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arxiv: 2403.00916 · v3 · submitted 2024-03-01 · 🌀 gr-qc · math-ph· math.MP· math.ST· quant-ph· stat.TH

Characterizing Signalling: Connections between Causal Inference and Space-time Geometry

Maarten Grothus , V. Vilasini This is my paper

Pith reviewed 2026-05-24 03:00 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.MPmath.STquant-phstat.TH
keywords causal inferenceaffects relationsconical space-timesno superluminal signallingfaithful modelsMinkowski space-timelight conesspace-time geometry
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The pith

A correspondence holds between conical space-times and faithful causal models, aligning their signalling constraints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper connects two notions of causality by refining how affects relations detect signalling in causal models, including cases where data is unfaithful. It introduces conicality as an order property of light cones that holds in Minkowski space with more than one spatial dimension but fails in one dimension. The central result proves that constraints from no superluminal signalling in space-time match those from information-theoretic causal inference exactly when space-times are conical and models are faithful. This link shows how geometric and informational causality can be studied together without assuming one reduces to the other.

Core claim

We improve the characterization of information-theoretic signalling through affects relations by giving conditions to identify redundancies, which enables causal inference even in unfaithful models using the absence of signalling. We define conicality and show it is satisfied by light cones in Minkowski space-times with d greater than 1 spatial dimensions but violated for d equals 1. We then prove a correspondence between conical space-times and faithful causal models under which the constraints imposed by no superluminal signalling align with those from purely information-theoretic causal inference.

What carries the argument

The proved correspondence between conical space-times (those satisfying the order-theoretic property of conicality) and faithful causal models (where observable data reflects underlying causal dependences), which makes no-superluminal-signalling constraints parallel the constraints from affects relations.

If this is right

  • Redundancies in affects relations can be detected, permitting causal inference from the absence of signalling between nodes.
  • Conicality distinguishes light-cone structures across different dimensions of Minkowski space.
  • Embedding causal models into space-time respects no-superluminal-signalling in general but produces matching constraints precisely in the conical-faithful case.
  • The parallel between no-superluminal-signalling and no-causal-loops principles can be examined uniformly across different space-time geometries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Experiments that measure signalling patterns in unfaithful settings could indirectly probe whether an underlying space-time is conical.
  • The same correspondence might be tested in discrete or quantum causal models to see whether conicality still separates the constraint types.
  • If the alignment extends to curved space-times, it could link information flow to local light-cone properties without global flatness assumptions.

Load-bearing premise

The prior formal connection between affects relations in causal models and relativistic no-superluminal-signalling principles holds in general physical theories.

What would settle it

A concrete space-time that satisfies conicality yet allows an embedding of a faithful causal model that violates no superluminal signalling, or a faithful model whose affects relations fail to match the light-cone structure of a conical space-time.

Figures

Figures reproduced from arXiv: 2403.00916 by Maarten Grothus, V. Vilasini.

Figure 1
Figure 1. Figure 1: Starting from the original (pre-intervention) causal structure, an intervention on the node [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Pre and post-intervention causal structures for Example [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Overview of causal inference concepts and relationships given by our results. Generally, an affects relation [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example for a fine-graining of a causal structure, as given in Example [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A light cone in 2+1-Minkowski space-time. For each point [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Light cones originating from three points located on a space-like line in Minkowski space-time with [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Schematic representation of light cones in Minkowski space-time for [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Sketch of a compatible non-degenerate embedding of Example [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Finite posets can be depicted using Hasse diagrams. Here, two nodes A and B are connected and B is shown above A if B covers A (cf. Definition B.3). In (a), we see a poset which is not a lattice, since B ≺ D ≻ C and B ≺ E ≻ C. Therefore, since D ̸⪯̸⪰ E, B and C have no least upper bound. In (b), we see a poset which forms a lattice. Here, H = B ∨ C. Proof: We prove this via contraposition. Let T be a conic… view at source ↗
Figure 10
Figure 10. Figure 10: Causal structures for various examples. see that (Y1 ⊥⊥ Y2|X)Gdo(X) holds, as this follows from the d-separation (Y1 ⊥d Y2|X)Gdo(X) in the causal structure. Noting that the only non-empty strict subsets sY in this case are {Y1} and {Y2}, the second condition is satisfied. Finally, notice that PGdo(X) (Y1|X) is deterministic and PGdo(X) (Y2|X) is uniform (since Y2 = X ⊕E2 with X independent of E2 and E2 un… view at source ↗
Figure 11
Figure 11. Figure 11: Combined representation of the causal structure of Example [PITH_FULL_IMAGE:figures/full_fig_p041_11.png] view at source ↗
read the original 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, 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. Here, information-theoretic causality is defined through a causal modelling approach. 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 for studying the relations between the principles of NSS and no causal loops in different space-time geometries and theories of information processing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper leverages a 2022 PRA framework connecting information-theoretic causality (via affects relations) to relativistic no-superluminal signalling (NSS) in general theories. It improves the characterization of signalling by giving conditions to identify redundancies in affects relations, enabling causal inference techniques for unfaithful models. It defines an order-theoretic property 'conicality' and shows it holds for light cones in Minkowski spacetime with d>1 spatial dimensions but fails for d=1. It then examines embeddings of causal models into spacetime respecting NSS and proves a correspondence between conical spacetimes and faithful causal models, under which NSS constraints and information-theoretic constraints exhibit a parallel.

