Temporal nonlocality from indefinite causal orders

Laurie Letertre

REVIEW 3 major objections 5 minor 18 references

Temporal nonlocality from indefinite causal orders

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

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Pith's one-line read A protocol built on two entangled quantum switches can violate temporal locality, provided the switch is read as a genuine cyclic causal loop whose backward influence changes earlier measurement outcomes.

desk verdict Careful conditional analysis of how indefinite causal orders might violate Adlam temporal locality, but the central inference rests on an unargued stipulation of output-impacting retrocausality. read the letter →

arxiv 2501.12870 v1 pith:HDWHCIWF submitted 2025-01-22 quant-ph physics.hist-ph

Laurie Letertre This is my paper · ORCID

classification quant-phphysics.hist-ph

keywords temporalnonlocalityindefinitecausalorderquantumswitchprocessmatrixformalismBellinequalityretrocausalitynonseparabilitylocality

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

The paper tries to show that a particular quantum process with no definite causal order—the quantum switch—can be used to test, and in one reading violate, the principle that all influences on a measurement outcome pass through the state of the world just before the measurement. The route runs through a known protocol with two entangled quantum switches whose correlations violate a temporal Bell inequality. Standard temporal tests need extra realism assumptions; this protocol avoids those, but the paper argues the inference to temporal nonlocality is model-dependent: it requires interpreting the switch as a definite but cyclic causal structure whose backward influences change earlier measurement outcomes. If the argument holds, temporal nonlocality becomes a live consequence of indefinite causal orders, and may point to a common retrocausal explanation of ordinary Bell nonlocality.

What carries the argument

The load-bearing object is the quantum switch, a process in which a control qubit in a superposition determines whether operation A precedes B or B precedes A, leaving the order itself indefinite. The protocol couples two spacelikely separated quantum switches through a common massive control, so the final target state is entangled and violates a Bell inequality under a set of explicit assumptions; the relevant step is rejecting the classical-order assumption. The paper connects that rejection to a proof that any causal structure compatible with an indefinite-order process must be cyclic, and then interprets the cyclic structure as retrocausal influence between well-localized events. Temporal locality fails precisely when the backward influence reaches the earlier party's measurement outcome rather than its input.

What would settle it

Show that a flat-spacetime implementation of the two-switch protocol involves four mutually exclusive event candidates instead of two definite operation events—for example with the event-count discrimination test discussed in the paper—and the cyclic retrocausal reading is ruled out. Alternatively, exhibit a complete model of the switch whose only backward influences are input-impacting; such a model keeps the temporal-locality equalities intact and falsifies the inference.

Watch

Extended reading notes

Core claim

The central claim is that a violation of the temporal Bell inequality in the two-switch protocol can be read as a violation of temporal locality, provided the indefinite causal order is not dismissed as a superposition of worldlines in a fixed spacetime. The paper argues that if the process is embedded as a definite but cyclic relativistic causal structure among well-localized events, then the absence of any acyclic order forces retrocausal influences between the operations. Those influences violate temporal locality only when they are output-impacting—when they change the outcome of the earlier measurement. The paper states plainly that this last step is model-dependent, and that other embeddings either make temporal locality inapplicable (simulations and metaphysically indefinite spacetimes) or fail to violate it (input-impacting retrocausality). It also contends that because the same retrocausal mechanism could account for ordinary Bell nonlocality, the distinction between spatial and temporal nonlocality may not mark a difference in physical underpinning.

Load-bearing premise

The argument depends on treating the quantum switch as a real loop of influence between well-defined events, with backward effects that alter the earlier measurement outcome rather than only the input that leads to it; if either part fails, the Bell violation no longer demonstrates temporal nonlocality.

Editorial extensions

If this is right

If the two-switch protocol is realized and its Bell inequality violated, the classical-order assumption must be dropped, forcing either retrocausal influence or a violation of the free-evolution assumption.
Under the cyclic retrocausal reading, that violation is a genuine violation of temporal locality, without the realism commitments carried by Leggett-Garg or temporal CHSH tests.
The same retrocausal mechanism that would produce temporal nonlocality can also account for standard Bell nonlocality, so the two phenomena may share one physical explanation.
A modest broadening of the definition—from a state immediately prior to a measurement to a state merely prior to it—would let input-impacting retrocausal models also count as temporally nonlocal.

