Extendability and complete extendability of Gaussian states

Motivated by the notions of $k$-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al (Phys. Rev. A, 69:022308), we introduce parallel definitions in the context of Gaussian states and derive necessary and sufficient conditions for their extendability. It is shown that every separable Gaussian state is completely extendable but the converse is still a conjecture. However, the converse is proved for two mode Gaussian states.