Complementarity in categorical 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.

15 pages; removed incorrect lemma and its dependents

References in corpus (2)

• Quantum Logic in Dagger Kernel Categories
• Complementarity in quantum systems

Cited by in corpus (3)

• The many classical faces of quantum structures
• Categories of Quantum and Classical Channels (extended abstract)
• Characterizations of categories of commutative C*-subalgebras