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Oxford Quantum Talks Archive

Convexity, Categorical Semantics and the Foundations of Physics

Ross Duncan, University of Oxford
Quantum Physics and Logic 2010, May 2010, University of Oxford

We consider symmetric monoidal categories of convex operational models, and adduce necessary and sufficient conditions for these to be compact-closed or dagger-compact. Compact closure amounts to the condition that all processes be implementable by means of a ``remote evaluation" protocol (generalizing standard conclusive quantum teleportation protocols), which amounts to a form of classical conditioning. This requires the existence, for each system, of a bipartite state involving a further system, whose corresponding conditioning map is an isomorphism, and an an effect whose corresponding map is the inverse of this isomorphism. Degenerate compact closure, in which systems act as their own duals in the compact structure, means that one may take this extension to be the system itself, so the isomorphism implies that systems are weakly self-dual as ordered vector spaces. Degenerate dagger compact categories emerge from a further restriction, namely, that the bipartite ``isomorphism" state and effect be symmetric.

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