Quantum association schemes
Daniel Gromada
TL;DR
This work extends classical association schemes to the quantum realm by formulating quantum coherent algebras as the Bose–Mesner tools for finite quantum spaces. It develops a robust duality theory, including translation quantum association schemes arising from finite quantum groups and their Fourier duals, thereby generalizing Pontryagin duality to noncommutative settings. The authors construct and analyze distance‑regular and strongly regular quantum graphs (e.g., cocommutative CQAS and Hadamard graphs) and introduce quantum Latin squares as a versatile construction method for SRGs, linking these combinatorial objects to Cayley‑type quantum graphs. The framework promises new combinatorial structures and insights for quantum information and quantum group theory, with explicit diagrammatic calculus and dualities underpinning the theory.
Abstract
We introduce quantum association schemes. This allows to define distance regular and strongly regular quantum graphs. We bring examples thereof. In addition, we formulate the duality for translation quantum association schemes corresponding to finite quantum groups.