🤖 AI Summary
This work aims to formalize the foundational principles of higher-order circuit diagrams and their correspondence with higher-order quantum theories. By introducing nested structures, spatiotemporal composition, and an equivalence between lower-order bipartite processes and higher-order bipartite states, the authors construct an axiomatic framework for higher-order circuits. Methodologically, they uniquely combine enrichment in symmetric multicategories with cotensorial structure and impose Frobenius-like coherence conditions. The primary contribution lies in establishing a rigorous categorical foundation for higher-order circuits and proving that any such theory can be faithfully embedded into the theory of strong profunctors, thereby delineating its theoretical upper bound and semantic boundaries.
📝 Abstract
We write down a series of basic laws for (strict) higher-order circuit diagrams. More precisely, we define higher-order circuit theories in terms of: (a) nesting, (b) temporal and spatial composition, and (c) equivalence between lower-order bipartite processes and higher-order bipartite states. In category-theoretic terms, these laws are expressed using enrichment and cotensors in symmetric polycategories, along with a frobenius-like coherence between them. We describe how these laws capture the salient features of higher-order quantum theory, and discover an upper bound for higher-order circuits: any higher-order circuit theory embeds into the theory of strong profunctors.