We present a new criterion that determines whether a fermionic state is a
convex combination of pure Gaussian states. This criterion is complete and
characterizes the set of convex-Gaussian states from the inside. If a state
passes a program it is a convex-Gaussian state and any convex-Gaussian state
can be approximated with arbitrary precision by states passing the criterion.
The criterion is presented in the form of a sequence of solvable semidefinite
programs. It is also complementary to the one developed by de Melo, Cwiklinski
and Terhal, which aims at characterizing the set of convex-Gaussian states from
the outside. Here we present an explicit proof that criterion by de Melo et al.
is complete, by estimating a distance between an n-extendible state, a state
that passes the criterion, to the set of convex-Gaussian states