Reading between the lines

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

If the model-dependence is resolved in favor of the cyclic reading, temporal locality becomes empirically testable in flat spacetime without gravitational switches, making the temporal/spatial distinction operational rather than metaphysical.
The same logic could be applied to other causally nonseparable processes: any process whose only compatible causal structure is cyclic would be a candidate witness for temporal nonlocality, not just the quantum switch.
A decisive experiment could use the event-count discrimination between two-event and four-event embeddings; showing four events would reclassify apparent violations as inapplicability rather than violation of temporal locality.

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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

Laurie Letertre's paper is a rare thing in this area: a genuinely clear conditional analysis. It takes Zych et al.'s two-switch temporal Bell protocol and asks what it would take to infer a violation of Adlam's temporal locality from the violation of the inequality. The answer, it argues, is a specific reading of ICOs as definite cyclic relativistic causal structures with output-impacting retrocausal influences. The paper is admirably transparent about its own model-dependence: it flags assumptions A5 and A6, discusses three distinct spacetime embeddings, and explicitly admits that the test is not model-independent. The distinction between input-impacting and output-impacting retrocausality is a useful conceptual contribution that I haven't seen drawn sharply before.

The soft spot is exactly where the stress-test note puts it. The Vilasini-Renner cyclic model of the quantum switch, which the paper relies on for the cyclic reading, delivers feedback loops in which each party's quantum output feeds the other's input. That is input-impacting, and as the author correctly notes, it does not violate Eq. (1) because the retrocausal influence is already included in the pre-measurement state \lambda(t_a). The output-impacting variant is introduced only as 'conceivable' in footnote 8, with no worked example, no consistency proof with the process-matrix probabilities, and no precedent in the retrocausal literature. So the conclusion that the protocol can violate Adlam's temporal locality is not derived from the ICO; it is assumed by choosing one of two empirically equivalent retrocausal models. The suggested modification of Adlam's definition from 'immediately prior' to 'prior' would make input-impacting retrocausality count, but that is a change of the principle under test, and the author leaves it open.

The paper does not pretend otherwise. It explicitly says the inference is model-dependent, and it works carefully through what would need to be true for a model-independent test. That is honest, but it means the central claim is conditional on a premise that has no independent support. For a philosophy-of-physics paper this is acceptable; it is a contribution to the conceptual geography rather than a proof. For an experiment or a mathematical derivation it would be a problem.

This is a paper for philosophers and foundations types, not for experimentalists. It deserves a serious referee — it already has one, since it is forthcoming in BJPS — and I would send it to review without hesitation. I'd bring it to a reading group if the group's interest includes temporal nonlocality or ICOs, because the output-impacting assumption is worth arguing about. I'd cite it for the input/output-impacting distinction, while being careful not to cite it as evidence of temporal nonlocality.

Referee Report

3 major / 5 minor

Summary. The paper proposes a protocol based on two entangled quantum switches (Zych et al. 2019) as a test of Adlam's principle of temporal locality. It argues that a violation of the temporal Bell inequality derived by Zych et al., under assumptions A1–A4, forces a choice between rejecting A5 (free-evolution/control correlations) and rejecting A6 (classical order). The paper contends that rejecting A6 can be read as implying retrocausal influences and, if those influences are 'output-impacting,' a violation of Adlam's temporal locality, without the realism assumptions of Leggett-Garg or temporal CHSH tests. It then reviews three possible spacetime embeddings of the quantum switch and concludes that only a 'proper ICO with a cyclic structure' yields temporal nonlocality, making the test model-dependent. A variant of Adlam's definition that would count input-impacting retrocausality is proposed but left open.

Significance. If the conditional claim were established, the paper would provide a genuine conceptual bridge between indefinite causal orders and temporal nonlocality, and it would sharpen the distinction between violating and merely making inapplicable a notion like temporal locality. The paper's strengths are its transparency and its careful taxonomy: it explicitly states at the end of Section 4.3 that the inference is 'highly model-dependent,' and Section 4.4 gives three distinct embeddings and identifies which one would support the conclusion. It also correctly notes that the Vilasini–Renner cyclic model is input-impacting and therefore does not violate Eq. (1). The main limitation is that the load-bearing notion of output-impacting retrocausality is introduced only as 'conceivable' (footnote 8), with no worked example, consistency proof, or precedent, so the paper establishes a conditional possibility rather than a derivation of temporal nonlocality from ICOs.

major comments (3)

