Quantum Information, Computation and Logic: Exploring New Connections - 2005

Introductory lecture summary: 1. Finite dimensional hilbert spaces and (complemented) modular lattices; infinite-dimensional hilbert spaces and orthomodularity. 2. von Neumann's QL; von Neumann-Birkhoff (briefly!); reconstruction of QM from P(H) 3. Mackey's programme; some early axiomatics (e.g., Zierler); QLs as OMPs + order-determining sets of states 4. Piron's Theorem; some discussion of Piron's axioms 5. Keller's examples (maybe just a mention, though I'd like to indicate how they come up); Soler's theorem (just the statement) 6. Abstract OMLs; Greechie Diagrams and the Loop Lemma; brief mention of Harding's results on decompositions 7. Tensor products (the F-R example, showing OMLs not stable under tensor products); orthoalgebras. 8. Orthoalgebras from test spaces

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.

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