  Among the many important geometric properties of quantum state space are:
transitivity of the group of symmetries of the cone of unnormalized states on
its interior (homogeneity), identification of this cone with its dual cone of
effects via an inner product (self-duality), and transitivity of the group of
symmetries of the normalized state space on the pure normalized states (pure
transitivity). Koecher and Vinberg showed that homogeneity and self-duality
characterize Jordan-algebraic state spaces: real, complex and quaternionic
quantum theory, spin factors, 3-dimensional octonionic quantum state space and
direct sums of these irreducible spaces. We show that self-duality follows from
homogeneity and pure transitivity. These properties have a more direct physical
and information-processing significance than self-duality. We show for instance
(extending results of Barnum, Gaebeler, and Wilce) that homogeneity is closely
related to the ability to steer quantum states. Our alternative to the
Koecher-Vinberg theorem characterizes nearly the same set of state spaces:
direct sums of isomorphic Jordan-algebraic ones, which may be viewed as
composites of a classical system with an irreducible Jordan-algebraic one.
There are various physically and informationally natural additional postulates
that are known to single out complex quantum theory from among these
Jordan-algebraic possibilities. We give various such reconstructions based on
the additional property of local tomography.
