Constructor Theory as Process Theory
Stefano Gogioso, Vincent Wang-Maścianica, Muhammad Hamza Waseem, Carlo Maria Scandolo, Bob Coecke
TL;DR
This work builds a bridge between constructor theory and process theory by casting conceivable and possible tasks within a symmetric monoidal, diagrammatic framework. It models conceivable tasks as relations in the category of sets and relations (Rel), defines possible tasks via a substrate quadruple (C, Σ, Γ) with a task-inducing process and a preserved constructor attribute, and proves that possible tasks form a sub-SMC of Rel closed under composition. It further develops a coarse-graining approach from microstates to macrostates, constructing a theory of coarse-grained tasks barC that is a sub-SMC of Rel and relating it to the original theory through embeddings, thereby unifying state- and attribute-based perspectives. The results provide a general, implementation-agnostic language for exploring constructor-theoretic ideas within process theories and lay groundwork for cross-disciplinary collaboration and future concrete case studies in quantum foundations and information processing.
Abstract
Constructor theory is a meta-theoretic approach that seeks to characterise concrete theories of physics in terms of the (im)possibility to implement certain abstract "tasks" by means of physical processes. Process theory, on the other hand, pursues analogous characterisation goals in terms of the compositional structure of said processes, concretely presented through the lens of (symmetric monoidal) category theory. In this work, we show how to formulate fundamental notions of constructor theory within the canvas of process theory. Specifically, we exploit the functorial interplay between the symmetric monoidal structure of the category of sets and relations, where the abstract tasks live, and that of symmetric monoidal categories from physics, where concrete processes can be found to implement said tasks. Through this, we answer the question of how constructor theory relates to the broader body of process-theoretic literature, and provide the impetus for future collaborative work between the fields.