Picturing general quantum subsystems
Octave Mestoudjian, Matt Wilson, Augustin Vanrietvelde, Pablo Arrighi
TL;DR
This work develops a diagrammatic framework for general quantum subsystems that extends beyond factor algebras by introducing splitting maps in dagger symmetric monoidal categories. It proves a bidirectional correspondence between diagrammatic subsystems and algebraic Von Neumann subalgebras via the comprehension preorder and the Artin–Wedderburn structure, showing that diagrammatic traces reproduce the Von Neumann trace. The results extend key locality-and-causality equivalences, such as semi-causality equaling semi-localisability, to non-factor subsystems through balanced splitting maps, and provide a unified treatment of tracing and non-signalling in this general setting. The framework lays groundwork for fully diagrammatic representations of general quantum subsystems and potential generalisations to higher categorical structures.
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.