[Section 4.3 (paragraph beginning 'One sees, however...' and footnote 8)] The central inference to temporal nonlocality is not derived from the indefinite causal order; it is assumed by stipulating an 'output-impacting' retrocausal model. The paper states that the Vilasini–Renner cyclic causal model of the quantum switch produces feedback loops in which each party's quantum output affects the other's input, and that this input-impacting model does not violate Eq. (1) because the retrocausal influence is already contained in λ(t_a). The output-impacting variant is introduced only as 'conceivable and could be posited with the same experimental predictions,' with no worked example, no proof of consistency with process-matrix probabilities, and no precedent in the retrocausal literature. Consequently the abstract's claim that the paper 'shows how Adlam's principle of temporal locality can be violated' is stronger than what the body supports; the body supports only a conditional claim. This gap is load-bearing because without output-impacting retrocausality the protocol does not violate Eq. (1).
[Section 4.4, scenarios (1)–(2), and Section 4.3, point (6)] The argument from violation of assumption A6 to 'causal influences happening in both directions of time' is valid only for scenario (3), where the quantum switch is embedded as a definite cyclic relativistic causal structure among well-localized events. The paper itself acknowledges in Section 4.4 that in scenario (1) (superposition of worldlines in a fixed spacetime) and scenario (2) (metaphysically indefinite spacetime) the events are not well-defined, so the notion of temporal locality is inapplicable rather than violated. This means that a violation of the temporal Bell inequality does not by itself establish temporal nonlocality; it requires a prior, model-dependent commitment to scenario (3), including the claim that the operations A_i and B_i actually take place as well-localized measurements. The paper says the latter is 'merely assumed' in Section 4.3, but that assumption is precisely what is needed for Eq. (1) and Eq. (2) to be applicable.
[Section 4.3, final two paragraphs] The proposed revision of Adlam's definition from 'immediately prior' to 'prior' is explicitly left open. If this revision is adopted, input-impacting retrocausality would count as a violation of the modified principle, but then the protocol would not be testing Adlam's original principle, so the paper's stated target would shift. If the revision is not adopted, the claim that the protocol can violate Adlam's temporal locality remains hostage to the unsupported output-impacting retrocausal model. The paper needs to take a definite position and argue for it; as written, the central claim is ambiguous between testing Adlam's principle and testing a close variant.

minor comments (5)

[Section 4.1, Fig. 3 caption] The control state in the caption is written as '1/2 [|0_c⟩ + |1_c⟩]', which lacks the correct normalization and should be '1/√2 (|0_c⟩ + |1_c⟩)' as in Eq. (4).
[Section 4.2, assumption A6] The parenthetical for the case 'B_i ⪯ A_i' repeats '(A_i is in the past causal cone of B_i)' from the previous clause; it should say that B_i is in the past causal cone of A_i or that A_i is in the future causal cone of B_i.
[Section 4.4, first paragraph] The phrase 'A model-independent guarantee that at least [A6] is violated via (output-impacting) retrocausal influences' is unclear, since A6 is an assumption in the Zych et al. derivation rather than a statement about retrocausality; consider rephrasing to say that the guarantee concerns the spatiotemporal embedding of the switch as a cyclic structure with output-impacting influences.
[Section 1, last paragraph of the introduction] The phrase 'the corresponding temporal Bell inequality' in the description of the Zych setup is not defined until Section 4.2; a forward reference to the inequality's derivation would help the reader.
[Section 6, Conclusion] The conclusion says the protocol 'could imply temporal nonlocality modulo model-dependent assumptions,' which is accurate, but it should be aligned with the stronger wording in the abstract; the abstract should carry the same conditional qualification.

Circularity Check

1 steps flagged · score 6.0 of 10

The central inference posits, rather than derives, output-impacting retrocausality; known cyclic models of the switch are input-impacting and do not violate Adlam's Eq. (1).

self definitional [Section 4.3, paragraph discussing Vilasini and Renner's cyclic causal structure and the introduction of output-impacting retrocausality; compare Section 2's statement about Eq. (1).]
"Importantly, Vilasini and Renner ([unpublished])'s framework considers feedback loops such that, in the case of the QS, the quantum outputs of each party causally influence the inputs of the other party. This model would yield no violation of the principle of temporal locality for reasons explained in section 2. However, a theoretical model for which these retrocausal influences impact the measurement's outputs of earlier parties (denoted 'output-impacting' retrocausality from now on), instead of their input, is conceivable and could be posited with the same experimental predictions."

