  We derive the invariant measure on the manifold of multimode quantum Gaussian
states, induced by the Haar measure on the group of Gaussian unitary
transformations. To this end, by introducing a bipartition of the system in two
disjoint subsystems, we use a parameterization highlighting the role of
nonlocal degrees of freedom -- the symplectic eigenvalues -- which characterize
quantum entanglement across the given bipartition. A finite measure is then
obtained by imposing a physically motivated energy constraint. By averaging
over the local degrees of freedom we finally derive the invariant distribution
of the symplectic eigenvalues in some cases of particular interest for
applications in quantum optics and quantum information.
