Introduction to Quantum Combinatorics

TL;DR

This paper develops a topos-theoretic framework for quantum sets, integrating classical logic, quantum observables, and quivers, providing new categorical insights into quantum algebra and state representations.

Contribution

It introduces a topos of quantum sets, extends quantum logic with Boolean algebra analogs, and categorifies quantum quivers and Leavitt path algebras.

Findings

Constructed a topos of quantum sets embedding classical sets.

Connected quantum states with internal algebra morphisms.

Categorified Leavitt path algebra via quantum quivers.

Abstract

We construct a topos of quantum sets and embed into it the classical topos of sets. We show that the internal logic of the topos of sets, when interpreted in the topos of quantum sets, provides the Birkhoff-von Neumann quantum propositional calculus of idempotents in a canonical internal commutative algebra of the topos of quantum sets. We extend this construction by allowing the quantum counterpart of Boolean algebras of classical truth values which we introduce and study in detail. We realize expected values of observables in quantum states in our topos of quantum sets as a tautological morphism from the canonical internal commutative algebra to a canonical internal object of affine functions on quantum states. We show also that in our topos of quantum sets one can speak about quantum quivers in the sense of Day-Street and Chikhladze. Finally, we provide a categorical derivation of the Leavitt path algebra of such a quantum quiver and relate it to the category of stable representations of the quiver. It is based on a categorification of the Cuntz-Pimsner algebra in the context of functor adjunctions replacing the customary use of Hilbert modules.