arxiv: 2605.03132 · v1 · submitted 2026-05-04 · 🪐 quant-ph

A missing causal principle: Coordination

Daniel Centeno , Antoine Coquet , Maria Ciudad Ala\~n\'on , Lucas Tendick , Marc-Olivier Renou , Elie Wolfe This is my paper

Pith reviewed 2026-05-08 18:14 UTC · model grok-4.3

classification 🪐 quant-ph

keywords coordination principlecommon causeReichenbach principleBell inequalitiesGHZ statecausal networksno-signalingquantum information

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The pith

Perfect coordination among parties on a uniformly random output requires them to share a common cause.

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

The paper introduces the coordination principle, stating that N parties can agree perfectly on the same uniformly random output only when a common cause is present in their causal structure. This serves as a multipartite generalization of Reichenbach's common cause principle and is proven to hold throughout quantum theory in any network. The authors derive corresponding noise-tolerant Bell-like inequalities that certify the existence of such a cause. They also construct a non-quantum operational theory obeying no-signaling and independence that permits perfect coordination without any common cause when intermediate transformations are allowed between preparations and measurements.

Core claim

The central claim is that perfect coordination among N parties—agreement on one uniformly random output—occurs only if those parties share a common cause. Quantum information theory satisfies the principle in arbitrary networks, which yields testable, noise-tolerant inequalities for certifying common causes. A concrete counterexample probabilistic theory is built that respects no-signaling yet allows perfect coordination without common cause; this construction requires fully general causal scenarios that include intermediate transformations. Preparation of a multipartite GHZ state is shown to demand a quantum common cause that the derived inequalities can detect.

What carries the argument

The coordination principle: perfect agreement on a uniformly random output among N parties is possible only if the parties share a common cause.

If this is right

Quantum theory obeys the principle in every network, so common causes can be certified by noise-tolerant inequalities.
Preparing a multipartite GHZ state requires a quantum common cause that can be verified experimentally.
The principle is independent of no-signaling and independence alone once intermediate operations are permitted.
Bell-like inequalities derived from the principle apply to arbitrary causal networks in quantum settings.

Where Pith is reading between the lines

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

The principle may supply new certification tools for multipartite quantum resources beyond standard Bell tests.
Approximate versions could quantify tolerable noise levels before coordination without common cause becomes possible.
The distinction between quantum and post-quantum theories might be sharpened by checking which obey the principle in general scenarios.

Load-bearing premise

The constructed operational probabilistic theory that allows perfect coordination without a common cause via intermediate transformations is a reasonable and physically motivated model.

What would settle it

An explicit quantum experiment or calculation demonstrating perfect coordination among three parties with no shared common cause and no intermediate transformations between preparations and measurements would falsify the claim that quantum theory obeys the principle.

Figures

Figures reproduced from arXiv: 2605.03132 by Antoine Coquet, Daniel Centeno, Elie Wolfe, Lucas Tendick, Marc-Olivier Renou, Maria Ciudad Ala\~n\'on.

**Figure 1.** Figure 1: FIG. 1. Various causal structures represented as directed acyclic graphs (DAGs). Squares denote observed nodes and circles view at source ↗

**Figure 2.** Figure 2: FIG. 2. Graphical representation of the view at source ↗

**Figure 3.** Figure 3: FIG. 3. Diagrammatic representation of a test connecting view at source ↗

**Figure 4.** Figure 4: FIG. 4. Diagrammatic representation of parallel composition. view at source ↗

**Figure 7.** Figure 7: FIG. 7. Different preparations available within the OPT view at source ↗

**Figure 5.** Figure 5: FIG. 5. Diagrammatic representation of sequential composition. view at source ↗

**Figure 6.** Figure 6: FIG. 6. Closed circuit with a preparation, a transformation and view at source ↗

**Figure 9.** Figure 9: FIG. 9. Different observations available within the OPT view at source ↗

**Figure 10.** Figure 10: FIG. 10. Closed circuit using all components of the OPT. Note that it presents the same causal structure as the one described by view at source ↗

**Figure 11.** Figure 11: FIG. 11. Closed circuit of a pair of observations that are tetrahedron-embeddable. Traced out systems and type numbers are view at source ↗

**Figure 12.** Figure 12: FIG. 12. A closed circuit for a single observation whose causal past differs from that in the view at source ↗

**Figure 13.** Figure 13: FIG. 13. The most general scenario in which four quantum observed nodes do not share a quantum common cause which we name view at source ↗

