2019/04/07 by Howard Barnum, Barnum, Howard, Joachim Hilgert +1 · 4 citations
Computer Science · Physics and Astronomy · #52A20 #81P16 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1904.03753
openalex publication_date 2019/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the strongly symmetric spectral convex compact sets are\nprecisely the normalized state spaces of finite-dimensional simple Euclidean\nJordan algebras and the simplices. Spectrality is the property that every state\nhas a convex decomposition into perfectly distinguishable pure states; strong\nsymmetry is transitivity, for each integer N, of the affine automorphism group\nof the state space on lists of N perfectly distinguishable pure states.\nAdditional assumptions combine with this theorem to give simple\ncharacterizations of finite-dimensional complex quantum state space.\n Important aspects of quantum and classical thermodynamics and of query\ncomplexity have been generalized to classes of general probabilistic theories\n(GPTs) satisfying natural postulates including or implying spectrality and\nstrong symmetry; our result shows that these apply to a narrower class of\ntheories than might have been hoped. Sorkin's notion of irreducibly k-th order\ninterference has been studied in the GPT framework and looked for in\nexperiments. Our result shows that the assumption of no higher-order (k > 2)\ninterference, used along with spectrality and strong symmetry to characterize\nthe same class of Jordan-algebraic convex sets by Barnum, Mueller, and Ududec\nin arXiv:1403.4147, was superfluous. It also implies that Lee and Selby's\nextension, on the assumption that interference has fixed maximal degree k, of\nthe important order square root of N lower bound on the quantum black-box query\ncomplexity of searching N possibilities for one having a desired property\n(which is achieved by Grover's quantum algorithm), to a class of theories\nsatisfying certain postulates allowing the formulation of a generalized notion\nof query algorithm, actually applies in the Jordan-algebraic setting where\nhigher-order interference is not possible.\n