Impossibility of superluminal signalling rules out causal loops in conical spacetimes

Abstract: 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 $d > 1$ 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.

• The paper shows that no-superluminal signalling in conical spacetimes enforces acyclicity by ruling out operationally detectable causal loops.
• It employs an affects framework to reduce higher-order signalling relations to standard forms, revealing strict geometric constraints.
• The results offer practical implications for designing robust protocols in quantum, classical, and post-quantum causal models by closing loopholes.

Impossibility of Superluminal Signalling and Causal Loops in Conical Spacetimes

Context and Problem Statement

The paper "Impossibility of superluminal signalling rules out causal loops in conical spacetimes" (2606.20476) addresses the long-standing question at the intersection of information-theoretic and spatiotemporal causal structures: Are operationally detectable causal loops compatible with no-superluminal-signalling (NSS) in higher-dimensional (d > 1) Minkowski or causally regular (conical) spacetimes, as they are in (1+1)-dimensional Minkowski spacetime? The prior work [7] demonstrated that in (1+1)-Minkowski spacetime, NSS does not necessarily imply the absence of causal loops. This paper resolves the open problem by proving that, in conical spacetimes—including higher-dimensional Minkowski—NSS unambiguously rules out all operationally detectable causal loops, independent of the underlying theory (classical, quantum, or post-quantum).

Theoretical Framework and Core Concepts

The analysis operates within the "affects framework," which unifies causal modelling using directed graphs (with observed and unobserved nodes) and the d-separation property to formalize conditional independence among random variables. The framework distinguishes between causation and signalling: fine-tuned models can exhibit underlying causal relations without observable signalling, making NSS and the prohibition of superluminal causation inequivalent in general.

Signalling is encapsulated through affects relations (including higher-order variants), with irreducibility constraints (Irred1 for the first argument, Irred3 for the third) to ensure robust inference. Embedding causal models in spacetime involves mapping observed random variables (RVs) to locations in a partially ordered set (poset) that reflects lightcone structure, enabling NSS to be framed as a compatibility condition between affects relations and this poset order.

A pivotal geometric ingredient is "conicality," an order-theoretic property of spacetime: the joint future region of a finite set of points uniquely identifies those points in conical spacetimes ($d > 1$ Minkowski, causally simple, future cohesive spacetimes). The absence of conicality in $d = 1$ is the source of previously demonstrated exceptions.

Main Theorem and Proof Structure

The central result asserts that in any conical spacetime, NSS is sufficient to rule out all operationally verifiable causal loops (ACLs). The proof proceeds by:

1. Reduction of Higher-Order Affects Relations: Technical tools show that any set of possibly higher-order affects relations can be reduced, without loss of generality, to a set of standard (0th-order) affects relations, preserving implications for causal inference and compatibility under conicality.
2. Compatibility Constraints and Embedding Geometry: For an ACL to be embedded non-degenerately, the future lightcones of participating RVs must satisfy strict subset inclusions. In conical spacetimes, this ultimately leads to a contradiction (a set and its strict subset having identical joint futures), rendering compatible embeddings necessarily degenerate.
3. Implications of Fine-Tuning: Non-conical embeddings (possible in non-conical spacetimes or finely-tuned posets) can support causal loops without superluminal signalling, but these are exceptional and fragile, requiring fine-tuning both in causal mechanisms and spacetime embedding.

Implications and Comparative Analysis

The result formalizes the intuition that the NSS principle in relativistic settings should preclude causal loops, but demonstrates that this is intimately tied to spacetime geometry. The correspondence between information-theoretic cyclic structures and spatiotemporal causality is thus not universal but dependent on conicality. Practically, this strengthens constraints for physical protocol design in quantum information, quantum foundations, and post-quantum scenarios: cyclic causal structures with operational signalling are forbidden by NSS in all conical spacetimes, closing loopholes available only in $d=1$ cases or through geometric fine-tuning.

Furthermore, the analysis generalizes to abstract posets and offers a bridge to DAG-based time emergence (cf. tensor network approaches (Ferradini et al., 5 Mar 2026)), suggesting that order-theoretic properties may underpin the emergence of acyclic time direction from operational primitives.

Strong Numerical Results and Notable Claims

The paper establishes a sharp dichotomy: in (d+1)-Minkowski spacetime with $d > 1$, NSS strictly enforces acyclicity, whereas in (1+1)-Minkowski, operationally detectable ACLs can exist with NSS compatibility. The result holds for arbitrary system cardinalities and does not depend on specifics of the underlying causal model (including quantum process matrix models, provided d-separation or related separation criteria apply).

Notably, the claim is uniform across theory classes: classical, quantum, and post-quantum cyclic causal models are equally constrained by conicality and NSS. The impossibility result is robust with respect to clustering in affects relations, where additional forms of information-theoretic fine-tuning are required for embedding ACLs without observable signalling.

Open Directions and Future Developments

Several avenues are delineated for future inquiry:

• Exploration of stronger causal inference rules when restricting to specific system cardinalities (e.g., binary RVs) or process matrices beyond d-separation, such as o-separation or p-separation.
• Investigation of the universality of the no-go result for cyclic causal models governed by alternative separation theorems in classical and quantum causal modeling (Ferradini et al., 6 Feb 2025).
• Expansion of the affects framework to study the emergence of acyclic causal order and time direction in the absence of a fixed spacetime background and applicability to operationally realisable quantum protocols.

Conclusion

This paper rigorously demonstrates that the impossibility of superluminal signalling in conical spacetimes rules out operationally detectable causal loops, establishing a geometric criterion for the correspondence between relativistic causality principles and cyclic information-theoretic structures. The results clarify the limitations of embedding cyclic causal models in higher-dimensional Minkowski and causally regular spacetimes and underscore the role of order-theoretic geometry in constraining information processing architectures across classical, quantum, and post-quantum domains. The approach offers a versatile foundation for future studies of causal order emergence and quantum information protocols within—and beyond—conventional spacetime formulations.