  There are several important abstract operator systems with the convex cone of
positive semidefinite matrices at the first level. Well-known are the operator
systems of separable matrices, of positive semidefinite matrices, and of block
positive matrices. In terms of maps, these are the operator systems of
entanglement breaking, completely positive, and positive linear maps,
respectively. But there exist other interesting and less well-studied such
operator systems, for example those of completely copositive maps, doubly
completely positive maps, and decomposable maps, which all play an important
role in quantum information theory. We investigate which of these systems is
finitely generated, and which admits a finite-dimensional realization in the
sense of the Choi-Effros Theorem. We answer this question for all of the
described systems completely. Our main contribution is that decomposable maps
form a system which does not admit a finite-dimensional realization, though
being finitely generated, whereas the system of doubly completely positive maps
is not finitely generated, though having a finite-dimensional realization. This
also implies that there cannot exist a finitary Choi-type characterization of
doubly completely positive maps.
