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Perspectives on signalling and causal order in physical theories

Maarten Grothus

Published Apr 13, 2026• 0 citations
OpenAlex

Abstract

Causality is pivotal to our understanding of the world, presenting itself in different forms: information-theoretic and relativistic, the former linked to the flow of information, and the latter to the structure of space-time.To operationally infer causal relations across a variety of different theories, the operational notion of signalling plays a central role.In particular, this includes the affects framework, formalising signalling in general physical theories using a causal modelling approach, and notions of signalling in quantum theory.In the first part of this thesis, we leverage the affects framework to improve the characterisation of information-theoretic signalling as defined through so-called affects relations. Specifically, we provide conditions for identifying redundancies in such a relation, introducing techniques for causal inference in unfaithful causal models (where the observable data does not “faithfully” reflect the causal dependences), and demonstrate how affects relations can be applied to describe signalling in quantum theory.Beyond capturing information-theoretic signalling alone, the affects framework allows one to study its interplay with relativistic principles. We define an order-theoretic property called conicality, showing that it is satisfied for Minkowski space-times with d > 1 spatial dimensions. Embedding of information-theoretic causal models in space-time without violating relativistic principles such as no superluminal signalling (NSS), we observe that constraints imposed by NSS in a space-time and those imposed by purely information-theoretic causal inference behave differently. We then prove a correspondence between conical space-times and faithful causal models: in both cases, there emerges a parallel between these two types of constraints. This indicates a connection between informational and geometric notions of causality, which we use to prove the impossibility of non-degenerate embeddings of causal loops into conical spacetimes.In the second part of this thesis, we study quantum processes with indefinite causal order (ICO), for which various frameworks have been proposed. In particular, quantum circuits with quantum control of causal order (QC-QCs) form a broad class of higher-order maps which are believed to represent all quantum processes physically realisable in a fixed spacetime. Complementarily, the formalism of routed quantum circuits introduces quantum operations constrained by “routes” to represent processes in terms of a more fine-grained routed circuit decomposition, representing the information flow in the respective process. However, the existence of routed circuit decompositions has only been established for a small set of processes so far.We remedy this fact by connecting these two frameworks, proving that for any given N, one can systematically obtain a routed circuit decomposition for any QC-QC with N parties. We detail this construction explicitly and contrast it with alternative routed circuit decompositions of QC-QCs.We additionally provide representations of subclasses of QC-QCs, including a proposed class of agent-localised processes and the QC-supFO class capturing processes with non-dynamical causal order. We conclude this part by applying affects relations to describe processes with dynamical causal order.

