Reproducing EPR correlations without superluminal signalling: backward conditional probabilities and Statistical Independence

Simon Friederich

REVIEW 3 major objections 5 minor 32 references

Reproducing EPR correlations without superluminal signalling: backward conditional probabilities and Statistical Independence

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper constructs probability models with backward-in-time conditional probabilities that preserve Statistical Independence and forbid superluminal signalling while reproducing Bell-state and GHZ correlations.

desk verdict A clean retrocausal construction that reproduces Bell correlations only as a mixture, not for a fixed prepared state, but still worth a serious referee. read the letter →

arxiv 2501.11064 v3 pith:KG37N5U7 submitted 2025-01-19 quant-ph physics.hist-ph

Simon Friederich This is my paper · ORCID

classification quant-phphysics.hist-ph

keywords Bell'stheoremEPRcorrelationsStatisticalIndependenceLocalCausalitybackward-in-timeconditionalprobabilitiesnosuperluminalsignallingGHZhiddenvariables

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Bell's theorem says no model with both Local Causality and Statistical Independence can reproduce the correlations of entangled quantum states. This paper tries to show that the way out is to keep Statistical Independence and drop Local Causality, using backward-in-time conditional probabilities as the mechanism. It constructs explicit probability models of this kind that reproduce the Bell-state correlations and, by the same recipe with three measurement wings, the GHZ predictions. The payoff, if the construction is accepted, is that a single-world realist account can match quantum correlations and still forbid superluminal signalling.

What carries the argument

The carrying object is a collider hidden variable $\lambda$ in a causal diagram: all arrows point into $\lambda$ from the measurement settings and outcomes. Equation (9) is the defining factorization, and Eq. (10) fixes the backward conditional $P(\lambda_i\mid a_1,a_2,\alpha_1,\alpha_2)$ to be proportional to the target quantum probability. Statistical Independence then functions as a fine-tuning condition: it demands that, after summing over outcomes, $\lambda$'s marginal distribution does not depend on $\alpha_1,\alpha_2$, which prevents distant settings from influencing local outcomes once $\lambda$ is conditioned on. Conditioning on a specific $\lambda_i$ recovers the quantum joint probabilities by Bayes' rule. For the GHZ case, Eq. (16) plays the same role, with $\lambda_0$ marking the GHZ-allowed outcome triples.

What would settle it

In a run prepared in the GHZ state $\lambda_0$, the model assigns zero probability to any outcome triple with $a_1 a_2 a_3 \neq (-1)^{\alpha_1+\alpha_2+\alpha_3}$; a single observed occurrence of such a triple would falsify the model.

Watch

Extended reading notes

Core claim

The paper's central claim is that a hidden variable $\lambda$ can serve as a collider receiving arrows from both measurement settings and outcomes, with the joint distribution factorizing as $P(a_1,a_2,\lambda\mid\alpha_1,\alpha_2)=P(a_1\mid\alpha_1)P(a_2\mid\alpha_2)P(\lambda\mid a_1,a_2,\alpha_1,\alpha_2)$. Statistical Independence is imposed by requiring the summed marginal $P(\lambda\mid\alpha_1,\alpha_2)$ to be independent of the settings; this fine-tuning also cancels the signaling paths that would otherwise run through the collider. Choosing $P(\lambda_i\mid a_1,a_2,\alpha_1,\alpha_2)=N P_i(a_1,a_2\mid\alpha_1,\alpha_2)$, with $P_i$ the quantum probability for Bell state $\lambda_i$, and conditioning on $\lambda_i$, yields exactly the quantum correlations. The same construction with a third wing reproduces the GHZ correlations by making $\lambda=\lambda_0$ exactly when the outcome triple satisfies the GHZ rule. Thus the model violates Local Causality rather than Statistical Independence and is explicitly non-signalling.

