We derive a necessary and sufficient condition for the separability of tripartite three-mode Gaussian states that is easy to check for any such state. We give a classification of the separability properties of those systems and show how to determine for any state to which class it belongs. We show that there exist genuinely tripartite bound entangled states and point out how to construct and prepare such states.

• Received 11 April 2001

DOI:https://doi.org/10.1103/PhysRevA.64.052303