Entanglement properties of Gaussian states

On the one hand we present necessary and sufficient conditions for both the separability and the distillability of bipartite Gaussian states of an arbitrary number of modes. On the other hand we classify tripartite three-mode Gaussian states according to their separability properties and provide necessary and sufficient conditions for separability of those states. All the conditions presented here can be easily checked by direct calculations, thus providing operational criteria. This solves the separability problem for all bipartite as well as tripartite three-mode Gaussian states and the distillability problem for all bipartite Gaussian states.