Complementarity in categorical quantum mechanics

TL;DR

This paper explores the concept of complementarity across different layers of quantum mechanics using a categorical framework, unifying algebraic, Hilbert space, and lattice perspectives.

Contribution

It introduces a point-free categorical approach linking von Neumann algebras, classical structures, and Boolean subalgebras in quantum theory.

Findings

Unified categorical framework for complementarity

Point-free definition of copyability

Connection between algebraic and lattice structures in quantum mechanics

Abstract

We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that (i) become (sub)endohomsets and (iii) become subobject lattices. By developing a `point-free' definition of copyability we link (i) commutative von Neumann subalgebras, (ii) classical structures, and (iii) Boolean subalgebras.