Picturing general quantum subsystems

TL;DR

This paper generalizes the process-theoretic approach to quantum subsystems by introducing splitting maps in dagger symmetric monoidal categories, enabling a unified treatment of locality, causality, and subsystem inclusion for complex quantum systems.

Contribution

It introduces splitting maps and the comprehension preorder in dagger symmetric monoidal categories, extending subsystem analysis to non-factor von Neumann algebras.

Findings

The comprehension preorder captures von Neumann algebra inclusion.

Splitting map trace aligns with von Neumann algebra trace.

Semi-causality and semi-localisability equivalence extends to all subsystems.

Abstract

We extend the usual process-theoretic view on locality and causality in subsystems (based on the tensor product case) to general quantum systems (i.e.\ possibly non-factor, finite-dimensional von Neumann algebras). To do so, we introduce a primitive notion of splitting maps within dagger symmetric monoidal categories. Splitting maps give rise to subsystems that admit comparison via a preorder called comprehension, and support an adaptation of the usual categorical trace. We show that the comprehension preorder precisely captures the inclusion partial order between von Neumann algebras, and that the splitting map trace captures the natural notion of von Neumann algebra trace. As a consequence of the development of these diagrammatic tools, we prove that the known equivalence between semi-causality and semi-localisability for factor subsystems extends to all (including non-factor) subsystems.