  We study the Hilbert-Schmidt measure on the manifold of mixed Gaussian states
in multi mode continuous variable quantum systems. An analytical expression for
the Hilbert-Schmidt volume element is derived. Its corresponding probability
measure can be used to study typical properties of Gaussian states. It turns
out that although the manifold of Gaussian states is unbounded, an ensemble of
Gaussian states distributed according to this measure still has a normalizable
distribution of symplectic eigenvalues, from which unitarily invariant
properties can be obtained. By contrast, we find that for an ensemble of
one-mode Gaussian states based on the Bures measure the corresponding
distribution cannot be normalized. As important applications, we determine the
distribution and the mean value of von Neumann entropy and purity for the
Hilbert-Schmidt measure.