**Figure 14.** Figure 14: FIG. 14. Scenario arising from measuring the observed nodes in view at source ↗

**Figure 15.** Figure 15: FIG. 15. Value of the inequality violation, i.e., value of the left-hand side of Eq. view at source ↗

**Figure 16.** Figure 16: FIG. 16. Value of the inequality violation derived from inequality view at source ↗

read the original abstract

We introduce the coordination principle, which states that perfect coordination, in the form of agreement on a uniformly random output, among N parties is possible only if they share a common cause. This principle is purely causal and can be viewed as a multipartite generalization of Reichenbach's common cause principle. We prove that quantum information theory satisfy the coordination principle in any network, and derive noise-tolerant Bell-like inequalities that certify the presence of a common cause. We further show that the principle is not a consequence of no-signaling and independence alone by constructing a concrete operational probabilistic theory that obeys both principles while still allowing perfect coordination without a common cause. This possibility arises only in fully general causal scenarios with intermediate transformations between preparations and measurements. We also formulate a genuinely quantum coordination task, showing that the preparation of a multipartite GHZ state requires a quantum common cause, which can be certified by Bell-like inequalities which are experimentally testable. Finally, we discuss the open problem of finding a quantitative, noise-tolerant version of the coordination principle that constrains approximate coordination in any reasonable causal theory. This work is the extended version of the more compact letter and provides all the technical details of the proofs.

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.

Desk Editor's Note private letter to a colleague

The paper defines a coordination principle requiring a common cause for perfect multipartite coordination on random outputs, proves quantum theory obeys it in networks, derives noise-tolerant inequalities, and gives a counterexample in another operational theory.

read the letter

The main point is that this paper defines a coordination principle requiring a common cause for perfect multipartite coordination on random outputs, proves quantum theory obeys it in networks, derives noise-tolerant inequalities, and gives a counterexample in another operational theory. Quantum satisfaction follows from the causal structure, and the inequalities are positioned as experimentally usable for certifying common causes. The counterexample respects no-signaling and independence yet permits coordination without the cause, but only when intermediate transformations are allowed between preparations and measurements. They also apply the idea to show that GHZ state preparation needs a quantum common cause, again with testable inequalities. The new content is the principle statement itself, the explicit proofs and counterexample, and the concrete GHZ case. The derivations are described in detail as an extended version. The soft spots are limited. The principle is stated only for perfect coordination, and the authors flag a quantitative version for approximate cases as an open problem. The counterexample framework is general and motivated in the paper, though its physical relevance may vary by reader. This is for people in quantum foundations and causal inference in networks who want new certification tools. It shows clear engagement with the literature and explicit constructions without circularity. The work has enough substance to deserve a serious referee. I would recommend sending it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper introduces the coordination principle, which states that perfect coordination (agreement on a uniformly random output) among N parties is possible only if they share a common cause; this is presented as a multipartite generalization of Reichenbach's common cause principle. It proves that quantum theory satisfies the principle in arbitrary networks, derives noise-tolerant Bell-like inequalities to certify common causes, constructs a counterexample operational probabilistic theory that obeys no-signaling and independence yet permits coordination without a common cause (possible only when intermediate transformations between preparations and measurements are allowed), formulates a quantum coordination task showing that GHZ-state preparation requires a quantum common cause (certifiable by testable inequalities), and identifies the open problem of a quantitative noise-tolerant version applicable to approximate coordination.

Significance. If the proofs and constructions hold, the work supplies a new, purely causal principle that strengthens the toolkit for analyzing causality in quantum networks and nonlocality. The explicit quantum satisfaction proof, the noise-tolerant inequalities, and the GHZ coordination task provide falsifiable, experimentally relevant predictions. The counterexample construction is a strength, as it demonstrates that the principle does not follow from no-signaling and independence alone and isolates the role of intermediate transformations. These elements advance the understanding of multipartite causal structures beyond standard assumptions.

major comments (1)

[Counterexample construction] Counterexample construction (the operational probabilistic theory section): the explicit verification that the theory respects no-signaling and independence without introducing a hidden common cause via the allowed intermediate transformations is load-bearing for the central claim that the coordination principle is independent of those two assumptions; without a self-contained check that no effective common cause is smuggled in by the transformation rules, the separation from no-signaling + independence remains unverified.

minor comments (2)

