Symplectic invariants, entropic measures and correlations of Gaussian states

Abstract

We present a derivation of the Von Neumann entropy and mutual information of arbitrarytwo-mode Gaussian states, based on the explicit determination of the symplecticeigenvalues of a generic covariance matrix. The key role of the symplectic invariants in sucha determination is pointed out. We show that the Von Neumann entropy depends on twosymplectic invariants, while the purity (or the linear entropy) is determined by only oneinvariant, so that the two quantities provide two different hierarchies of mixed Gaussianstates. A comparison between mutual information and entanglement of formation forsymmetric states is considered, taking note of the crucial role of the symplecticeigenvalues in qualifying and quantifying the correlations present in a generic state.