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PIRSA Number:

05070092

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.

©2012 Perimeter Institute for Theoretical Physics