Load-bearing premise

The load-bearing premise is that a hidden variable $\lambda$ can stand for the prepared quantum state while all of its conditional probabilities depend on later settings and outcomes, so stipulating $P(\lambda_i\mid a_1,a_2,\alpha_1,\alpha_2) = N P_i(a_1,a_2\mid\alpha_1,\alpha_2)$ is a legitimate model-building move rather than a restatement of the correlations one wants to explain.

Editorial extensions

If this is right

Bell's theorem is bypassed by dropping Local Causality while keeping Statistical Independence.
No superluminal signalling is possible: for each prepared state $\lambda_i$, each local outcome distribution is $1/2$ regardless of the distant setting.
The GHZ correlations are obtained without attributing pre-existing values to all possible spin components.
The model is necessarily fine-tuned, violating Faithfulness, in line with known results about Bell-compatible models without signaling.
Because it can also reproduce non-quantum no-signaling correlations such as Popescu-Rohrlich boxes, the construction gives no account of the Tsirelson bound.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

I infer that the explanatory weight is carried entirely by Eq. (10): because $\lambda$'s conditional distribution is set equal to the quantum probability, the construction shows which assumption must fall, namely Local Causality, rather than why the correlations have the quantum form.
I infer that the same method would reproduce any no-signaling distribution, so additional physical constraints on $\lambda$'s dynamics are needed if the approach is to explain why Nature respects the Tsirelson bound.
I infer that extending the model to non-maximally entangled states requires separating a forward-going preparation variable from the backward-collider $\lambda$, converting the pure backward-arrow diagram into a hybrid two-time structure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper is worth a serious look if you work on Bell’s theorem and retrocausality. What is new is the specific combination: a collider DAG with backward arrows from settings and outcomes into λ, with Statistical Independence imposed as a fine-tuning condition rather than violated. The Section IV algebra is internally consistent. Equation (9) is a valid factorization, Eq. (10) with N=1 satisfies normalization and SI, and Eq. (11) correctly recovers the quantum probabilities. The no-signaling check is also correct: with P(a_i|α_i)=1/2, changing α2 leaves a1’s distribution unchanged after conditioning on λ. This is a genuine, if modest, addition to the literature, and it is clearly positioned relative to Wood and Spekkens and Price and Wharton.

The soft spot is the one the author half-concedes in Section VI. The model treats λ as a posterior label over a uniform ensemble of all four Bell states, not as a prepared state. If you actually prepare |ψ1>, so P(λ1)=1, then Eq. (10) sets P(λ1|a1,a2,α) = N P1(a1,a2|α), which cannot equal 1 for all outcomes. The GHZ case is worse: Eq. (16) forces P(λ0|allowed)=2P(λ0), so with P(λ0)=1 the conditional probability exceeds 1. The construction therefore reproduces EPR correlations only for a mixture, not for a fixed prepared state. The author’s suggested separate preparation variable P is the right fix, but as it stands the advertised claim that the model accounts for EPR correlations is overstated.

This does not sink the paper. It is an honest, clearly written consistency proof for a fine-tuned retrocausal probability model, and the author flags the main limitation himself. What is missing is a derivation rather than a stipulation: Eq. (10) essentially inserts the target quantum probabilities by hand, so the “reproduction” is by construction. The paper is a useful starting point for thinking about what a retrocausal single-world model would need, not a working model.

I would give it a conditional acceptance if I were refereeing: the technical core is sound, the novelty is real, and the limitations are acknowledged. It deserves a serious referee, and should be cited in discussions of measurement independence and retrocausality.

Referee Report

3 major / 5 minor

Summary. The paper proposes a hidden-variable model with backward-in-time conditional probabilities that preserves Statistical Independence (SI) as a fine-tuning condition, with the aim of reproducing the EPR/Bell and GHZ correlations while forbidding superluminal signalling. The central construction is Eq. (9), a factorization in which the hidden variable λ acts as a collider depending on settings and outcomes; Eq. (10), which sets the posterior P(λ_i|a1,a2,α1,α2) proportional to the target quantum probability; and Eq. (11), which inverts this to recover the quantum correlations when conditioning on λ_i. The paper also discusses why the model violates Local Causality but satisfies no-signaling, and gives a GHZ analog. The formal algebra is transparent and internally consistent as a consistency exercise, but the advertised physical account is undermined by the absence of a preparation variable: the construction fails for a fixed prepared state and, in the GHZ case, produces an impossible conditional probability.

