Bosonic Gaussian channel and Gaussian witness entanglement criterion of continuous variables

We use quantum entanglement witnesses derived from Gaussian operators to study the separable criteria of continuous variable states. For bipartite system, we transform the validity of a Gaussian witness to a bosonic Gaussian channel problem. It follows that the maximal means of two-mode and some four-mode Gaussian operators over product pure states are achieved by vacuum (or coherent states and squeezed states) according to the properties of bosonic Gaussian channels. We demonstrate necessary and sufficient criteria of separability for two-mode and some four-mode Gaussian quantum states. We also propose multipartite Gaussian witnesses based on numeric evidence. The entanglement and multipartite entanglement detecting powers of the Gaussian witness criteria are illustrated by the mixture of two Gaussian states. We apply the Gaussian witness criterion to a Gaussian Fock state, which is prepared by passing a product Fock state through bosonic Gaussian channel. We show that the necessary criterion of separability for a Gaussian Fock state does not rely on the initial product Fock state, it relies on the channel.