Cyclic quantum causal modelling with a graph separation theorem
Carla Ferradini, Victor Gitton, V. Vilasini
TL;DR
This work develops a general framework for cyclic causal modelling in finite-dimensional quantum and classical systems, addressing the long-standing challenge of defining probabilities and a graph-separation principle beyond acyclic graphs. It introduces a robust probability rule for cyclic graphs by mapping arbitrary cyclic causal models to families of acyclic post-selected graphs via post-selected teleportation, ensuring the resulting probabilities are well-defined and independent of the teleportation implementation. The central graph-theoretic contribution is p-separation, a sound and complete separation criterion that reduces to d-separation in the acyclic limit, thereby enabling graph-based causal reasoning in cyclic quantum networks. The paper also situates its framework within broader causality formalisms (e.g., BLO, CS, and process matrices), establishes connections to classical counterparts via a companion paper, and demonstrates the framework through explicit acyclic examples such as prepare-and-measure and Bell scenarios. Overall, this work lays the groundwork for general cyclic causal discovery and causal inference in quantum information processing, including potential extensions to indefinite causal order and post-selected closed timelike curves.
Abstract
Causal modelling frameworks link observable correlations to causal explanations, which is a crucial aspect of science. These models represent causal relationships through directed graphs, with vertices and edges denoting systems and transformations within a theory. Most studies focus on acyclic causal graphs, where well-defined probability rules and powerful graph-theoretic properties like the d-separation theorem apply. However, understanding complex feedback processes and exotic fundamental scenarios with causal loops requires cyclic causal models, where such results do not generally hold. While progress has been made in classical cyclic causal models, challenges remain in uniquely fixing probability distributions and identifying graph-separation properties applicable in general cyclic models. In cyclic quantum scenarios, existing frameworks have focussed on a subset of possible cyclic causal scenarios, with graph-separation properties yet unexplored. This work proposes a framework applicable to all consistent quantum and classical cyclic causal models on finite-dimensional systems. We address these challenges by introducing a robust probability rule and a novel graph-separation property, p-separation, which we prove to be sound and complete for all such models. Our approach maps cyclic causal models to acyclic ones with post-selection, leveraging the post-selected quantum teleportation protocol. We characterize these protocols and their success probabilities along the way. We also establish connections between this formalism and other classical and quantum frameworks to inform a more unified perspective on causality. This provides a foundation for more general cyclic causal discovery algorithms and to systematically extend open problems and techniques from acyclic informational networks (e.g., certification of non-classicality) to cyclic causal structures and networks.