\(H^\ast\)-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics (Q2918872)

This paper focuses on the categorical quantum mechanics and shows how one can use monoidal dagger categories and the possible structures definable within them to axiomatize large parts of quantum mechanics. Up to now, this project has been done on finite-dimensional Hilbert spaces with respect to a certain class of Frobenius algebras. But in this paper, the authors introduce the key notion of \(H^\ast\)-algebras in the general setting of symmetric monoidal dagger categories (see Section 3) and show that this notion is the right suitable algebraic notion to characterize orthonormal bases in arbitrary dimension (see Section 4). They offer several equivalent characterizations of coincidence of \(H^\ast\)-algebras and nonunital Frobenius algebras in the category of Hilbert space. In Section 5, they further investigate \(H^\ast\)-algebras in the categories of generalized relations and positive matrices and show that if no phenomena of ``destructive interference'' (see p. 17) arise, then \(H^\ast\)-algebras and nonunital Frobenius algebras always coincide. The authors close the paper in Section 6 with an outlook which shows the main lines of research for continuing a development of categorical quantum mechanics applicable to infinite-dimensional cases.NEWLINENEWLINEFor the entire collection see [Zbl 1245.00037].