Self-duality and Jordan structure of quantum theory follow from homogeneity and pure transitivity

Abstract

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.