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PERIMETER INSTITUTE RECORDED SEMINAR ARCHIVE

Operational Quantum Logic
Speaker(s): Howard Barnum
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.
Date: 17/07/2005 - 4:30 pm
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