Quantum quirks, classical contexts: Towards a Bohrification of effect algebras
Leo Lobski
Abstract:
Two results of the following general form are proved: a functor from a category of algebras to the category of posets is essentially injective on objects above a certain size. The first result is for Boolean algebras and the functor taking each Boolean algebra to its poset of finite subalgebras. This strengthens and provides a novel proof for a result by Sachs, Filippov and Grätzer, Koh & Makkai. The second result is for finite MV-algebras and the functor taking each such algebra to its poset of partitions of unity.
The second result uses the dual equivalence of finite MV-algebras with finite multisets, as well as the correspondence between partition posets of finite multisets and setoid quotients. Thus the equivalence is constructed via the powerset functor for multisets, and setoid quotients are introduced. The equivalence is a special case of a more general duality proved by Cignoli, Dubuc and Mundici.
The primary interest of this work lies in algebras describing quantum observables, hence both results are viewed as statements about effect algebras. Since MV-algebras contain ‘unsharp’ (i.e. self-orthogonal) elements, the second result shows that sharpness is not a necessary condition for essential injectivity of the partitions of unity functor. Physically, this means that there are systems with unsharp effects (namely, those represented by finite MV-algebras) which can be faithfully reconstructed from all the possible measurements.