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A classical gravitational interaction can entangle two quantum masses by exchanging virtual matter particles, so at Planck-scale masses entanglement alone no longer proves gravity is quantum.

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2026-08-04 08:38 UTC pith:UTUM7ZDD

load-bearing objection A plausible but not airtight loophole in the BMV entanglement witness: the virtual-matter calculation is serious, but two unaddressed technical points keep the central claim conditional. the 3 major comments →

arxiv 2510.19714 v3 pith:UTUM7ZDD submitted 2025-10-22 quant-ph

Classical theories of gravity produce entanglement

Joseph Aziz , Richard Howl This is my paper
classification quant-ph
keywords classical gravityquantum entanglementvirtual particlesquantum communicationgravitationally induced entanglementsemiclassical gravityLOCCquantum field theory
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper argues that the standard no-go result—classical gravity can only act through local operations and classical communication, hence cannot entangle—misses a process available when matter is treated as a quantum field. Even with a purely classical gravitational field that is identical in every branch of the superposition, the interaction Hamiltonian contains vertices that allow virtual matter particles to travel between the two masses. At fourth order in perturbation theory these virtual-matter exchanges yield a branch-dependent amplitude that creates spin entanglement. The effect scales differently from the quantum-gravity phase, so for low masses near-future experiments are unaffected, but for Planck-scale masses entanglement can no longer be treated as unambiguous evidence of quantum gravity.

Core claim

In a version of the two-superposed-masses experiment, with each mass described by an N-particle quantum-field-theory wavepacket and gravity by a classical gravitational potential that is the same in every branch, the leading entangling process is not the classical analogue of the graviton diagram but a fourth-order diagram in which the two objects exchange two virtual matter particles. The amplitude for branch (i,j) is β^(4)_ij ≈ (6/25 i G^2 m^2 M^3 R t / (ħ^3 d_ij))^2. Because the virtual matter propagator involves |x−y|, the distance between the spherical wavepackets depends on the branch, so β differs across branches even though the classical potential does not; this branch-dependence is

What carries the argument

The central object is the fourth-order perturbative amplitude for virtual-matter exchange under a classical gravitational interaction, built from: (i) N-particle non-relativistic wavepacket states localized in spheres of radius R; (ii) a classical potential sourced by the expectation value of the matter energy-momentum tensor, hence identical in all superposition branches; and (iii) Feynman propagators of a massive complex scalar field connecting the two objects. The key identity is β^(4)_ij proportional to [∫ dx dy Φ(x)Φ(y) θ_{1i}(x)θ_{2j}(y)/|x−y|]^2, whose d_ij^{-2} branch dependence generates the entanglement. It does this by letting virtual matter particles traverse different distances

Load-bearing premise

The paper's spin-entanglement witness computes amplitudes from states that keep the particle number N fixed in each object; if the states |N±k⟩|N∓k⟩ that the same interaction creates are not truly negligible or are not traced out, they could carry which-branch information and spoil the predicted spin entanglement.

What would settle it

Recompute the final state without truncating to N-particle sectors, or measure spin negativity in a Planck-mass configuration where the predicted classical amplitude is about 0.1: if the number-transfer components destroy the spin witness, or no entanglement appears despite the predicted amplitude, the central claim is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For the near-term low-mass parameter regime (M≈10^-14 kg, t≈2 s), the classical-gravity entanglement amplitude is far below the quantum-gravity phase, so observing entanglement there retains its status as evidence for quantum effects in gravity.
  • At masses approaching the Planck mass and above, the classical amplitude reaches order 0.1 even at short times, so entanglement observation in that regime would not discriminate classical from quantum gravity.
  • The two amplitudes scale differently with mass, radius, and distance, so scanning these parameters can separate the classical-gravity contribution from the quantum-gravity contribution in a single experiment.
  • The LOCC-based theorems are not generic: classical gravity models with quantum-field-theoretic matter are not restricted to classical communication, because the matter sector itself provides a quantum channel.
  • In stochastic classical-gravity models with fundamental decoherence, the virtual-matter entanglement can still dominate when its generation rate exceeds the gravitational decoherence rate, so entanglement is not unique to semiclassical sourcing.