[Introduction / Definition of the principle] Definition of perfect coordination: the precise meaning of 'uniformly random output' and the operational criterion for 'agreement' in the N-party case should be stated formally at the outset (e.g., as a joint probability distribution) to prevent ambiguity when generalizing Reichenbach's principle.
[Derivation of inequalities] Noise-tolerant inequalities: confirm in the text that the derived inequalities reduce exactly to the standard Bell inequalities in the zero-noise limit and state whether they are tight for the coordination task.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment, and constructive feedback on our manuscript. We address the major comment below and will incorporate the suggested clarification in the revised version.

read point-by-point responses

Referee: [Counterexample construction] Counterexample construction (the operational probabilistic theory section): the explicit verification that the theory respects no-signaling and independence without introducing a hidden common cause via the allowed intermediate transformations is load-bearing for the central claim that the coordination principle is independent of those two assumptions; without a self-contained check that no effective common cause is smuggled in by the transformation rules, the separation from no-signaling + independence remains unverified.

Authors: We agree that an explicit, self-contained verification strengthens the central claim. In the revised manuscript we will add a dedicated paragraph (or short appendix subsection) that directly computes the joint output probabilities for the constructed theory under the allowed intermediate transformations. We will show that these probabilities factorize according to no-signaling and independence while still permitting perfect coordination without any shared common cause, thereby confirming that no effective common cause is introduced by the transformation rules. This addition will make the separation from no-signaling + independence fully transparent without altering the existing construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines the coordination principle explicitly as a new multipartite causal statement and then proves it holds in quantum networks via direct derivations from the causal structure of quantum theory. It separately constructs an explicit operational probabilistic theory obeying no-signaling and independence yet permitting coordination without common cause, thereby showing the principle is not entailed by those assumptions alone. No derivation step reduces by construction to a fitted parameter, self-citation, or renamed input; the counterexample and quantum proofs are independent of the principle itself. The argument chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central contribution is the introduction of the coordination principle itself together with proofs that quantum theory obeys it and that it is independent of no-signaling plus independence. No numerical free parameters are introduced or fitted. The principle functions as an ad-hoc-to-paper axiom whose validity is then checked against quantum mechanics and other theories.

axioms (2)

ad hoc to paper Coordination principle: perfect coordination among N parties on a uniformly random output is possible only if they share a common cause
This is the load-bearing statement introduced by the authors and then shown to hold in quantum theory.
domain assumption Standard no-signaling and independence assumptions in operational probabilistic theories
Invoked when constructing the counterexample theory that still obeys these but violates coordination.

pith-pipeline@v0.9.0 · 5517 in / 1491 out tokens · 84097 ms · 2026-05-08T18:14:11.904649+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation / J(x)=½(x+x⁻¹)−1 washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

P_{A_1,...,A_N} = (1/2)([0...0]+[1...1])

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Take all the largest sets of observed nodes that share a common cause and, for each of them, add a nonclassical latent node which has all the nodes in the set as children
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Find the largest sets of common descendants for all sets of theaddednonclassical common causes (note that as we are considering common descendants, the sets of common causes has size two or more). For each distinct maximal set of common descendants of size larger or equal than two, Si (where i is an index for each of these sets), take the set of all their...
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Proof of Theorems 6 and 1 (noise-robust version) Finally, we give the formal proof of Theorems 6 and 1 (noise-robust version). We write it only for the N-partite case, as they are completely analogous. Proof. By Lemma 9, any inequality that we derive for the set of probability distributions compatible with g∗ N is an inequality that must be satisfied by a...
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We write it only for the N-partite case, as they are completely analogous

Proof of Theorems 6 and 1 (noise-robust version) Finally, we give the formal proof of Theorems 6 and 1 (noise-robust version). We write it only for the N-partite case, as they are completely analogous. Proof. By Lemma 9, any inequality that we derive for the set of probability distributions compatible with g∗ N is an inequality that must be satisfied by a...
[48]

N vmin,N fmin,N 4 0.9417 0.9454 5 0.9617 0.9629 6 0.9730 0.9735 7 0.9800 0.9802 8 0.9846 0.9847 9 0.9878 0.9878 10 0.9901 0.9901 TABLE VI

Note that indeed only for N =4 the minimum value for positivity is smaller for the inequality provided in this appendix. N vmin,N fmin,N 4 0.9417 0.9454 5 0.9617 0.9629 6 0.9730 0.9735 7 0.9800 0.9802 8 0.9846 0.9847 9 0.9878 0.9878 10 0.9901 0.9901 TABLE VI. Numerical values (up to the fourth decimal) of vmin,N and fmin,N derived from inequality C1 for d...