Using a result of H. Hanche-Olsen, we show that (subject to fairly natural
constraints on what constitutes a system, and on what constitutes a composite
system), orthodox finite-dimensional complex quantum mechanics with
superselection rules is the only non-signaling probabilistic theory in which
(i) individual systems are Jordan algebras (equivalently, their cones of
unnormalized states are homogeneous and self-dual), (ii) composites are locally
tomographic (meaning that states are determined by the joint probabilities they
assign to measurement outcomes on the component systems) and (iii) at least one
system has the structure of a qubit. Using this result, we also characterize
finite dimensional quantum theory among probabilistic theories having the
structure of a dagger-monoidal category