Where Pith is reading between the lines

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

  • The same fourth-order virtual-matter mechanism should operate for any classical field coupled to quantum matter—for instance a classical electromagnetic potential—so the ability of a classical field to entangle is not peculiar to gravity; tabletop experiments may need to control classical electromagnetic backgrounds at the same perturbative order.
  • The material dependence of the classical amplitude (it scales with the constituent particle mass and radius) versus the quantum-gravity phase's independence of those parameters offers a direct experimental discriminator: changing the material at fixed total mass should change the classical contribution but not the quantum one.
  • The calculation truncates to fixed-N sectors; including the |N±k⟩|N∓k⟩ number-transfer channels could either introduce decoherence that suppresses the spin entanglement or open additional witnesses, so a full calculation is needed before Planck-scale experiments are designed.
  • If the effect is real, the experimental question shifts from 'does entanglement occur?' to 'does the entanglement amplitude scale as predicted?', making classical-sourcing models directly testable through the predicted dependence on mass, radius, time, and separation.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper challenges the consensus that a classical gravitational interaction cannot entangle two quantum matter systems. Its central claim is that when matter is treated at QFT level, a classical (unquantized) gravitational interaction permits quantum communication through virtual matter, and this can generate entanglement in a Feynman-type experiment. The authors compute the fourth-order Dyson amplitude for two N00N-state mass distributions under semiclassical Einstein gravity and obtain β^(4)_RL ≈ [(6/25) G² m² M³ R t/(ℏ³ d_RL)]², concluding that for Planck-scale masses the observation of entanglement alone cannot be taken as evidence of quantum gravity. The paper includes extensive appendices on relativistic, momentum-space, stochastic, and boundary treatments.

Significance. If the technical concerns below are resolved, this is an important result. It provides a concrete, quantitative mechanism by which a local classical gravity model — semiclassical Einstein gravity — can produce gravitationally mediated entanglement, thereby sharpening the conditions under which entanglement-based quantum-gravity witnesses are valid. The strength of the paper is its explicit QFT calculation: the Dyson-series derivation is detailed and cross-checked in momentum space (Appendix B.3), a quantum-gravity counterpart is computed (Appendix B.6), and the final amplitude gives a falsifiable mass/time/distance scaling. The main weaknesses are the inconsistent sign convention in the central amplitude and an unquantified postselection assumption connecting the computed amplitude to the observable spin entanglement.

major comments (3)
  1. [Section 5.2, Eqs. (79)–(80), Fig. 6] There is an internal inconsistency in the definition of ϑ. Eq. (79) states β^(4)_ij = ( (6/25) i G² m² M³ R t / (ℏ³ d_ij) )². Since i² = −1, this is a negative real number, not a pure phase or positive parameter. Eq. (80) then sets β^(4)_RL =: ϑ and writes √ϑ = (6/25) G² m² M³ R t/(ℏ³ d_RL), treating ϑ as a positive real number. This is not a harmless notational slip: the sign and imaginary unit determine whether the branch amplitude α_RL is 1 + ϑ, 1 − ϑ, or 1 + i times a real quantity, which affects the entanglement witness and the ϑ/φ comparison in Fig. 6. The authors should either define ϑ := |β^(4)_RL| and adjust the text, or restore the i consistently and explain how a negative real branch amplitude enters the final state.
  2. [Footnote 39; Section 5.2, Eqs. (17)–(18)] The conversion of the computed amplitude β into an observable spin-entanglement witness assumes that the post-measurement state is the pure state (18) with support only on |N⟩_{1i}|N⟩_{2j}. However, the same interaction Hamiltonian (43) at fourth order also produces number-transfer sectors |N+k⟩_{1i}|N−k⟩_{2j} and |N−k⟩_{1i}|N+k⟩_{2j}. Footnote 39 dismisses these as 'not seen in the experiment' because of different Stern-Gerlach deflections, but no amplitude, overlap, or postselection error estimate is supplied. If these components are not cleanly separated from the |N,N⟩ port, or if they are traced out, the reduced spin state is not generically the pure entangled state used in the witness, and the claimed entanglement can disappear. This is load-bearing: it is the only step that turns the formal fourth-order amplitude into a concrete experimental prediction. Please quantify the discarde
  3. [Abstract and Section 6] The title and abstract state that 'classical theories of gravity' produce entanglement, but the detailed calculation in Section 5.2 is performed for one specific sourcing prescription: semiclassical Einstein gravity with the Newtonian potential given by the expectation value (63). Appendix B.10 discusses stochastic variants, and Appendix D reviews consistency issues, but the universal claim is broader than the demonstrated case. If the paper is intended as a counterexample to the claim that no local classical gravity can entangle, this should be stated explicitly; if it is intended as a general theorem, additional argument is needed to show that the virtual-matter mechanism survives across the full class of classical theories.
minor comments (3)
  1. [Fig. 6 caption] The caption reads 'The ratio ϑ/φ ... is identical to ϑ/φ.' The second occurrence is presumably meant to be something else (e.g., the ratio of quantum and classical effects).
  2. [Around Eq. (16)] The notation 'κj≠λj' for non-overlapping spheres is confusing; it should be 'κ≠λ' or 'i≠j' depending on which indices label the objects and branches.
  3. [Acknowledgements] The text says 'Note that this preprint has not undergone peer review. The Version of Record of this article is published in Nature'. These two statements are inconsistent in a submission context. If this is a reprint of a published article, that should be disclosed; if it is a fresh submission, the preprint boilerplate should be removed.