The paper's own Section 2 establishes that only retrocausal influences acting on the measurement outcome of an earlier party violate Eq. (1); retrocausal influences on inputs are screened off by the state lambda(t_a) and do not. The known cyclic model of Vilasini and Renner is explicitly input-impacting, so it yields no violation. The output-impacting variant is then introduced as 'conceivable' and, a few lines later, adopted: 'For the time being, we adopt the model in which retrocausal influences impact the measurement's outcomes of earlier parties, from which a violation of temporal locality can be inferred.' This is exactly the conclusion to be established: a model defined as producing retrocausal influence on outcomes is by construction a model violating Eq. (1).

full rationale

The paper is transparent about its model-dependence and does not hide the gap: it concedes that the known cyclic reading of the quantum switch is input-impacting and hence does not violate Adlam's temporal locality, and that the output-impacting variant is only 'conceivable and could be posited with the same experimental predictions.' Nevertheless, the advertised result—that indefinite causal orders can yield a violation of Adlam's principle—is not derived from the ICO structure itself. It is obtained by adopting a retrocausal model whose defining feature is precisely the violation of Eq. (1). Since the paper defines temporal nonlocality in Section 2 as retrocausal influence on measurement outcomes, the inference from 'output-impacting retrocausality' to 'temporal nonlocality' is true by definition, not by derivation. The experiment and the ICO framework do not force this model; they are compatible with the input-impacting model that the paper agrees yields no violation. This is a partial circularity in the central claim, although the paper's explicit caveats and the external Zych et al. inequality derivation prevent it from being a fully concealed or score-8/10 circularity. The suggested modification of Adlam's definition from 'immediately prior' to 'prior' is another route to the same conclusion, but the paper leaves that question open and does not rely on it for the main argument.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The paper's central claim is conditional on several non-trivial assumptions from prior literature (process matrix formalism, Zych et al.'s theorem, Vilasini and Renner's cyclic interpretation) and on two explicitly model-dependent choices: the cyclic spacetime embedding and the output-impacting nature of the retrocausal influences. There are no fitted parameters or experimental data in the paper; the ledger reflects the conceptual assumptions that the argument requires.

assumptions (6)

domain assumption Adlam's definition of temporal locality: all influences on a measurement outcome are mediated by the state of the world immediately prior to the measurement (Eqs. 1 and 2 in Section 2).
Adopted as the principle to be tested; the paper does not derive it independently but uses it as the target of the protocol's analysis.
domain assumption The process matrix formalism describes all valid quantum processes, including causally nonseparable ones such as the quantum switch.
The paper relies on this formalism to define indefinite causal orders and to present the quantum switch in Section 3.
domain assumption Zych et al.'s theorem that the two-switch setup, under assumptions A1-A6, produces an entangled final state that violates a Bell inequality.
This external result is the starting point of the paper's argument; it is cited and not re-derived in Section 4.2.
domain assumption Vilasini and Renner's result that an indefinite causal order is equivalent to a definite but cyclic information-theoretic causal structure.
Used in Section 4.3 to support the retrocausal reading of the quantum switch; the reference is marked unpublished.
ad hoc to paper The cyclic information-theoretic causal structure translates into a cyclic relativistic causal structure with well-localized spacetime events.
This is one of three possible embeddings of the quantum switch discussed in Section 4.4; the paper argues only this one yields temporal nonlocality, while admitting it is model-dependent.
ad hoc to paper The retrocausal influences are output-impacting, affecting earlier measurement outcomes rather than only inputs.
Introduced in Section 4.3 as a conceivable model that produces a violation of Adlam's temporal locality; no independent experimental handle is provided.

invented entities (1)

Output-impacting retrocausality
purpose: A hypothesized type of retrocausal influence that affects the measurement outcomes of earlier parties, thereby producing a violation of Adlam's temporal locality.
The paper explicitly says such a model 'is conceivable and could be posited with the same experimental predictions' (Section 4.3), and provides no independent falsifiable way to distinguish it from input-impacting retrocausality.

