Infinite-dimensional Categorical Quantum Mechanics

Abstract

We use non-standard analysis to define a category ${}^{\star }\mathrm{Hilb}$ suitable for categorical quantum mechanics in arbitrary separable Hilbert spaces, and we show that standard bounded operators can be suitably embedded in it. We show the existence of unital special commutative $†$-Frobenius algebras, and we conclude ${}^{\star }\mathrm{Hilb}$ to be compact closed, with partial traces and a Hilbert-Schmidt inner product on morphisms. We exemplify our techniques on the textbook case of 1-dimensional wavefunctions with periodic boundary conditions: we show the momentum and position observables to be well defined, and to give rise to a strongly complementary pair of unital commutative $†$-Frobenius algebras.

In Proceedings QPL 2016, arXiv:1701.00242

References in corpus (4)

• Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing
• The Topology of Quantum Algorithms
• An approach to nonstandard quantum mechanics
• Categorical Semantics for Schrödinger's Equation

Cited by in corpus (14)

• A no-go theorem on the nature of the gravitational field beyond quantum theory
• Reconstructing quantum theory from diagrammatic postulates
• The Category of Von Neumann Algebras
• Frobenius structures over Hilbert C*-modules
• Towards Quantum Field Theory in Categorical Quantum Mechanics
• Light-Matter Interaction in the ZXW Calculus
• Dagger linear logic for categorical quantum mechanics
• Quantum Supermaps are Characterized by Locality
• A computer scientist's reconstruction of quantum theory
• Effectuses in Categorical Quantum Foundations
• Quantum Field Theory in Categorical Quantum Mechanics
• Causality in Higher Order Process Theories
• ZX Graphical Calculus for Continuous-Variable Quantum Processes
• Hopf-Frobenius Algebras and a Simpler Drinfeld Double