  We investigate different geometries and invariant measures on the space of
mixed Gaussian quan- tum states. We show that when the global purity of the
state is held fixed, these measures coincide and it is possible, within this
constraint, to define a unique notion of volume on the space of mixed Gaussian
states. We then use the so defined measure to study typical non-classical
correlations of two mode mixed Gaussian quantum states, in particular
entanglement and steerability. We show that under the purity constraint alone,
typical values for symplectic invariants can be computed very elegantly,
irrespectively of the non-compactness of the underlying state space. Then we
consider finite volumes by constraining the purity and energy of the Gaussian
state and compute typical values of quantum correlations numerically.
