Convex structures and operational quantum mechanics

Abstract

A general mathematical framework called a convex structure is introduced. This framework generalizes the usual concept of a convex set in a real linear space. A metric is constructed on a convex structure and it is shown that mappings which preserve the structure are contractions. Convex structures which are isomomorphic to convex sets are characterized and for such convex structures it is shown that the metric is induced by a norm and that structure preserving mappings can be extended to bounded linear operators.Convex structures are shown to give an axiomatization of the states of a physical system and the metric is physically motivated. We demonstrate how convex structures give a generalizing and unifying formalism for convex set and operational methods in axiomatic quantum mechanics.