Compositional Quantum Logic

TL;DR

This paper develops a new compositional framework for quantum logic using order-theoretic structures derived from dagger symmetric monoidal categories, enabling a more general understanding of composite quantum systems.

Contribution

It introduces a canonical, compositional approach to quantum logic that generalizes C*-algebras within dagger symmetric monoidal categories, addressing the lack of a standard composite system description.

Findings

The framework yields projection lattices of finite-dimensional C*-algebras.

Models show that noncommutativity does not always imply nondistributivity.

The approach is entirely compositional without extra assumptions.

Abstract

Quantum logic aims to capture essential quantum mechanical structure in order-theoretic terms. The Achilles' heel of quantum logic is the absence of a canonical description of composite systems, given descriptions of their components. We introduce a framework in which order-theoretic structure comes with a primitive composition operation. The order is extracted from a generalisation of C*-algebra that applies to arbitrary dagger symmetric monoidal categories, which also provide the composition operation. In fact, our construction is entirely compositional, without any additional assumptions on limits or enrichment. Interpreted in the category of finite-dimensional Hilbert spaces, it yields the projection lattices of arbitrary finite-dimensional C*-algebras. Interestingly, there are models that falsify standardly assumed correspondences, most notably the correspondence between noncommutativity of the algebra and nondistributivity of the order.