Scott Continuity in Generalized Probabilistic Theories

Robert Furber (Aalborg University)

Scott continuity is a concept from domain theory that had an unexpected previous life in the theory of von Neumann algebras. Scott continuous states are known as normal states, and normal states characterize the states coming from density matrices. Similarly Scott continuous *-isomorphisms are the “Heisenberg picture” of what unitary maps are in the “Schrödinger picture”. Given these well-known applications, it is natural to ask whether this carries over to generalized probabilistic theories. We show that the answer is no – there are infinite-dimensional convex sets for which the set of Scott continuous states on the corresponding set of 2-valued POVMs does not recover the original convex set, but is strictly larger. This shows the necessity of the use of topologies for state-effect duality in the general case, rather than order theory.