  In the search of appropriate riemannian metrics on quantum state space the
concept of statistical monotonicity, or contraction under coarse graining, has
been proposed by Chentsov. The metrics with this property have been classified
by Petz. All the elements of this family of geometries can be seen as quantum
analogues of Fisher information. Although there exists a number of general
theorems sheding light on this subject, many natural questions, also stemming
from applications, are still open. In this paper we discuss a particular member
of the family, the Wigner-Yanase information.
  Using a well-known approach that mimics the classical pull-back approach to
Fisher information, we are able to give explicit formulae for the geodesic
distance, the geodesic path, the sectional and scalar curvatures associated to
Wigner-Yanase information. Moreover we show that this is the only monotone
metric for which such an approach is possible.