Significance. If the correspondence result holds, the work supplies a concrete bridge between causal-inference methods and spacetime geometry that could inform studies of signalling in curved or discrete spacetimes. The redundancy-identification techniques for unfaithful models constitute a usable addition to causal inference, while the conicality definition offers a geometric diagnostic that is falsifiable in low-dimensional Minkowski space. These elements are independent of the prior framework and would remain valuable even if the embedding correspondence requires further qualification.

major comments (1)
  1. [Abstract (final paragraph) and the section presenting the embedding and correspondence results] The central correspondence (abstract, final paragraph) between conical spacetimes and faithful causal models is stated to follow from the 2022 PRA framework's connection between affects relations and NSS. The manuscript does not re-derive or independently verify the key linking propositions of that framework; any gaps in the formalization of faithfulness or redundancy identification in the 2022 work therefore propagate directly into the claimed parallel. This is load-bearing for the embedding and correspondence claims.
minor comments (2)
  1. [Section introducing conicality] The definition of conicality is introduced without an explicit comparison to standard order-theoretic properties (e.g., interval orders or causal orders) already used in the relativistic causality literature; a short paragraph situating the new notion would improve readability.
  2. [Sections on affects relations and on embedding] Notation for affects relations and the redundancy conditions is introduced in the first technical section but is not cross-referenced when the same objects appear in the embedding discussion; consistent equation numbering or a short notation table would reduce ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying this point about the manuscript's reliance on the 2022 PRA framework. We address the concern directly below.

read point-by-point responses

  1. Referee: [Abstract (final paragraph) and the section presenting the embedding and correspondence results] The central correspondence (abstract, final paragraph) between conical spacetimes and faithful causal models is stated to follow from the 2022 PRA framework's connection between affects relations and NSS. The manuscript does not re-derive or independently verify the key linking propositions of that framework; any gaps in the formalization of faithfulness or redundancy identification in the 2022 work therefore propagate directly into the claimed parallel. This is load-bearing for the embedding and correspondence claims.

    Authors: We agree that the correspondence between conical spacetimes and faithful causal models is an application of the connection between affects relations and NSS established in the 2022 PRA paper, and the present manuscript does not re-derive those foundational linking propositions. This is intentional, as the work is positioned as an extension. The independent contributions are (i) the new conditions for identifying redundancies within affects relations, which enable causal inference techniques specifically for unfaithful models, and (ii) the order-theoretic definition of conicality together with its verification that the property holds for light cones in Minkowski spacetime when the number of spatial dimensions d > 1 but fails for d = 1. The correspondence result then shows how these new elements interact with the prior framework to produce a parallel between NSS constraints and information-theoretic constraints. To make the logical dependence explicit and thereby address the referee's concern, we will revise the final paragraph of the abstract and the section on embeddings and correspondence to state clearly that the result leverages the 2022 framework without re-deriving its core propositions. We believe this clarification will render the structure of the argument transparent without requiring a full re-derivation, which would be outside the scope of the present paper. revision: partial

