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

Geometry of mixed states for a q-bit and the quantum Fisher information tensor