A compositional approach to quantum functions

Abstract

We introduce a notion of quantum function, and develop a compositional framework for finite quantum set theory based on a 2-category of quantum sets and quantum functions. We use this framework to formulate a 2-categorical theory of quantum graphs, which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero-error communication, and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups. We show that the 2-categories of quantum sets and quantum graphs are semisimple and characterise existing notions of quantum permutations and quantum graph isomorphisms as dagger-dualisable 1-morphisms in these 2-categories.

57 pages, final version

References in corpus (6)

• Biunitary constructions in quantum information
• The Morita theory of quantum graph isomorphisms
• Constructing Mutually Unbiased Bases from Quantum Latin Squares
• Higher Quantum Theory
• The Quantum Monad on Relational Structures
• Quantum relations

Cited by in corpus (30)

• Bigalois extensions and the graph isomorphism game
• The Morita theory of quantum graph isomorphisms
• The quantum-to-classical graph homomorphism game
• Orthogonality for Quantum Latin Isometry Squares
• Quantum sets
• Quantum graphs: different perspectives, homomorphisms and quantum automorphisms
• Some examples of quantum graphs
• Classification of Quantum Graphs on $M_2$ and their Quantum Automorphism Groups
• Random quantum graphs
• Unitary transformations of fibre functors
• Quantum Semigroups from Synchronous Games
• Quantum teleportation in the commuting operator framework
• The Frucht property in the quantum group setting
• Covariant quantum combinatorics with applications to zero-error communication
• Entanglement-symmetries of covariant channels
• A state-space approach to quantum permutations
• A category of quantum posets
• Discrete quantum structures
• Quantum graphs, subfactors and tensor categories I
• Quantum symmetry vs nonlocal symmetry
• Non-local games and quantum symmetries of quantum metric spaces
• The finite dual coalgebra as a quantization of the maximal spectrum
• Entanglement-invertible channels
• Algebraic connectedness and bipartiteness of quantum graphs
• Quantum partial automorphisms of finite graphs
• Transitive Nonlocal Games
• Modelling questions for transitive quantum groups
• Complete Classification of Directed Quantum Graphs on M2
• Quantum Suplattices
• Duals of finite quantum permutation groups