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The relation between the symplectic and Lorentz groups is explored toinvestigate entanglement features in a two-mode bipartite Gaussian state. Weverify that the correlation matrix of arbitrary Gaussian states can beassociated to a hyperbolic space with a Minkowski metric, which is divided intwo regions - separablelike and entangledlike, in equivalence to timelike andspacelike in special relativity. This correspondence naturally allows thedefinition of two insightful invariant squared distances measures - one relatedto the purity and another related to amount of entanglement. The seconddistance allows us to define a measure for entanglement in terms of theinvariant interval between the given state and its closest separable state,given in a natural manner without the requirement of a minimization procedure.

Minkowski structure for purity and entanglement of Gaussian bipartite states