Categorical Equivalences from State-Effect Adjunctions

Robert Furber
(Aalborg University)

From every pair of adjoint functors it is possible to produce a (possibly trivial) equivalence of categories by restricting to the subcategories where the unit and counit are isomorphisms. If we do this for the adjunction between effect algebras and abstract convex sets, we get the surprising result that the equivalent subcategories consist of reflexive order-unit spaces and reflexive base-norm spaces, respectively. These are the convex sets that can occur as state spaces in generalized probabilistic theories satisfying both the no-restriction hypothesis and its dual. The linearity of the morphisms is automatic. If we add a compact topology to either the states or the effects, we can obtain a duality for all Banach order-unit spaces or all Banach base-norm spaces, but not both at the same time.

In Peter Selinger and Giulio Chiribella: Proceedings of the 15th International Conference on Quantum Physics and Logic (QPL 2018), Halifax, Canada, 3-7th June 2018, Electronic Proceedings in Theoretical Computer Science 287, pp. 107–126.
Published: 31st January 2019.

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