Circularity Check

0 steps flagged

No significant circularity; new results on conicality and signalling are independent

full rationale

The paper cites its own prior 2022 framework (PRA 106, 032204) to connect affects relations with NSS, but this citation supports background context rather than forcing the central results. The work introduces independent content: conditions for redundancies in affects relations, the conicality property (satisfied in Minkowski d>1 but not d=1), and a proof of correspondence between conical space-times and faithful causal models. No derivation step reduces by construction to the prior work or to fitted inputs; the self-citation is not load-bearing for the new theorems. The paper is self-contained against external benchmarks for its novel contributions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based solely on the abstract; the central claims rest on the validity of the 2022 framework and the new definitions introduced here.

axioms (1)
  • domain assumption The framework introduced in PRA, 106, 032204 (2022) formally connects information-theoretic causality (affects relations) and relativistic causality in general physical theories.
    Invoked in the first sentence of the abstract as the basis for the entire study.
invented entities (1)
  • conicality no independent evidence
    purpose: An order-theoretic property of space-times that holds for light cones in Minkowski space with d>1 spatial dimensions but is violated for d=1.
    Newly defined in the paper to characterize space-times and link to faithful causal models.

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discussion (0)

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Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages · 3 internal anchors

  1. [1]

    Spirtes, C

    P. Spirtes, C. Glymour, and R. Scheines.Causation, Prediction, and Search. Springer New York,

    work page

  2. [2]

    doi: 10.1007/978-1-4612-2748-9

    isbn: 9781461227489. doi: 10.1007/978-1-4612-2748-9

    work page doi:10.1007/978-1-4612-2748-9

  3. [3]

    J. Pearl. Causality: Models, Reasoning and Inference. 2nd ed. Cambridge University Press, 2009. doi: 10.1017/CBO9780511803161

    work page doi:10.1017/cbo9780511803161 2009

  4. [4]

    A graphical causal model for resolving species identity effects and biodiversity–ecosystem function correlations

    D. R. Schoolmaster, C. R. Zirbel, and J. P. Cronin. “A graphical causal model for resolving species identity effects and biodiversity–ecosystem function correlations”. In:Ecology 101.8 (June 2020). doi: 10.1002/ecy.3070

    work page doi:10.1002/ecy.3070 2020

  5. [5]

    A biologist’s guide to model selection and causal inference

    Z. M. Laubach et al. “A biologist’s guide to model selection and causal inference”. In:Proceedings of the Royal Society B: Biological Sciences288.1943(Jan.2021),p.20202815. doi:10.1098/rspb.2020.2815

    work page doi:10.1098/rspb.2020.2815 1943

  6. [6]

    A review of causal inference for biomedical informatics

    S. Kleinberg and G. Hripcsak. “A review of causal inference for biomedical informatics”. In:Journal of Biomedical Informatics44.6(2011),pp.1102–1112. issn:1532-0464. doi:10.1016/j.jbi.2011.07.001

    work page doi:10.1016/j.jbi.2011.07.001 2011

  7. [7]

    Big Data, Data Science, and Causal Inference: A Primer for Clinicians

    Y. Raita et al. “Big Data, Data Science, and Causal Inference: A Primer for Clinicians”. In: Frontiers in Medicine8 (July 2021).doi: 10.3389/fmed.2021.678047

    work page doi:10.3389/fmed.2021.678047 2021

  8. [8]

    Research Trend of Causal Machine Learning Method: A Literature Review

    S. Arti, I. Hidayah, and S. S. Kusumawardhani. “Research Trend of Causal Machine Learning Method: A Literature Review”. In:IJID (International Journal on Informatics for Development) 9.2 (Dec. 2020), pp. 111–118.doi: 10.14421/ijid.2020.09208. 27

    work page doi:10.14421/ijid.2020.09208 2020

  9. [9]

    Quantum Common Causes and Quantum Causal Models

    J.-M. A. Allen et al. “Quantum Common Causes and Quantum Causal Models”. In:Phys. Rev. X 7 (July 2017), p. 031021.doi: 10.1103/PhysRevX.7.031021

    work page doi:10.1103/physrevx.7.031021 2017

  10. [10]