Significance. If the construction worked as advertised, it would be a notable conceptual contribution: a backward-arrow, SI-preserving model that reproduces Bell-state and GHZ correlations while avoiding superluminal signaling. The equations are simple enough to be checked by hand, and the paper is explicit about several limitations, including the Tsirelson bound and the need for a preparation variable. However, the claimed significance is currently overstated: because the hidden-variable posterior is stipulated to be proportional to the desired quantum probability, the reproduction is built into the model, and the lack of a preparation variable causes concrete normalization failures for prepared states. The paper is therefore best read as a formal consistency exercise rather than a fully specified account of EPR correlations.

major comments (3)

[Section IV, Eqs. (10)–(11)] The claimed reproduction of the EPR correlations holds only for an equiprobable mixture over the four λ_i, not for a run in which a definite Bell state is prepared. If P(λ_1)=1, then Bayes's theorem gives P(λ_1|a1,a2,α1,α2)=1 for every possible outcome, whereas Eq. (10) with N=4P(λ_1)=4 sets this posterior to 4P_1(a1,a2|α1,α2)=1±a1a2 cos(α1−α2), which takes values between 0 and 2. The author's own Section VI concedes that a separate preparation variable P must be introduced and that this is needed for non-maximally entangled states; until that is done, the central claim of accounting for a prepared EPR pair is not established.
[Section V, Eq. (16)] The GHZ extension suffers from a normalization failure when the GHZ state is the prepared state. With P(a_i|α_i)=1/2, the coefficient in Eq. (16) is N=8P(λ_0). For any GHZ-allowed triple, P_GHZ=1/4, so Eq. (16) gives P(λ_0|a1,a2,a3,α1,α2,α3)=2P(λ_0). If λ_0 is the only hidden-variable value and the state is prepared, P(λ_0)=1 and this conditional probability equals 2, which is impossible. The model is therefore internally inconsistent for the fixed-state GHZ scenario; consistency requires P(λ_0)≤1/2, i.e. an ensemble interpretation rather than a single prepared state.
[Section IV, Eq. (10)] The key move is a stipulation, not a derivation: the hidden-variable posterior P(λ_i|a1,a2,α1,α2) is set equal to N times the target quantum probability P_i(a1,a2|α1,α2), and Eq. (11) then recovers that same probability by Bayesian inversion. This is a valid consistency check, but it means the model's ability to reproduce the EPR correlations is built into its definition rather than following from the backward-arrow structure plus SI. The paper should state this more cautiously; as it stands, the abstract's claim that the model 'accounts for' EPR correlations overstates the logical status of the construction.

minor comments (5)

[Section V, Eq. (15)] The argument list in P|ψ⟩GHZ (a1,a2,a2|α1,α2,α3) contains a2 twice; the third outcome should be a3.
[References] The reference list contains two entries both numbered [9] (Evans 2015 and Evans 2018); the citations should be renumbered consistently.
[Various] There are typographical errors, e.g. 'meausrement' in Section II and 'simultaneouslky' in the concluding paragraph.
[Section IV, Eq. (11)] The text says 'fix N so that P(λ1)/[P(a1|α1)P(a2|α2)] = N'; it would be clearer to state explicitly that SI and the normalization of Eq. (10) force P(λ_i)=1/4 and hence N=1 for the Bell-state case, so the reader can see the consistency conditions.
[Section IV, Eq. (10)] The notation P1;2;3;4 is ambiguous: it should be clear that the subscript ranges over the four Bell states and that the corresponding λ_i are equipped with equal priors.

