  This paper charts a very direct path between the categorical approach to
quantum mechanics, due to Abramsky and Coecke, and the older convex-operational
approach based on ordered vector spaces (recently reincarnated as "generalized
probabilistic theories"). In the former, the objects of a symmetric monoidal
category C are understood to represent physical systems and morphisms, physical
processes. Elements of the monoid C(I,I) are interpreted somewhat
metaphorically as probabilities. Any monoid homomorphism from the scalars of a
symmetric monoidal category C gives rise to a covariant functor V_o from C to a
category of dual-pairs of ordered vector spaces. Specifying a natural
transformation u from V_o to 1 (where 1 is the trivial such functor) allows us
to identify normalized states, and, thus, to regard the image category V_o(C)
as consisting of concrete operational models. In this case, if A and B are
objects in C, then V_o(A x B) defines a non-signaling composite of V_o(A) and
V_o(B). Provided either that C satisfies a "local tomography" condition, or
that C is compact closed, this defines a symmetric monoidal structure on the
image category, and makes V_o a (strict) monoidal functor.