    & Oreshkov, O

    J.Barrett,R.Lorenz,andO.Oreshkov. Quantum Causal Models.2019. doi:10.48550/arXiv.1906.10726. arXiv: 1906.10726 [quant-ph]

    work page doi:10.48550/arxiv.1906.10726 2019

  11. [12]

    Embedding cyclic causal structures in acyclic spacetimes: no-go results for process matrices,

    V. Vilasini and R. Renner.Embedding cyclic causal structures in acyclic spacetimes: no-go results for process matrices. Mar. 2022.doi: 10.48550/arXiv.2203.11245. arXiv:2203.11245 [quant-ph]

    work page doi:10.48550/arxiv.2203.11245 2022

  12. [13]

    Impossibility of Superluminal Signaling in Minkowski Spacetime Does Not Rule Out Causal Loops

    V. Vilasini and R. Colbeck. “Impossibility of Superluminal Signaling in Minkowski Spacetime Does Not Rule Out Causal Loops”. In:Phys. Rev. Lett. 129 (11 Sept. 2022), p. 110401. doi: 10.1103/PhysRevLett.129.110401

    work page doi:10.1103/physrevlett.129.110401 2022

  13. [14]

    Theories with no superluminal signaling have greater information-processing power than theories with no superluminal causation

    V. Vilasini and R. Colbeck. Information-processing in theories constrained by no superluminal causation vs no superluminal signalling. 2024.doi: 10.48550/arXiv.2402.12446. arXiv:2402.12446 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2402.12446 2024

  14. [15]

    Quantum secret sharing

    M. Hillery, V. Bužek, and A. Berthiaume. “Quantum secret sharing”. In:Phys. Rev. A59 (3 Mar. 1999), pp. 1829–1834.doi: 10.1103/PhysRevA.59.1829

    work page doi:10.1103/physreva.59.1829 1999

  15. [16]

    Theory of quantum secret sharing

    D. Gottesman. “Theory of quantum secret sharing”. In:Phys. Rev. A61 (4 Mar. 2000), p. 042311. doi: 10.1103/PhysRevA.61.042311

    work page doi:10.1103/physreva.61.042311 2000

  16. [17]

    Quantum entanglement for secret sharing and secret splitting

    A. Karlsson, M. Koashi, and N. Imoto. “Quantum entanglement for secret sharing and secret splitting”. In:Phys. Rev. A59 (1 Jan. 1999), pp. 162–168.doi: 10.1103/PhysRevA.59.162

    work page doi:10.1103/physreva.59.162 1999

  17. [18]

    Stabiliser Codes and Quantum Error Correction

    D. E. Gottesman. “Stabiliser Codes and Quantum Error Correction”. PhD thesis. California Insti- tute of Technology, 1997.doi: 10.48550/arXiv.2102.02393

    work page doi:10.48550/arxiv.2102.02393 1997

  18. [19]

    How to Share a Quantum Secret

    R. Cleve, D. Gottesman, and H.-K. Lo. “How to Share a Quantum Secret”. In:Phys. Rev. Lett.83 (3 July 1999), pp. 648–651.doi: 10.1103/PhysRevLett.83.648

    work page doi:10.1103/physrevlett.83.648 1999

  19. [20]

    A no-summoning theorem in relativistic quantum theory

    A. Kent. “A no-summoning theorem in relativistic quantum theory”. In:Quantum Information Processing12.2 (July 2012), pp. 1023–1032.issn: 1573-1332. doi: 10.1007/s11128-012-0431-6

    work page doi:10.1007/s11128-012-0431-6 2012

  20. [21]

    Quantum tasks in Minkowski space

    A. Kent. “Quantum tasks in Minkowski space”. In:Classical and Quantum Gravity29.22 (Oct. 2012), p. 224013.issn: 1361-6382. doi: 10.1088/0264-9381/29/22/224013

    work page doi:10.1088/0264-9381/29/22/224013 2012

  21. [22]

    Summoning information in spacetime, or where and when can a qubit be?

