Kahler fibrations in quantum information theory

We discuss the fibre bundle of co-adjoint orbits of compact Lie groups, and show how it admits a compatible Kahler structure. The case of the unitary group allows us to reformulate the geometric framework of quantum information theory. In particular, we show that the Fisher information tensor gives rise to a structure that is sufficiently close to a Kahler structure to generalise some classical result on co-adjoint orbits.