Strongly symmetric spectral convex bodies are Jordan algebra state spaces

We show that the strongly symmetric spectral convex compact sets are precisely the normalized state spaces of finite-dimensional simple Euclidean Jordan algebras and the simplices. Spectrality is the property that every state has a convex decomposition into perfectly distinguishable pure states; strong symmetry is transitivity, for each integer N, of the affine automorphism group of the state space on lists of N perfectly distinguishable pure states. Additional assumptions combine with this theorem to give simple characterizations of finite-dimensional complex quantum state space. Important aspects of quantum and classical thermodynamics and of query complexity have been generalized to classes of general probabilistic theories (GPTs) satisfying natural postulates including or implying spectrality and strong symmetry; our result shows that these apply to a narrower class of theories than might have been hoped. Sorkin's notion of irreducibly k-th order interference has been studied in the GPT framework and looked for in experiments. Our result shows that the assumption of no higher-order (k > 2) interference, used along with spectrality and strong symmetry to characterize the same class of Jordan-algebraic convex sets by Barnum, Mueller, and Ududec in arXiv:1403.4147, was superfluous. It also implies that Lee and Selby's extension, on the assumption that interference has fixed maximal degree k, of the important order square root of N lower bound on the quantum black-box query complexity of searching N possibilities for one having a desired property (which is achieved by Grover's quantum algorithm), to a class of theories satisfying certain postulates allowing the formulation of a generalized notion of query algorithm, actually applies in the Jordan-algebraic setting where higher-order interference is not possible.