Circularity Check

0 steps flagged

No significant circularity: the central amplitude is derived from the stated Hamiltonian via a standard Dyson-series calculation and is not fitted to the target conclusion.

full rationale

The paper's derivation chain is self-contained. The central object, β^(4)_RL, is obtained by expanding the interaction-picture unitary (9) in a Dyson series, taking the classical-gravity Hamiltonian (5)/(43), specifying the semiclassical sourcing Φ(x) from ⟨T00⟩ (63), and evaluating the fourth-order amplitude (53) with stated non-relativistic and long-time approximations to reach Eq. (79). No parameter is fitted to the claimed outcome, and the target statement (that classical gravity can generate entanglement) is not assumed among the inputs. The quantum-gravity comparison amplitude γ^(2)_ij is likewise re-derived from the graviton propagator, not imported as a conclusion. Self-citations appearing in the paper (e.g., refs. [19], [24], [33], [49]) provide experimental context or prior comparative results, but the load-bearing perturbative calculation does not reduce to those citations. The discussion of |N±k⟩|N∓k⟩ states in footnote 39 and Appendix B.5 is a possible operational limitation concerning postselection in the spin witness, not a circularity: it concerns whether the computed amplitude translates to an observable entangled spin state, not whether that amplitude was assumed. The scope gap between 'classical theories' in the title and the specific semiclassical model analyzed is a breadth/evidence limitation, not a circular dependence of the derivation on its conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The calculation has no fitted free parameters; all experimental parameters are inputs. It rests on the validity of QFT in curved spacetime and on a specific classical-gravity sourcing assumption (semiclassical Einstein gravity). The most fragile adopted axiom is the discarding of number-transfer final states in the entanglement witness.

axioms (5)
  • domain assumption QFT in a fixed classical background metric (h_μν a c-number) is a valid description of a classical theory of gravity at low energies.
    Section 4, Eq. (5): the interaction Hamiltonian H_int^(CG) = -1/2 ∫ h_μν T̂^μν treats h as classical.
  • domain assumption The classical gravitational potential is sourced by the expectation value of the matter energy-momentum tensor and is identical in every superposition branch (semiclassical Einstein gravity).
    Section 5.2, Eqs. (62)-(64): Φ(x)=ΦC1+ΦC2 with each the average of L/R potentials; the same Φ in all branches is asserted as the meaning of classical gravity.
  • domain assumption Non-relativistic wavepacket approximations R ≫ ħ/(mc), t ≪ 2mR²/ħ, and ct ≫ d_ij hold and justify replacing ω_k by mc²/ħ and using delta-function time integrals.
    Section 5.1, around Eqs. (28) and (57)-(60).
  • ad hoc to paper Final states with |N±k⟩|N∓k⟩ matter-number transfers can be discarded when computing the spin entanglement witness.
    Eq. (19) projects onto |N⟩|N⟩; footnote [39] dismisses |N-k⟩|N+k⟩ states as not seen in the experiment.
  • standard math Standard QFT Wick contraction, Feynman propagators, and Dyson series are valid for the amplitudes.
    Used throughout Section 5 and Appendix B.

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read the original abstract

The unification of gravity and quantum mechanics remains one of the most profound open questions in science. With recent advances in quantum technology, an experimental idea first proposed by Richard Feynman is now regarded as a promising route to testing this unification for the first time. The experiment involves placing a massive object in a quantum superposition of two locations and letting it gravitationally interact with another mass. In modern versions of the experiment, if the two objects subsequently become entangled, this is considered unambiguous evidence that gravity obeys the laws of quantum mechanics. This conclusion derives from theorems that treat a classical gravitational interaction as a local interaction capable of only transmitting classical, not quantum, information. Here, we argue that the classical gravitational interaction can transmit quantum information, and thus generate entanglement through physically local processes. The effects are found to scale differently to the considered quantum gravity effect, providing information on the form of the experiment required to evidence the quantum nature of gravity.

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Forward citations

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