Flow of dynamical causal structures with an application to correlations

TL;DR

The paper addresses the challenge of dynamical causal order in classical-deterministic processes by introducing the flow of causal structures, a tool that visualizes how causal relations evolve under interventions. It provides two constructions: a flow (model-parameter aware) and a parameter-free superflow, with correctness and termination proofs and an exponential-time complexity analysis. A key result shows that if every leaf in a flow is trivial, the resulting correlations are causal, reinforcing prior findings about cycles with chords and offering a practical criterion to identify causal structures. The work also discusses the quantum generalization's obstacles, highlighting how dynamical causal order could be studied within relativistic contexts and outlining future directions for quantum causal models and computational bounds.

Abstract

Causal models capture cause-effect relations both qualitatively - via the graphical causal structure - and quantitatively - via the model parameters. They offer a powerful framework for analyzing and constructing processes. Here, we introduce a tool - the flow of causal structures - to visualize and explore the dynamical aspect of classical-deterministic processes, arguably like those present in general relativity. The flow describes all possible ways in which the causal structure of a process can evolve. We also present an algorithm to construct its supergraph - the superflow - from the causal structure only, without invoking the model parameters. As an application, we show that if all leaves of a flow are trivial, then the corresponding process produces causal correlations only, i.e., correlations where future data cannot influence past events. This strengthens the result that processes, where every directed cycle in their causal structure is chordless, establish causal correlations only. We also discuss the main difficulties for the quantum generalization of the present algorithms.

Figures (7)

• Figure 1: (a) Causal models: The model parameters with the directed graph constitute a causal model for the correlations $P_{X,Y,Z}.$ (b) Causal structure of the quantum switch: The control qubit specified in the global past ($\mathbf P$) influences the global future ($\mathbf F$). Simultaneously, the target system may evolve through $\mathbf A$ and then through $\mathbf B$, or vice versa. This potentiality of transmitting a signal from $\mathbf A$ to $\mathbf B$ or from $\mathbf B$ to $\mathbf A$ is reflected by the directed cycle in the causal structure. We use bold letters to indicate that the vertices are split-nodes.
• Figure 2: Example of a flow for the model parameters given in Eq. \ref{['eq:examplemodparam']} (without dashed part), and the superflow computed by Algorithm \ref{['alg:agnostic']} (including dashed part).
• Figure 3: These causal structures give causal correlations only.
• Figure 4: All connected four-node cyclic digraphs with at lest one source and at least one cycle with a chord.
• Figure 5: Superflow of the example given in Fig. \ref{['subfig:exn4']} (Fig. \ref{['subfig:n4gb']} in the Appendix).

Theorems & Definitions (6)

• Definition 1: Causal model, consistency, and faithfulness qcmadmissibility
• Theorem 1: Reduction
• proof
• Theorem 2: Admissibility admissibility
• Theorem 3: Causal correlations
• proof