  We discuss the fibre bundle of co-adjoint orbits of compact Lie groups, and
show how it admits a compatible K\"ahler 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 K\"ahler structure to generalise
some classical result on co-adjoint orbits.