    P. Hayden and A. May. “Summoning information in spacetime, or where and when can a qubit be?” In:Journal of Physics A: Mathematical and Theoretical49.17 (Mar. 2016), p. 175304.issn: 1751-8121. doi: 10.1088/1751-8113/49/17/175304

    work page doi:10.1088/1751-8113/49/17/175304 2016

  22. [23]

    Quantum causal modelling

    F. Costa and S. Shrapnel. “Quantum causal modelling”. In:New Journal of Physics18.6 (June 2016), p. 063032.issn: 1367-2630. doi: 10.1088/1367-2630/18/6/063032

    work page doi:10.1088/1367-2630/18/6/063032 2016

  23. [24]

    and Lu, W

    D. Geiger, T. Verma, and J. Pearl. “Identifying independence in bayesian networks”. In:Networks 20.5 (1990), pp. 507–534.doi: 10.1002/net.3230200504

    work page doi:10.1002/net.3230200504 1990

  24. [25]

    Identifying Independencies in Causal Graphs with Feedback

    J. Pearl and R. Dechter. Identifying Independencies in Causal Graphs with Feedback. 2013. doi: 10.48550/arXiv.1302.3595. arXiv: 1302.3595 [cs]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1302.3595 2013

  25. [26]

    Cyclic quantum causal models

    J. Barrett, R. Lorenz, and O. Oreshkov. “Cyclic quantum causal models”. In:Nature Communica- tions 12.1 (Feb. 2021).doi: 10.1038/s41467-020-20456-x

    work page doi:10.1038/s41467-020-20456-x 2021

  26. [27]

    Blasi, A

    J. Henson, R. Lal, and M. F. Pusey. “Theory-independent limits on correlations from generalized Bayesian networks”. In:New Journal of Physics16.11 (Nov. 2014), p. 113043.doi: 10.1088/1367- 2630/16/11/113043

    work page doi:10.1088/1367- 2014

  27. [28]

    Reichenbach

    H. Reichenbach. The Direction of Time. University of California Press., 1956

    work page 1956

  28. [29]

    Foundations of structural causal models with cycles and latent variables

    S. Bongers et al. “Foundations of structural causal models with cycles and latent variables”. In: The Annals of Statistics49.5 (Oct. 2021).issn: 0090-5364. doi: 10.1214/21-aos2064

    work page doi:10.1214/21-aos2064 2021

  29. [30]

    Manca, S

    M. Gachechiladze, N. Miklin, and R. Chaves. “Quantifying Causal Influences in the Presence of a Quantum Common Cause”. In:Phys. Rev. Lett.125 (23 Dec. 2020), p. 230401.doi: 10.1103/Phys- RevLett.125.230401. 28

    work page doi:10.1103/phys- 2020

  30. [31]

    Jamming nonlocal quantum correlations

    J. Grunhaus, S. Popescu, and D. Rohrlich. “Jamming nonlocal quantum correlations”. In:Phys. Rev. A53 (6 June 1996), pp. 3781–3784.doi: 10.1103/PhysRevA.53.3781

    work page doi:10.1103/physreva.53.3781 1996

  31. [32]

    & Colbeck, R

    V.VilasiniandR.Colbeck. A causal modelling analysis of Bell scenarios in space-time: implications of jamming non-local correlations for relativistic causality principles.2023. doi:10.48550/arXiv.2311.18465. arXiv: 2311.18465 [quant-ph]

    work page doi:10.48550/arxiv.2311.18465 2023

  32. [33]

    R. D. Sorkin. Impossible Measurements on Quantum Fields. Feb. 1993. doi: 10.48550/arXiv.gr- qc/9302018. arXiv: gr-qc/9302018 [gr-qc]

    work page doi:10.48550/arxiv.gr- 1993

  33. [34]

    Impossible measurements require impossible apparatus

    H. Bostelmann, C. J. Fewster, and M. H. Ruep. “Impossible measurements require impossible apparatus”. In:Phys. Rev. D103 (2 Jan. 2021), p. 025017.doi: 10.1103/PhysRevD.103.025017

    work page doi:10.1103/physrevd.103.025017 2021

  34. [35]

    Onthestructureofcausalspaces

    E.H.KronheimerandR.Penrose.“Onthestructureofcausalspaces”.In: Mathematical Proceedings of the Cambridge Philosophical Society63.2(Apr.1967),pp.481–501. doi:10.1017/s030500410004144x

    work page doi:10.1017/s030500410004144x 1967

  35. [36]

