A general framework for cyclic and fine-tuned causal models and their compatibility with space-time

TL;DR

A causal modelling framework that allows operational and relativistic notions of causality to be independently defined and for connections between them to be established in a theory-independent manner, and gives an operational way to study causation that allows for cyclic, fine-tuned and non-classical causal influences.

Abstract

Causal modelling is a tool for generating causal explanations of observed correlations and has led to a deeper understanding of correlations in quantum networks. Existing frameworks for quantum causality tend to focus on acyclic causal structures that are not fine-tuned i.e., where causal connections between variables necessarily create correlations between them. However, fine-tuned causal models (which permit causation without correlation) play a crucial role in cryptography, and cyclic causal models can be used to model physical processes involving feedback and may also be relevant in exotic solutions of general relativity. Here we develop a causal modelling framework capable of dealing with these general scenarios. The key feature of our framework is that it allows operational and relativistic notions of causality to be independently defined and for connections between them to be established in a theory-independent manner. The framework first gives an operational way to study causation that allows for cyclic, fine-tuned and non-classical causal influences. We then consider how a causal model can be embedded in a space-time structure (modelled as a partial order) and propose a compatibility condition for ensuring that the embedded causal model does not allow signalling outside the space-time future. We identify several distinct classes of causal loops that can arise in our framework, showing that compatibility with a space-time can rule out only some of them. We discuss conditions for preventing superluminal signalling within arbitrary (and possibly cyclic) causal structures and consider models of causation in post-quantum theories admitting so-called jamming correlations. Finally, this work introduces the concept of a "higher-order affects relation", which is useful for causal discovery in fined-tuned causal models.

Figures (15)

• Figure 1: (a) The bipartite Bell causal structure: $\Lambda$ represents a bipartite state (classical, quantum or that of a generalised probabilistic theory) shared by two non-communicating parties Alice and Bob who measure their subsystems locally using classical measurement settings $A$ and $B$ to obtain classical outcomes $X$ and $Y$. (b) A variation of (a) in which the settings $A$ and $B$ are both causes of both outcomes.
• Figure 2: Causal structures for the motivating examples described in the main text: (a) Friedman's thermostat (b) Traitorous Lieutenant (c) One-time pad. Note that there may be additional causal influences. For example, in (b), we will later see that an additional common cause between $M_1$ and $M_2$ will be required to fully explain the correlations (cf. Figure \ref{['fig: jamming']}).
• Figure 3: Jamming correlations in the tripartite Bell scenario: Three parties Alice, Bob and Charlie share a tripartite system $\Lambda$, they measure their subsystem using the freely chosen measurement settings $A$, $B$ and $C$, producing the outcomes $X$, $Y$ and $Z$ respectively, without communicating. (a) Space-time configuration for the jamming scenario Grunhaus1996Horodecki2019: the measurement of the three parties are pairwise space-like separated with the future of Bob's input $B$ containing the joint future of Alice's and Charlie's outputs $X$ and $Z$ (blue region). Here, it is argued that allowing $B$ to signal to $X$ and $Z$ jointly but not individually is consistent with the principle of "no signalling outside the future lightcone", since the joint signalling can only be verified in the blue region which is in the future of $B$. Such correlations form a larger set as compared to the standard tripartite no-signalling correlations, which forbid individual as well as joint signalling from the inputs of any set of parties to the outputs of a complementary set of parties Salazar2020. To model the joint signalling through jamming, a new variable $C_{XZ}$ was introduced in Horodecki2019, located at the earliest point in the joint future of $X$ and $Z$ and representing the correlations between $X$ and $Z$. (b) Causal structure for the usual tripartite Bell experiment.
• Figure 4: Pre-intervention, augmented and post-intervention causal structures: Taking the original, pre-intervention causal structure, $\mathcal{G}$, to be that of (a), parts (b) and (c) of this figure illustrate the augmented causal structure, $\mathcal{G}_{I_X}$, and post-intervention causal structure, $\mathcal{G}_{\mathrm{do}(X)}$, for intervention on $X$. In $\mathcal{G}_{I_X}$, the variable $I_X$ can take values in the set $\{\mathrm{idle},\{\mathrm{do}(x)\}_{x\in X}\}$ while in $\mathcal{G}_{\mathrm{do}(X)}$, it can only take the values $\{\mathrm{do}(x)\}_{x\in X}$ corresponding to an active intervention. Conditioned on $I_X=\mathrm{idle}$, we effectively obtain the original causal model (a) which corresponds to no intervention being performed, as specified by Equation \ref{['eq: intervention1']}.
• Figure 5: Causal structure of Example \ref{['example:affects_transitive']}

Theorems & Definitions (111)

• Definition 2.1: Cause
• Example 2.1: Sets of compatible correlations in the bipartite Bell causal structure $\mathcal{G}_B$
• Definition 2.2: Blocked paths
• Definition 2.3: d-separation
• Proposition 3.1
• proof
• Definition 4.1: Compatibility of observed distribution with a causal structure
• Definition 4.2: Causal model
• Lemma 4.1
• Theorem 4.1