Quantum State Spaces over Associative Composition Algebras

Wootters pointed out that a theory of quantum mechanics could be formulated without probability amplitudes [1]. Usual quantum theory over the complex field is among those in a hierarchy of theories indexed by their capacity, as defined in [2]. We explore the structure of quantum state spaces over associative composition algebras within the general Quantum Bayesian framework proposed by Fuchs and Schack [2]. We consider the possibility of expanding self-adjoint operators in terms of symmetric informationally complete bases for different algebraic modules. We chart the geometry of quantum state space on the corresponding probability simplexes by imposing a self- adjoint positive semi-definite nature to the pure states and their convex hull. \\[4pt] [1] W. K. Wootters, ``Quantum Mechanics without Probability Amplitudes,'' Foundations of Physics, Vol. 16, No. 4 (1985)\\[0pt] [2] C. A. Fuchs and R. Schack, ``Quantum-Bayesian Coherence,'' arXiv:0906.2187v1 [quant-ph] (2009)