  We consider the geometrization of quantum mechanics. We then focus on the
pull-back of the Fubini-Study metric tensor field from the projective Hibert
space to the orbits of the local unitary groups. An inner product on these
tensor fields allows us to obtain functions which are invariant under the
considered local unitary groups. This procedure paves the way to an algorithmic
approach to the identification of entanglement monotone candidates. Finally, a
link between the Fubini-Study metric and a quantum version of the Fisher
information metric is discussed.
