I'll give theorems characterizing finite-dimensional quantum theory's framework of  density matrices (states) and POVM elements (measurement outcomes) for describing systems  by simple postulates whose physical and informational meaning and appeal is clear.  Each theorem first characterizes a class of Euclidean Jordan-algebraic (EJA) systems.  This is already only a slightly larger class than the usual quantum theory: real, complex, and quaternionic quantum theory, systems whose state spaces are balls, and 3-dimensional octonionic quantum theory. Complex quantum theory then follows from "local tomography", or  "energy observability": the generators of continuous symmetries of the state space (potential reversible dynamics) are observables.

The first characterization (with Cozmin Ududec and John van de Wetering) uses:   
(1) Homogeneity: any strictly positive element of the cone of unnormalized states may be taken to any other by a symmetry of this cone, 
(2) Pure Transitivity: any pure state may be taken to any other pure state by a symmetry of the normalized state space

Time permitting, I'll discuss a second characterization (with Joachim Hilgert), using:  (1) Spectrality: every state is a convex combination of perfectly distinguishable pure states, (2) Strong Symmetry: every set of perfectly distinguishable pure states may be taken to any other such set (of the same size) by a symmetry of the state space.

The physical, informational, and operational significance of the postulates will be discussed.

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