  The notion of partial trace of a density operator is essential for the
understanding of the entanglement and separability properties of quantum
states. In this paper we investigate these notions putting an emphasis on the
geometrical properties of the covariance ellipsoids of the reduced states. We
thereafter focus on Gaussian states and we give new and easily numerically
implementable sufficient conditions for the separability of all Gaussian
states. Unlike the positive partial transposition criterion, none of these
conditions is however necessary.