    The class of continuous timelike curves determines the topology of spacetime

    D. B. Malament. “The class of continuous timelike curves determines the topology of spacetime”. In: Journal of Mathematical Physics18.7 (July 1977), pp. 1399–1404.doi: 10.1063/1.523436

    work page doi:10.1063/1.523436 1977

  36. [37]

    Causal structure of spacetime and geometric algebra for quantum gravity

    I. Dukovski. “Causal structure of spacetime and geometric algebra for quantum gravity”. In:Phys. Rev. D87 (6 Mar. 2013), p. 064022.doi: 10.1103/PhysRevD.87.064022

    work page doi:10.1103/physrevd.87.064022 2013

  37. [38]

    Bombelli, J

    L. Bombelli et al. “Space-time as a causal set”. In:Phys. Rev. Lett.59 (5 Aug. 1987), pp. 521–524. doi: 10.1103/PhysRevLett.59.521

    work page doi:10.1103/physrevlett.59.521 1987

  38. [39]

    The causal set approach to quantum gravity

    S. Surya. “The causal set approach to quantum gravity”. In:Living Reviews in Relativity22.1 (Sept. 2019). issn: 1433-8351. doi: 10.1007/s41114-019-0023-1

    work page doi:10.1007/s41114-019-0023-1 2019

  39. [40]

    2. Causality and Chronology

    R. Penrose. “2. Causality and Chronology”. In:Techniques in Differential Topology in Relativity. Society for Industrial and Applied Mathematics, Jan. 1972, pp. 11–17

    work page 1972

  40. [41]

    8. Causal Structure

    R. M. Wald. “8. Causal Structure”. In:General Relativity. University of Chicago Press, 1984, pp. 188–2010.doi: 10.1017/9781108837996.009

    work page doi:10.1017/9781108837996.009 1984

  41. [42]

    Lorentzian causality theory

    E. Minguzzi. “Lorentzian causality theory”. In:Living Reviews in Relativity22.1 (June 2019).doi: 10.1007/s41114-019-0019-x

    work page doi:10.1007/s41114-019-0019-x 2019

  42. [43]

    Forré and J

    P. Forré and J. M. Mooij.Markov Properties for Graphical Models with Cycles and Latent Variables

    work page

  43. [44]

    Markov Properties for Graphical Models with Cycles and Latent Variables

    doi: 10.48550/arXiv.1710.08775. arXiv:1710.08775 [math]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1710.08775

  44. [45]

    Ferradini and V

    C. Ferradini and V. Vilasini.In preparation. ETH Zurich. 2024

    work page 2024

  45. [46]

    Grothus and V

    M. Grothus and V. Vilasini. In preparation based on arXiv:2211.03593 (MGs Master’s thesis). 2024

    work page arXiv 2024

  46. [47]

    Quantum gravity from causal dynamical triangulations: a review

    R. Loll. “Quantum gravity from causal dynamical triangulations: a review”. In:Classical and Quantum Gravity37.1 (Dec. 2019), p. 013002.issn: 1361-6382. doi: 10.1088/1361-6382/ab57c7

    work page doi:10.1088/1361-6382/ab57c7 2019

  47. [48]

    On Deducing Conditional Independence from d-Separation in Causal Graphs with Feedback (Research Note)

    R. M. Neal. “On Deducing Conditional Independence from d-Separation in Causal Graphs with Feedback (Research Note)”. In:Journal of Artificial Intelligence Research12 (Mar. 2000), pp. 87–

    work page 2000

  48. [49]

    doi: 10.1613/jair.689

    work page doi:10.1613/jair.689

  49. [50]

    Birkhoff

    G. Birkhoff. Lattice Theory. American Mathematical Society Colloquium Publications. Providence, RI: American Mathematical Society, July 1940

    work page 1940

  50. [51]

    B. A. Davey and H. A. Priestley.Introduction to Lattices and Order. 2nd. Cambridge: Cambridge University Press, 2002

    work page 2002

  51. [52]

