2020/04/09 by Andre Kornell, Kornell, Andre · 1 voice
Computer Science · Mathematics · Physics and Astronomy · #46L89 #Advanced Algebra and Logic #Advanced Operator Algebra Research #FOS: Mathematics #FOS: Physical sciences #Logic (math.LO) #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Mechanics and Applications #math-ph #math.LO #math.MP #math.OA #msc:46L89
paper · pdf · doi:10.48550/arxiv.2004.04377
64 pages; corrected statement of Theorem 1.2.1 only
openalex publication_date 2020/04/09 · arxiv published 2020/04/09 · arxiv created 2022/03/08 · arxiv updated 2022/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A majority of established quantum generalizations of discrete structures are shown to be instances of a single quantum generalization. In particular, the quantum graphs of Duan, Severini and Winter, the quantum metric spaces of Kuperberg and Weaver, the quantum isomorphisms of Atserias, Mančinska, Roberson, Šámal, Severini and Varvitsiotis and the quantum groups of Woronowicz that are all discrete in the sense that the underlying von Neumann algebra is hereditarily atomic are shown to be subclasses of a single class of discrete quantum structures. Such a discrete quantum structure is defined to be a discrete quantum space equipped with relations and functions of various arities. Weaver's quantum predicate logic, a generalization of the quantum propositional logic of Birkhoff and von Neumann, provides canonical quantum generalizations for a large class of properties. The equality relation on discrete quantum spaces that is introduced here plays a central role in this approach to mathematical quantization.