Supermaps on generalised theories

2026-02-27

AI Summary

This work proposes a categorical framework applicable to arbitrary circuit theories to resolve the long-standing ambiguity in the definition of higher-order supermaps within generalized physical theories. By leveraging channel–state duality, the framework constructs explicit representations of supermaps and introduces a generalized Yoneda lemma to eliminate definitional ambiguities. This approach not only provides a rigorous foundation for higher-order processes in real quantum theory but also successfully incorporates higher-order operations from non-standard theories—such as Boxworld—into a unified formalism. Consequently, it establishes a universal and unambiguous definition of higher-order operations within the broader context of generalized probabilistic theories.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Categorical supermaps generalise higher-order quantum operations from finite-dimensional quantum theory to arbitrary circuit theories. In this paper, we establish the Yoneda lemma for categorical supermaps, which states that whenever a physical theory has a suitable notion of channel-state duality, then categorical supermaps on that theory can be concretely represented in terms of that duality. This lemma eliminates any guesswork or ambiguity when defining the appropriate notion of supermap for these theories. As a concrete application, we show that the recently proposed higher-order processes on boxworld can be obtained as a particular instance of categorical supermaps, and put forward a stable definition of higher-order real quantum theory.

Problem

• supermaps
• higher-order operations
• generalised physical theories
• channel-state duality
• categorical quantum theory

Innovation

• categorical supermaps
• Yoneda lemma
• channel-state duality
• higher-order processes
• generalised physical theories