Generalised Process Theories (2502.10368v2)

Abstract: Process theories provide a powerful framework for describing compositional structures across diverse fields, from quantum mechanics to computational linguistics. Traditionally, they have been formalized using symmetric monoidal categories (SMCs). However, various generalizations, including time-neutral, higher-order, and enriched process theories, do not naturally conform to this structure. In this work, we propose an alternative formalization using operad algebras, motivated by recent results connecting SMCs to operadic structures, which captures a broader class of process theories. By leveraging the string-diagrammatic language, we provide an accessible yet rigorous formulation that unifies and extends traditional process-theoretic approaches. Our operadic framework not only recovers standard process theories as a special case but also enables new insights into quantum foundations and compositional structures. This work paves the way for further investigations into the algebraic and operational properties of generalised process theories within an operadic setting.

• The paper introduces an operadic framework that generalizes process theories by modeling acyclic, time-neutral, and causal compositions.
• It demonstrates an equivalence between acyclic wiring operads and symmetric monoidal categories, bypassing forced identity and swap constructs.
• The findings extend the applicability of process models to quantum mechanics and resource theories, offering versatile tools for complex system analysis.

Overview of "Generalised Process Theories"

The paper "Generalised Process Theories" by Selby, Stasinou, Wilson, and Coecke presents an advanced theoretical framework that broadens the conventional notion of process theories. This framework is designed to offer a more versatile perspective on the composition and structure of systems, specifically by transitioning from the conventional symmetric monoidal categories (SMCs) to a more flexible operadic approach. The work establishes the premise that this enriched framework can encapsulate various atypical process theories that are not well-suited to traditional SMCs, including time-neutral theories and certain higher-order processes.

Key Contributions:

1. Acyclic Wiring Operads and Traditional Process Theories: The authors reinforce that process theories can be precisely modeled as operad algebras for the acyclic wiring operad. They show an equivalence between these algebras and SMCs, thus providing a novel perspective on the conventional process theories without necessitating forced constructs like identity and swap morphisms within categories.
2. Time-Neutral Process Theories: The paper introduces the concept of time-neutral process theories, where systems do not have an input/output distinction. This is implemented through a different type of wiring operad, termed the wiring operad of dots, allowing for compositions that do not conform to typical time-order constraints. Examples such as tensor networks highlight the relevance of this framework in practical applications.
3. Higher-Order Processes: By employing operads that allow for single-wire compositions, the framework flexibly accommodates higher-order processes. These extensions represent structures that are more complex than typical categorical models and address theoretical questions related to temporal and causal structures in quantum mechanics.
4. Causal Process Theories and Operadic Disc: The authors introduce causal wiring operads to integrate causality within process theories. This involves modifying the operadic definitions to incorporate operations reflecting the causal disposability of systems, hence aligning the theoretical constructs with physical intuitions about causality.
5. Enrichment and Generalised Process Theories: The operadic approach is shown to encompass enriched process theories by varying the codomain of the operad algebra beyond sets to more structured categories, such as vector spaces or other mathematical constructs. This flexibility makes it possible to model theories that involve convexity or other enrichments that are prominent in physical processes, particularly in quantum mechanics.

Implications and Future Directions:

The implications of this work are significant for both theoretical and practical advancements in the foundations of physics and computer science. The operadic framework allows for:

• Cross-disciplinary Applications: The generalised framework can potentially unify diverse domains such as computational linguistics, quantum field theory, or even areas dealing with systems biology and network theory.
• Resource Theory Development: By offering a flexible account of compositionality, this framework could enhance the paper of resource theories in a variety of contexts, particularly those where current categorical frameworks are insufficient.
• Enhanced Modeling of Quantum Theories: The operadic approach provides a robust formalism for exploring quantum theories, particularly in scenarios where traditional process theories are inadequate, such as those involving indefinite causal structures or complex entanglement scenarios.

The paper opens up avenues for deeper exploration of operadic structures in representing complex system interactions and encourages further research into novel algebraic and categorical structures that could follow from this foundational work.