Discrete quantum structures

Abstract

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.

arXiv comment64 pages; corrected statement of Theorem 1.2.1 only

References in corpus (8)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Estimating quantum chromatic numbers
• Categorical formulation of quantum algebras
• Graph Homomorphisms for Quantum Players
• A compositional approach to quantum functions
• The Morita theory of quantum graph isomorphisms
• Deciding the existence of perfect entangled strategies for nonlocal games
• Hereditarily antisymmetric operator algebras

Cited by in corpus (2)

• A natural deduction system for orthomodular logic
• The finite dual coalgebra as a quantization of the maximal spectrum