P-representable subset of all bipartite Gaussian separable states

Abstract

P-representability is a necessary and sufficient condition for separability of bipartite Gaussian states only for the special subset of states whose covariance matrix are $Sp(2,R)\otimes Sp(2,R)$ locally invariant. Although this special class of states can be reached by a convenient $Sp(2,R)\otimes Sp(2,R)$ transformation over an arbitrary covariance matrix, it represents a loss of generality, avoiding inference of many general aspects of separability of bipartite Gaussian states.