Bound entangled gaussian states

We discuss the entanglement properties of bipartite states with Gaussian Wi gner functions. For the separability, and the positivity of the partial tra nspose, we establish explicit necessary and sufficient criteria in terms of the covariance matrix of the state. It is shown that, for systems composed of a single oscillator for Alice and an arbitrary number for Bob, positivi ty of the partial transpose implies separability. However, this implication fails with two oscillators on each side, as we show by constructing a five parameter family of bound entangled Gaussian states.