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

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 modelling causation in 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. 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. ContentsI. Introduction II. Preliminaries: Acyclic and faithful causal models III. Motivation for analysing fine-tuned and cyclic causal models A. Friedman's thermostat and the one-time pad B. Jamming non-local correlations IV. The framework, Part 1: Causality A. Cyclic and fine-tuned causal models B. Interventions and affects relations C. Conditional and higher-order affects relations D. Relationships between concepts V. The framework, Part 2: Space-time A. Space-time structure B. Embedding of a causal model in a space-time structure C. Compatibility of a causal model with an embedding in space-time D. Necessary and sufficient conditions for compatibility VI. Causal loops and their space-time embeddings A. Different classes of causal loops B. Possibility of compatibly embedding causal loops in space-time

Is There Causation in Fundamental Physics? New Insights from Process Matrices and Quantum Causal Modelling

“…However, we would still caution that the expectation that the underlying structure should be causal (albeit not acyclic) should be treated with care, since these sorts of phenomena do not involve causal order and therefore there is no obvious reason to expect them to be best explicable in the context of a causal formalism, even one that has been generalized in certain ways. For example, ref [73] continues to maintain that the Bell correlations should be understood in terms of 'causation without signalling (i.e., fine-tuned causal influences)' whereas we have argued here that this is not the right way to think of the Bell correlations. This does not mean that research on cyclic causal structures should not be pursued, but it does mean that we should be prepared to be quite flexible about these kinds of models: assumptions that they must have certain prototypically causal features should be carefully examined, and discarded if they are not working well.…”

“…Another interesting line of research involves extending quantum causal models to allow for cyclic causal structures [70,73]. In light of the arguments we have made here, this seems like a promising approach, since one way to model symmetric relationships which can't be decomposed into a strict partial order would be to write them as cyclic structures in the context of a broader causal description.…”

Compatibility of Cyclic Causal Structures with Spacetime in General Theories with Free Interventions

“…Causality is a concept of utmost importance in science [1]. In contrast to just observing correlations in arbitrary scenarios, it allows to ask why these occur, differentiating between cause and effect.…”

“…Over the following years, the respective approaches developed and diversified, bringing forth remarkable insights, some of which we will review over the course of this work. Nonetheless, to the best of the author's knowledge, [1] suggests the first framework which clearly separates the notions of relativistic and information-theoretic causality to explore their connections.…”

“…This work builds particularly on generalized notions of signalling for models with free interventions, which have been thoroughly studied in [1]. This approach is almost entirely theory independent, focusing mainly on the operational interface of the theory which is in-principle accessible via experiments.…”