We present a derivation of the Von Neumann entropy and mutual information of
arbitrary two--mode Gaussian states, based on the explicit determination of the
symplectic eigenvalues of a generic covariance matrix. The key role of the
symplectic invariants in such a determination is pointed out. We show that the
Von Neumann entropy depends on two symplectic invariants, while the purity (or
the linear entropy) is determined by only one invariant, so that the two
quantities provide two different hierarchies of mixed Gaussian states. A
comparison between mutual information and entanglement of formation for
symmetric states is considered, remarking the crucial role of the symplectic
eigenvalues in qualifying and quantifying the correlations present in a generic
state.