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Cite this review

Pith. "Pith review of Temporal nonlocality from indefinite causal orders." pith.science (2026). https://pith.science/paper/HDWHCIWF

@misc{pith2026250112870,
  author       = {Pith},
  title        = {Pith review of: Temporal nonlocality from indefinite causal orders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HDWHCIWF}},
  note         = {Machine review of arXiv:2501.12870}
}

read the original abstract

A temporal counterpart to Bell nonlocality would intuitively refer to the presence of non-classical correlations between timelike-separated events. The hypothesis of temporal nonlocality has received recent support in the literature, and its existence would likely influence the future development of physical theories. This paper shows how Adlam's principle of temporal locality can be violated within a protocol involving indefinite causal orders. While the derivations of Leggett-Garg inequalities or the temporal CHSH inequality are said to involve problematic assumptions preventing a targeted probing of a well-defined notion of temporal nonlocality, the present test is free from such worries. However, it is shown that the test, in its current formulation, fails to be fully model-independent. We provide several considerations regarding the physicality of ICOs that could help alleviate this drawback. In the present work, a specific physical interpretation of ICOs in terms of retrocausal influences would explain the presence of temporally nonlocal correlations. It is argued that, as the physical underpinnings of temporal nonlocality might also account for standard Bell nonlocality, focusing on the former as a consequence of ICOs might support under-explored strategies to make sense of the latter.

Figures

Figures reproduced from arXiv: 2501.12870 by the authors.

**Figure 2.** (a) A process matrix W representing a (quantum) process as a map applying n local (quantum) operations (denoted Aj with j going from 1 to n) over a global operation. The labels I and F represent the input and output of the process, respectively, while Ij and Fj represent the input and output of the local operation Aj , respectively. (b) Process describing one and the same system (initially described by the quantum… view at source ↗

**Figure 3.** Schematic representation of a quantum switch. A preparation procedure prepares the target system in state |ψt0 ⟩ and the control system in state |ψc⟩ = 1 2 [|0c⟩ + |1c⟩]. The target system undergoes an operation in party A, according to the map UA, and another operation in party B, according to the map UB. The order between these operations is indefinite. These occurrences of the operations are denoted E1 and E2, … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

[1] [1]

[2018]: ‘Spooky Action at a Temporal Distance’, Entropy, 20(1), pp

Adlam, E. [2018]: ‘Spooky Action at a Temporal Distance’, Entropy, 20(1), pp

work page 2018

[2] [9]

and Eisenberg, H

Megidish, E., Halevy, A., Shacham, T., Dvir, T., Dovrat, L. and Eisenberg, H. [2013]: ‘Entan- glement Swapping between Photons that Have Never Coexisted’, Physical Review Letters, 110(21), pp. 210403. Mjelva, J. K. [2024]: ‘Delayed-choice Entanglement Swapping Experiments: No Evidence for Timelike Entanglement’, Studies in History and Philosophy of Scienc...

work page 2013

[3] [28]

[2023]: ‘Is There Causation in Fundamental Physics? New Insights from Process Matrices and Quantum Causal Modelling’, Synthese, 201(5), pp

26 Adlam, E. [2023]: ‘Is There Causation in Fundamental Physics? New Insights from Process Matrices and Quantum Causal Modelling’, Synthese, 201(5), pp

work page 2023

[4] [37]

[2023]: ‘Temporal Global Correlations in Time-Symmetric Collapse Models’, Foundations of Physics, 53(3), pp

Rodr´ıguez-Warnier, P. [2023]: ‘Temporal Global Correlations in Time-Symmetric Collapse Models’, Foundations of Physics, 53(3), pp

work page 2023

[5] [41]

[2019]: ‘Fundamental?’, in A

Adlam, E. [2019]: ‘Fundamental?’, in A. Aguirre, B. Foster and Z. Merali ( eds), What is Fundamental?, Springer International Publishing, Switzerland, pp. 5–12. Adlam, E. [2022]: ‘Laws of Nature as Constraints’, Foundations of Physics, 52(1), pp

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Special State

Rubino, G., Rozema, L. A., Feix, A., Ara ´ujo, M., Zeuner, J. M., Procopio, L. M., Brukner, ˇC. and Walther, P. [2017]: ‘Experimental Verification of an Indefinite Causal Order’, Science Advances, 3(3), pp. e1602589. Schulman, L. S. [1997]: Time’s Arrows and Quantum Measurement , Cambridge University Press, Cambridge, U.K. Schulman, L. S. [2012]: ‘Experim...

arXiv doi:10.48550/arxiv.2203.11245 2017

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Athalye, V ., Roy, S. S. and Mahesh, T. S. [2011]: ‘Investigation of the Leggett-Garg Inequality for Precessing Nuclear Spins’, Phys. Rev. Lett., 107, pp. 130402. Baas, A. and Le Bihan, B. [2023]: ‘What Does the World Look like According to Superdeter- minism?’, The British Journal for the Philosophy of Science, 74(3), pp. 555–72. Barrett, J., Lorenz, R. ...

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