A link between quantum logic and categorical quantum mechanics
\textit{S. Abramsky} and \textit{B. Coecke} [Proc. 19th Annual IEEE Symp. on Logic in Computer Sci, pp. 415--425, IEEE Comput. Soc., New York, (2004)] have introduced an approach to finite-dimensional quantum mechanics in the abstract setting of strongly compact closed categories with biproducts. The author of the paper under review aims to connect this approach to the quantum logic approach initiated by Birkhoff and von Neumann. He demonstrates that a number of features of the quantum logic approach fit nicely in the categorical approach. Many results obtained by Abramsky and Coecke are shown to be valid in the more general dagger (alias involutive) biproduct categories, some other require also a compatible symmetric monoidal tensor. Besides a general introduction, the paper contains sections which provides a background on digger biproduct categories and such categories with a symmetric monoidal tensor. In other sections, weak projections and projections of an object \(A\) in a digger biproduct category are introduced. Weak projections form an orthomodular projection, while projections form even an orthoalgebra \(\text{proj}A\). Then symmetric monoidal tensors compatible with both dagger and biproducts are introduced in the categories considered. The endomorphisms of the tensor unit in dagger biproduct symmetric monoidal categories form a commutative semiring, the semiring of scalars. The orthoalgebra \(\text{proj}(A \otimes B)\) satisfies several physically motivated conditions. Finally, the discussed notions are considered in the category of sets and relations, in the category of finite dimensional Hilbert spaces and in the category of matrices over a field. [Reviewer's remark: This paper could be of interest both to people working in categorical foundations of quantum mechanics and to those working in quantum logic. Another categorical approach to quantum theory initiated by Butterfield and Isham involves topos theory -- see a recent paper by \textit{C. Heunen, N. Landsman} and \textit{B. Spitters} [Commun. Math. Phys. 291, No.~1, 63--110 (2009; Zbl 1209.81147).]
scientific article; zbMATH DE number 5612771