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Causality is pivotal to our understanding of the world, presenting itself in different forms: information-theoretic and relativistic, the former linked to the flow of information, and the latter to the structure of space-time.To operationally infer causal relations across a variety of different theories, the operational notion of signalling plays a central role.In particular, this includes the affects framework, formalising signalling in general physical theories using a causal modelling approach, and notions of signalling in quantum theory.In the first part of this thesis, we leverage the affects framework to improve the characterisation of information-theoretic signalling as defined through so-called affects relations. Specifically, we provide conditions for identifying redundancies in such a relation, introducing techniques for causal inference in unfaithful causal models (where the observable data does not “faithfully” reflect the causal dependences), and demonstrate how affects relations can be applied to describe signalling in quantum theory.Beyond capturing information-theoretic signalling alone, the affects framework allows one to study its interplay with relativistic principles. We define an order-theoretic property called conicality, showing that it is satisfied for Minkowski space-times with d > 1 spatial dimensions. Embedding of information-theoretic causal models in space-time without violating relativistic principles such as no superluminal signalling (NSS), we observe that constraints imposed by NSS in a space-time and those imposed by purely information-theoretic causal inference behave differently. We then prove a correspondence between conical space-times and faithful causal models: in both cases, there emerges a parallel between these two types of constraints. This indicates a connection between informational and geometric notions of causality, which we use to prove the impossibility of non-degenerate embeddings of causal loops into conical spacetimes.In the second part of this thesis, we study quantum processes with indefinite causal order (ICO), for which various frameworks have been proposed. In particular, quantum circuits with quantum control of causal order (QC-QCs) form a broad class of higher-order maps which are believed to represent all quantum processes physically realisable in a fixed spacetime. Complementarily, the formalism of routed quantum circuits introduces quantum operations constrained by “routes” to represent processes in terms of a more fine-grained routed circuit decomposition, representing the information flow in the respective process. However, the existence of routed circuit decompositions has only been established for a small set of processes so far.We remedy this fact by connecting these two frameworks, proving that for any given N, one can systematically obtain a routed circuit decomposition for any QC-QC with N parties. We detail this construction explicitly and contrast it with alternative routed circuit decompositions of QC-QCs.We additionally provide representations of subclasses of QC-QCs, including a proposed class of agent-localised processes and the QC-supFO class capturing processes with non-dynamical causal order. We conclude this part by applying affects relations to describe processes with dynamical causal order.
Abstract Causality is pivotal to our understanding of the world, presenting itself in different forms: information-theoretic and relativistic, the former linked to the flow of information, and the latter to the structure of space-time. Leveraging a framework introduced in PRA, 106, 032204 (2022), which formally connects these two notions in general physical theories, we study their interplay. The framework defines information-theoretic causality through a causal modelling approach, which enables the inference of causal connections through agents’ interventions and correlations. First, we improve the characterization of information-theoretic signalling as defined through so-called affects relations. Specifically, we provide conditions for identifying redundancies in different parts of such a relation, introducing techniques for causal inference in unfaithful causal models (where the observable data does not ‘faithfully’ reflect the causal dependences). In particular, this demonstrates the possibility of causal inference using the absence of signalling between certain nodes. Second, we define an order-theoretic property called conicality, showing that it is satisfied for light cones in Minkowski space-times with d > 1 spatial dimensions but violated for d = 1. Finally, we study the embedding of information-theoretic causal models in space-time without violating relativistic principles such as no superluminal signalling (NSS). In general, we observe that constraints imposed by NSS in a space-time and those imposed by purely information-theoretic causal inference behave differently. We then prove a correspondence between conical space-times and faithful causal models: in both cases, there emerges a parallel between these two types of constraints. This indicates a connection between informational and geometric notions of causality, and offers new insights and tools for studying the relations between the principles of NSS and no causal loops in different space-time geometries and theories of information processing.
By relating and ordering events, causality constitutes a pivotal feature of our world. On the one hand, there are information-theoretic notions of causality defined in terms of the information processing ability of agents and on the other hand, there are relativistic notions of causality tied to a spacetime. In this thesis, we improve upon a framework introduced by V. Vilasini and R. Colbeck in PRA, 106, 032204 (2022) and PRL, 129, 110401 (2022) for connecting these notions, where the possibility of operationally detectable causal loops embedded in (1+1)-Minkowski spacetime without superluminal signalling was demonstrated. In the first part, we take the information-theoretic point of view, where the concept of higher-order (HO) affects relations was proposed to generically model signalling in the presence of cyclic, fine-tuned and non-classical causal influences. We establish new properties of HO affects relations and apply them to infer causal structures. We then demonstrate a complete and constructive way to detect causal loops from a set of HO affects relations. In the second part, we study the embedding of information-theoretic causal structures into partially ordered spacetimes. We propose stability conditions on the spacetime embedding to rule out a class of operationally detectable loops that cannot be ruled out by the principle of no-signalling (outside the relativistic future) alone. We then propose novel order-theoretic properties that we conjecture to hold in Minkowski spacetime with d≥2d \geq 2d≥2 spatial dimensions. This would imply that in contrast to (1+1)-Minkowski spacetime, in higher dimensions, the no-signalling principle is indeed sufficient for ruling out this class of loops. Finally, we introduce a relation which allows for causal inference through knowledge about the absence of signalling relations and we relate it to novel order-theoretic features of spacetime.
