The mixture of quantum states

Convex prestructures, a mathematical framework that extends the usual concept of convex subset of a real linear space, are employed to generalize methods used in the study of axiomatic quantum mechanics. A brief summary of the mathematical framework of convex prestructures is given. Convex prestructures are classified, and those which are isomorphic to a convex subset of a real linear space are characterized. The operational quantum mechanics of Davies and Lewis is generalized within the framework of convex prestructures and the existence of a physically motivated orthomodular poset is given. Mielnik's beams and filters are also discussed within the framework of convex prestructures. An error in Mielnik's formulation of beam mixtures is pointed out and it is shown that his beam mixtures and one classification of convex prestructures are equivalent. Also his concept of a filter is generalized in the framework of convex prestructures and geometric requirements needed on the set of normalized states so that they may correspond to a physical system are investigated. Finally, Mackey's axioms are discussed and reformulated in the language of P‐convex structures.