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Dusko Pavlovic: "(Modular) effect algebras are equivalent to (Frobenius) antispecial algebras"

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Published on Jun 30, 2016

Talk given as part of QPL2016: http://qpl2016.cis.strath.ac.uk/
Slides: http://qpl2016.cis.strath.ac.uk/pdfs/...

Abstract: Effect algebras are one of the generalizations of Boolean algebras proposed in the quest for a *quantum logic*. Frobenius algebras are a tool of *categorical quantum mechanics*, used to present various families of observables in abstract, often nonstandard frameworks. Both effect algebras and Frobenius algebras capture their respective fragments of quantum mechanics by elegant and succinct axioms; and both come with their conceptual mysteries. A particularly elegant and mysterious constraint, imposed on Frobenius algebras to characterize a class of tripartite entangled states, is the *antispecial* law. A particularly contentious issue on the quantum logic side is the *modularity* law, proposed by von Neumann to mitigate the failure of distributivity of quantum logical connectives. We show that, if quantum logic and categorical quantum mechanics are formalized in the same framework, then the antispecial law corresponds in effect algebras to the natural requirement that the units are each other's single complements; and that the modularity law corresponds to the Frobenius condition. These correspondences lead to the equivalence announced in the title. Aligning the two formalisms, at the very least, sheds new light on the concepts that are more clearly displayed on one side than on the other (such as e.g. orthogonality). Beyond that, it may also open up new approaches to deep and important problems of quantum mechanics (such as the classification of complementary observables).

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