  We use matched quantum entanglement witnesses to study the separable criteria
of continuous variable states. The witness can be written as an identity
operator minus a Gaussian operator. The optimization of the witness then is
transformed to an eigenvalue problem of a Gaussian kernel integral equation. It
follows a separable criterion not only for symmetric Gaussian quantum states,
but also for non-Gaussian states prepared by photon adding to or/and
subtracting from symmetric Gaussian states. Based on Fock space numeric
calculation, we obtain an entanglement witness for more general two-mode
states. A necessary criterion of separability follows for two-mode states and
it is shown to be necessary and sufficient for a two mode squeezed thermal
state and the related two-mode non-Gaussian states. We also connect the witness
based criterion with Werner-Wolf criterion and refine the Werner-Wolf
criterion.