Abstract Formalisms for higher order quantum processes provide a theoretical formalisation of quantum processes where the order of agents’ operations need not be definite and acyclic, but may be subject to quantum superpositions. This has led to the concept of indefinite causal structures (ICS) which have garnered much interest. However, the interface between these information-theoretic approaches and spatiotemporal notions of causality is less understood, and questions relating to the physical realisability of ICS in a spatiotemporal context persist despite progress in their information-theoretic characterisation. Further, previous work suggests that composition of processes is not so straightforward in ICS frameworks, which raises the question of how this connects with the observed composability of physical experiments in spacetime. To address these points, we compare the formalism of quantum circuits with quantum control of causal order (QC-QC), which models an interesting class of ICS processes, with that of causal boxes, which models composable quantum information protocols in spacetime. We incorporate the set-up assumptions of the QC-QC framework into the spatiotemporal perspective and show that every QC-QC can be mapped to a causal box that satisfies these set up assumptions and acts on a Fock space while reproducing the QC-QC’s behaviour in a relevant subspace defined by the assumptions. Using a recently introduced concept of fine-graining, we show that the causal box corresponds to a fine-graining of the QC-QC, which unravels the original ICS of the QC-QC into a set of quantum operations with a well-defined and acyclic causal order, compatible with the spacetime structure. Our results also clarify how the composability of physical experiments is recovered, while highlighting the essential role of relativistic causality and the Fock space structure.
In PRL 129, 110401 it was shown that it is theoretically possible to have operationally detectable causal loops without violating the principle of no superluminal signalling (NSS) in (1+1)-Minkowski spacetime. Whether or not such causal loops are also possible in d>1d > 1d>1 spatial dimensions, has remained a key open question. We resolve this question by showing that in a wide class of "conical" spacetimes, including Minkowski with d > 1, NSS does rule out all operationally detectable causal loops, in classical, quantum and post-quantum theories. This establishes that the relationship between the relativistic principles of NSS and no causal loops depends inherently on the geometry of spacetime.
Quantum theory challenges classical causal explanations for physical phenomena, calling for a quantum generalisation of causal structure. This thesis advances the study of causal decompositions, a research programme that aims to bridge two prominent, yet conceptually distinct types of structural assumption found across the quantum causality literature. The first is in terms of the global dependency relations between input and output systems of quantum transformations, which in the unitary case we shall refer to as causal structure; the second is in terms of the compositional structure of quantum circuits or networks. Concretely, a causal decomposition is a circuit representation of a unitary transformation in which no-influence relations that hold between its inputs and outputs are made manifest in the absence of paths through the circuit connecting them. Applications include translating between different quantum causal modelling frameworks; equipping a framework based on unitary causal structure with a local, bottom-up description via generalised autonomous causal mechanisms; and relating notions of compatibility between quantum transformations and relativistic causality. Whether causal decompositions always exist remains an open problem. The contributions of this thesis are threefold. First, we develop an order-theoretic account of circuit syntax and demonstrate that every circuit with a given input-output connectivity can be rewritten into a one of a canonical form given by the concept lattice construction from lattice theory. This systematises the search for causal decompositions by providing the shape of the quantum circuit one should look for. Second, we establish a combinatorial condition that fully characterises when causal decompositions into unitary quantum circuits exist without requiring the generalisation to routed quantum circuits prompted by earlier research. This solves a well-defined subproblem of the broader quest, and our proof, an interplay of order theory and operator algebra, lays a foundation for future extensions. Third, we detail the application to relativistic causality. Within a multipartite framework generalising existing literature, we emphasise that the mere lack of superluminal signalling through a quantum channel is not sufficient to conclude its realisability by dynamics that are locally consistent with relativity theory. We conjecture that, on the other hand, a lack of superluminal causal influences through its unitary dilations is sufficient—at least up to arbitrarily small errors. We demonstrate this in various special cases and show that the general statement reduces to the existence of a weak version of causal decompositions.
In recent years, various frameworks have been proposed for the study of quantum processes with indefinite causal order. In particular, quantum circuits with quantum control of causal order (QC-QCs) form a broad class of physical supermaps obtained from a bottom-up construction and are believed to represent all quantum processes physically realisable in a fixed spacetime. Complementarily, the formalism of routed quantum circuits introduces quantum operations constrained by "routes" to represent processes in terms of a more fine-grained routed circuit decomposition. This decomposition, formalised using a so-called routed graph, represents the information flow within the respective process. However, the existence of routed circuit decompositions has only been established for a small set of processes so far, including both certain specific QC-QCs and more exotic processes as examples. In this work, we remedy this fact by connecting these two frameworks. We prove that for any given NNN, one can use a single routed graph to systematically obtain a routed circuit decomposition for any QC-QC with NNN parties. We detail this construction explicitly and contrast it with other routed circuit decompositions of QC-QCs, which we obtain from alternative routed graphs. We conclude by pointing out how this connection can be useful to tackle various open problems in the field of indefinite causal order, particularly establishing circuit representations of subclasses of QC-QCs.