On the geometry of mixed states and the Fisher information tensor

In this paper, we will review the co-adjoint orbit formulation of finite dimensionalquantum mechanics, and in this framework, we will interpret the notion of quantumFisher information index (and metric). Following previous work of part of theauthors, who introduced the definition of Fisher information tensor, we will showhow its antisymmetric part is the pullback of the natural Kostant-Kirillov-Souriausymplectic form along some natural diffeomorphism. In order to do this, we willneed to understand the symmetric logarithmic derivative as a proper 1-form, settlingthe issues about its very definition and explicit computation. Moreover, the fibrationof co-adjoint orbits, seen as spaces of mixed states, is also discussed.