The quantum analogue of the Fisher information metric of a
probability simplex is searched and several Riemannian metrics
on the set of positive definite density matrices are studied.
Some of them appeared in the literature in connection with
Cram{\'e}r-Rao type inequalities or the generalization of the Berry
phase to mixed states. They are shown to be stochastically monotone here.
All stochastically monotone Riemannian metrics are characterized
by means of operator monotone functions and it is proven that there
exist a maximal and a minimal among them. A class of metrics can
be extended to pure states and the Fubini-Study metric shows up
there.

Keywords: Fisher information, Cramer-Rao-type inequalities, stochastically monotone Riemannian metrics