  After a review of the pure state case, we discuss from a geometrical point of
view the meaning of the quantum Fisher metric in the case of mixed states for a
two-level system, i.e. for a q-bit, by examining the structure of the fiber
bundle of states, whose base space can be identified with a co-adjoint orbit of
the unitary group. We show that the Fisher Information metric coincides with
the one induced by the metric of the manifold of SU(2), i.e. the 3-dimensional
sphere $S^3$, when the mixing coefficients are varied. We define the notion of
Fisher Tensor and show that its anti-symmetric part is intrinsically related to
the Kostant Kirillov Souriau symplectic form that is naturally defined on
co-adjoint orbits, while the symmetric part is nontrivially again represented
by the Fubini Study metric on the space of mixed states, weighted by the mixing
coefficients.