Circularity Check

3 steps flagged · score 8.0 of 10

EPR and GHZ reproductions are stipulated in Eqs. (10) and (16) as the λ-posterior, so the recovered correlations equal the input by construction; the independent no-signaling check does not make the derivation non-circular.

self definitional [Section IV, Eqs. (10)-(11)]
"To reproduce the quantum correlations, we further set P (λ_{1;2;3;4}|a_1,a_2,α_1,α_2) = N P_{1;2;3;4}(a_1,a_2|α_1,α_2) (10) where P_{1;2;3;4} is the desired quantum probability ... Under these assignments, one recovers exactly the quantum predictions by conditioning on a particular λ_i."

The target quantum distribution P_i is inserted directly on the right-hand side of Eq. (10) as the defining conditional probability of λ_i, and Eq. (11) simply applies Bayes's rule to the factorization (9) to bring that same P_i back out on the left. The 'recovered' EPR correlations are therefore not derived from the model; they are stipulated into the model by construction. What remains non-trivial is only the consistency check that this stipulation can coexist with SI and no-signaling, not the reproduction of the correlations themselves.
self definitional [Section V, Eq. (16)]
"To reproduce these probabilities using a model of the form in Fig. 3 (but with an extra wing for α_3,a_3), we define P (λ_0|a_1,a_2,a_3,α_1,α_2,α_3) = N P_{|ψ⟩GHZ}(a_1,a_2,a_3|α_1,α_2,α_3), (16) where λ_0 denotes the hidden variable label for the GHZ state."

The GHZ construction repeats the EPR move: the GHZ quantum probabilities are used to define the conditional distribution over λ_0, and the paper then concludes that non-GHZ-allowed triples are simply not observed under λ_0. The GHZ predictions are inputs to the model, not outputs derived from backward arrows or SI. No independent derivation of the GHZ correlations is given; the defining equation already contains the desired result.

1 more flagged steps

other [Section VI (Summary and Outlook), second stated limitation]
"Second, the backward-arrow approach treats λ purely as an effect of later measurement variables and outcomes. In practice, λ labels the prepared (Bell) state, which plausibly depends at least partially on earlier experimental actions. Consequently, it seems physically implausible to have only backward arrows into λ. More complete models should separate the collider variable λ from a preparation variable P."

This is the paper's own admission that λ has been defined inconsistently: in Eq. (10) λ_i is said to label which Bell state is prepared, yet all defining arrows into λ come from future settings and outcomes. For a fixed prepared state, P(λ_1)=1, and consistency would require P(λ_1|a_1,a_2,α_1,α_2)=1 for every outcome pair, which Eq. (10) cannot supply because P_1(a_1,a_2|α_1,α_2) is non-constant. The advertised reproduction of EPR correlations therefore holds only for the uniform λ_i mixture forced by normalization, not for a run preparing one Bell state. This acknowledged gap reinforces that the 'account' is a consistency exercise on a stipulated posterior rather than a derivation for prepared states.

full rationale

The central derivation chain in Sections IV and V is circular in the specific sense identified above. Eq. (10) defines the hidden-variable posterior by P(λ_i|a_1,a_2,α_1,α_2)=N P_i(a_1,a_2|α_1,α_2), where P_i is explicitly called 'the desired quantum probability.' Eq. (11) then applies Bayes's rule to the factorization (9) and the same Eq. (10) to display P_i(a_1,a_2|α_1,α_2) as P(a_1,a_2|α_1,α_2,λ_i). The reproduced EPR correlations are therefore the input distribution itself, moved from the posterior over λ to the conditional on λ by algebra; no independent derivation of Bell correlations is supplied. The GHZ construction (Eq. (16)) repeats the same move for P_GHZ. The paper's independent contribution is the demonstration that, once those probabilities are stipulated, imposing SI with P(a_i|α_i)=1/2 blocks superluminal signalling; that part is non-trivial and not circular, which prevents a maximal score. But the headline claim that the model 'accounts for' EPR/GHZ predictions is undercut by the stipulation. The paper's Section VI further concedes that λ cannot plausibly be both the prepared state and a pure future effect; with a fixed pure preparation, P(λ_1)=1, Eqs. (9)-(10) cannot hold because Eq. (10) would make P(λ_1|a_1,a_2,α_1,α_2) proportional to the non-constant P_1, while P(λ_1)=1 forces this conditional to be 1 for all outcomes. Thus the reproduction works only for the uniform λ_i mixture, not for a definite prepared Bell state. No load-bearing self-citation chain or imported uniqueness theorem is involved; the circularity is internal to the defining equations.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The model introduces no new particles or forces. It relies on one hand-chosen probability function (the hidden-variable posterior) and several domain assumptions about causal structure and no-signaling. The central construction, Eq. (10), is the main ad hoc element.

