Orthomodularity in dagger biproduct categories
by
John Harding
New Mexico State University

Abramsky and Coecke recently introduced an approach to finite dimensional quantum mechanics based on strongly compact closed categories with biproducts. We show that the projections of any object A in such a category forms an orthoalgebra Proj A. Sufficient conditions are given to ensure this orthoalgebra is an orthomodular poset. A notion of a preparation for such an object is given by Abramsky and Coecke, and it is shown that each preparation induces a finitely additive map from Proj A to the unit interval of the semiring of scalars for this category. The tensor product for the category is shown to induce an orthoalgebra bimorphism Proj A ×Proj B → Proj (A⊗B) that shares some of the properties required of a tensor product of orthoalgebras.

These results are established in a setting more general than that of strongly compact closed categories. Many are valid in dagger biproduct categories, others require also a symmetric monoidal tensor compatible with the dagger and biproduct structure. Examples are considered for several familiar strongly compact closed categories.

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Date received: May 29, 2008