    Virtual Time and Global States of Distributed Systems

    F. Mattern. “Virtual Time and Global States of Distributed Systems”. In:Proceedings of the International Workshop on Parallel and Distributed Algorithms. 1989.url: https://www.vs.inf. ethz.ch/publ/papers/VirtTimeGlobStates.pdf

    work page 1989

  52. [53]

    The logic of causally closed spacetime subsets

    H. Casini. “The logic of causally closed spacetime subsets”. In:Classical and Quantum Gravity 19.24 (Nov. 2002), pp. 6389–6404.doi: 10.1088/0264-9381/19/24/308

    work page doi:10.1088/0264-9381/19/24/308 2002

  53. [54]

    Petkov, ed

    V. Petkov, ed. Minkowski Spacetime: A Hundred Years Later. Springer Netherlands, 2010. doi: 10.1007/978-90-481-3475-5

    work page doi:10.1007/978-90-481-3475-5 2010

  54. [55]

    Causal structure in the presence of sectorial con- straints, with application to the quantum switch

    N. Ormrod, A. Vanrietvelde, and J. Barrett. “Causal structure in the presence of sectorial con- straints, with application to the quantum switch”. In:Quantum 7 (June 2023), p. 1028. issn: 2521-327X. doi: 10.22331/q-2023-06-01-1028. 29 A Further details on the causal modelling framework In this section, we provide a formal definition for the technical conc...

    work page doi:10.22331/q-2023-06-01-1028 2023

  55. [56]

    opposite

    proposes another graph separation property calledp-separation that is valid for all fine-dimensional, possibly cyclic quantum causal models (and therefore for all finite and discrete variable classical causal models). The extension of our results to these alternative graph separation properties is an interesting avenue for future work. B Properties of par...

    work page

  56. [57]

    X ⊨Y |do(Z),W =⇒X ⊨YW

    work page

  57. [58]

    X ⊨Y |do(Z),W is Irred1 =⇒X ⊨YW is Irred1

    work page

  58. [59]

    X ⊨YW |do(Z) ⇐⇒X ⊨Y |do(Z),W ∨X ⊨W|do(Z). By the same argument as for Lemma E.2.2., one can also derive: Corollary E.3 For a causal model over a setS of RVs, whereX,Y,Z,W ⊂S disjoint, X ⊨Y |do(Z),W is Irred3 =⇒X ⊨YW |do(Z) is Irred3. We continue with a general result about conditional affects relations that will be useful, this is essentially a generaliza...

    work page

  59. [60]

    c2(t−t0)2 = ∑ 1<i≤d (xi−xi 0)2

    are thed-dimensional spatial co-ordinates of the point (in some chosen co-ordinate system). c2(t−t0)2 = ∑ 1<i≤d (xi−xi 0)2. (55) Observe that for each time slice (fixed value oft), the above equation corresponds to ad-dimensional sphere with radiusc2(t−t0)2. Whenever d≥2, for a fixedt, given any finite portion of this spherical boundary, we can determine ...

    work page

  60. [61]

    Hence, ¯Fs(X ) = ¯Fs(XY i) (64) =⇒¯Fs(X )⊆¯Fs(Yi) ∀i∈I (65) ⇐⇒¯Fs(X )⊆ ⋂ i∈I ¯Fs(Yi) = ¯Fs (⋃ i∈I Yi ) , (66) yielding the first alternative of Equation (60)

    Case 1 (∃i :O(X ) =O(span(XY i))) : In this case, we can deduce that the same holds for allYk with k∈I, due to Equation (62). Hence, ¯Fs(X ) = ¯Fs(XY i) (64) =⇒¯Fs(X )⊆¯Fs(Yi) ∀i∈I (65) ⇐⇒¯Fs(X )⊆ ⋂ i∈I ¯Fs(Yi) = ¯Fs (⋃ i∈I Yi ) , (66) yielding the first alternative of Equation (60)

    work page

  61. [62]

    Case 2 (∀i : O(X )̸= O(span(XY i))) : This implies thatO(span(XY i))\O(X )̸=∅. We have already established in Equation (63) that in conical space-times where Equation (62) is satisfied, O(span(XY k)) must be identical for alli∈I, this implies the same holds forO(span(XY i))\O(X ), and yields the second alternative of Equation (60). □ We conclude by provid...

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