[Paper Review] A categorical semantics for causal structure

Aleks Kissinger, Sander Uijlen|arXiv (Cornell University)|Jun 20, 2017

Quantum Mechanics and Applications15 references28 citations

TL;DR

This paper introduces a novel categorical framework that models both definite and indefinite causal structures within process theories like classical and quantum mechanics. By encoding fine-grained causal relationships through diagrammatic axioms, it derives the operational behavior of processes such as one-way signalling, non-signalling maps, quantum n-combs, and the quantum switch, unifying diverse causal structures under a single formalism.

ABSTRACT

We present a categorical construction for modelling both definite and indefinite causal structures within a general class of process theories that include classical probability theory and quantum theory. Unlike prior constructions within categorical quantum mechanics, the objects of this theory encode fine-grained causal relationships between subsystems and give a new method for expressing and deriving consequences for a broad class of causal structures. To illustrate this point, we show that this framework admits processes with definite causal structures, namely one-way signalling processes, non-signalling processes, and quantum n-combs, as well as processes with indefinite causal structure, such as the quantum switch and the process matrices of Oreshkov, Costa, and Brukner. We furthermore give derivations of their operational behaviour using simple, diagrammatic axioms.

Motivation & Objective

• To develop a unified categorical framework for modeling causal structures in process theories.
• To capture both definite and indefinite causal relationships between subsystems in a fine-grained way.
• To provide a diagrammatic derivation method for operational behaviors of causal processes.
• To extend categorical quantum mechanics by incorporating causal relationships as structural features of objects.

Proposed method

• The framework constructs a category where objects represent subsystems with explicit causal relationships.
• It employs diagrammatic reasoning based on axioms that capture causal dependencies and signaling constraints.
• The construction generalizes existing structures like quantum combs and process matrices by embedding causal order into the categorical objects.
• It defines processes such as one-way signalling and non-signalling maps as morphisms within this category.
• It derives the behavior of processes like the quantum switch through simple, compositional diagrammatic rules.
• The approach is applicable to both classical probability theory and quantum theory, ensuring broad generality.

Experimental results

Research questions

• RQ1 How can causal structures—both definite and indefinite—be uniformly modeled within a categorical framework for process theories?
• RQ2 What diagrammatic axioms are sufficient to derive the operational behavior of processes with complex causal orders?
• RQ3 How can the quantum switch and process matrices be derived from a unified categorical construction?
• RQ4 In what way do the objects in this framework encode fine-grained causal relationships between subsystems?
• RQ5 Can the framework reproduce known causal processes like quantum n-combs and non-signalling maps using a consistent set of principles?

Key findings

• The framework successfully models one-way signalling processes by encoding directional information in the categorical objects.
• Non-signalling processes are naturally captured as morphisms that respect causal independence between subsystems.
• Quantum n-combs are derived as specific instances of the categorical construction, showing consistency with known operational behavior.
• The quantum switch is derived diagrammatically from the axioms, demonstrating its operational behavior without additional assumptions.
• Process matrices from Oreshkov, Costa, and Brukner are shown to be realizable within this framework, confirming its generality.
• The entire construction is consistent across classical and quantum theories, enabling a unified treatment of causal structures.