assumptions (6)

standard math Standard probability calculus, including Bayes' theorem, is applicable to the backward-conditioned model.
Used in Eqs. (9)-(11) to factorize the joint distribution and invert conditional probabilities.
domain assumption The joint distribution can be written as P(a1,a2,λ|α1,α2)=P(a1|α1)P(a2|α2)P(λ|a1,a2,α1,α2) (Eq. 9).
This factorized form is the defining structure of the model and encodes the absence of direct arrows between a1 and a2.
domain assumption Statistical Independence P(λ|α1,α2)=P(λ) is imposed as a fine-tuning condition.
Central assumption needed to block superluminal signaling and avoid interventionist retrocausality.
domain assumption Each local outcome is fair: P(ai|αi)=1/2.
Required to match quantum marginals for maximally entangled states and to satisfy SI with the chosen P(λ|...).
ad hoc to paper The hidden-variable posterior is chosen as P(λ_i|a1,a2,α1,α2)=N P_i(a1,a2|α1,α2), where P_i is the target quantum probability.
This choice is the entire mechanism that makes the model reproduce quantum correlations; it is not derived from deeper principles.
domain assumption No-superluminal-signaling is defined by P(a1|α1,α2,λ)=P(a1|α1,α2',λ) for all λ.
Standard hidden-variable no-signaling condition, checked in Section IV.

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Cite this review

Pith. "Pith review of Reproducing EPR correlations without superluminal signalling: backward conditional probabilities and Statistical Independence." pith.science (2026). https://pith.science/paper/KG37N5U7

@misc{pith2026250111064,
  author       = {Pith},
  title        = {Pith review of: Reproducing EPR correlations without superluminal signalling: backward conditional probabilities and Statistical Independence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KG37N5U7}},
  note         = {Machine review of arXiv:2501.11064}
}

read the original abstract

Bell's theorem states that no model that respects Local Causality and Statistical Independence can account for the correlations predicted by quantum mechanics via entangled states. This paper proposes a new approach, using backward-in-time conditional probabilities, which relaxes conventional assumptions of temporal ordering while preserving Statistical Independence as a "fine-tuning condition. It is shown how such models can account for EPR/Bell correlations and, analogously, the GHZ predictions while nevertheless forbidding superluminal signalling.

Figures

Figures reproduced from arXiv: 2501.11064 by the authors.

**Figure 3.** DAG with backward arrows, second type Various models corresponding to [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗

**Figure 3.** By imposing Statistical Independence on these “backward-arrow” models, one can avoid superluminal signaling while still violating Local Causality and thus recovering a Bell-CHSH inequality violation. The models proposed here two notable limitations. First, they do not address why quantum correlations respect the Tsirelson bound. Although no quantum prediction can exceed 2√ 2 on the left-hand side of Eq. (4), one c… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Bell, J. S. [1964], Physics, 1:195-200, repr. in: J. S. Bell, Speakable and Unspeakable in Quantum Mechanics , 2nd ed., 2004, pp. 14-21

work page 1964

[2] [2]

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work page 1988

[3] [3]

F., Horne, M

Clauser, J. F., Horne, M. A., Shimony, A. and Holt, R. A. [1969], Phys. Rev. Lett. , 23, pp. 880-884

work page 1969

[4] [4]

Drummond, P. D. [2021], Phys. Rev. Res. , 3, p. 013240

work page 2021

[5] [5]

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