An Introduction to Infinite-Dimensional Categorical Quantum Computing

Abstract: Category theory has proven promising in capturing the logic of quantum information processing at a fairly high level, in similar fashion to Boolean logic and classical computing. In particular, quantum state evolution and quantum teleportation have been able to be depicted by the category of finite-dimensional Hilbert spaces together with linear transformations. Since all categories behave identically by definition, we can then view quantum computation in a highly intuitive, diagrammatic language. By generalizing the category of finite-dimensional Hilbert spaces to the category of infinite-dimensional Hilbert spaces, we can begin to represent categorically the mathematics of quantum mechanics, which involves observables and bases in arbitrary dimension. We show that this generalization might be achieved by expanding our use of unital Frobenius algebras to nonunital Frobenius algebras and H*-algebras.