Operational Quantum Logic

          @misc{ pirsa_PIRSA:05070092,
            doi = {10.48660/05070092},
            url = {https://pirsa.org/05070092},
            author = {Barnum, Howard},
            keywords = {},
            language = {en},
            title = {Operational Quantum Logic },
            publisher = {Perimeter Institute},
            year = {2005},
            month = {jul},
            note = {PIRSA:05070092 see, \url{https://pirsa.org}}
          }
          

Abstract

Introductory lecture summary: Operational Quantum Logic I: Effect Algebras, States, and Basic Convexity • Effect algebras, effect test-spaces, PAS's (partial abelian semigroups). • Morphisms, states, dynamics. Classes of effect algebras whose state-set has nice properties. • Operational derivation of effect algberas, summarized. • "Theories"--- Effect-state systems. • Tensor product (defined, existence result stated). • Some notions of sharpness in EA's, examples that separate them, conditional equivalences that are interesting. • Convex cones/sets, ordered linear space basics. Partially ordered abelian groups. Operational Quantum Logic II: Convexity, Representations, and Operations • Convex cones and convex sets. Extremality. Krein-Milman. Caratheodory. Affine maps. • Positive maps. Automorphisms. Dual space, Dual cone. Adjoint map. Faces. Exposed faces. Lattices of faces. • Interval EA's, representations on partially ordered abelian groups, unigroups. Analogues of Naimark's theorem, open problems. • Convex EA's. Observables, "generalized" observables. Representation theorem for convex EA's. Relation of observables to effects formulation. • State representation theorem for finite-d homogeneous self-dual cones (statement). • Homogeneous cones as slices of positive semidefinite cones (statement). • Axioms concerning the face lattice.