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arxiv: 2605.00967 · v1 · submitted 2026-05-01 · 🪐 quant-ph

Gravity-induced Entanglement under Constrained Dynamics

Hollis Williams This is my paper

Pith reviewed 2026-05-09 19:39 UTC · model grok-4.3

classification 🪐 quant-ph
keywords gravity-induced entanglementconstrained dynamicsshort-time approximationphase accumulationinterferometryquantum gravity testspendulum systems
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The pith

Constrained systems with short-time inertial dynamics reproduce the gravitational phase accumulation of free-fall interferometers.

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

The paper argues that gravity-induced entanglement generation does not require free-fall of massive superpositions; any system whose motion stays effectively inertial over the short interferometer timescale will accumulate the same gravitational phase. This equivalence holds because deviations from the free-fall phase appear only at second order in the small ratio of interferometer time t to the natural period T of the constrained motion. For a concrete case of carbon nanotube pendula, the paper calculates that this correction stays below one part in a million under realistic parameters, leaving the interference visibility used to witness entanglement essentially unchanged. If the claim is correct, tests of whether gravity has quantum features can proceed with mechanically constrained setups that avoid the most stringent experimental demands of pure free fall.

Core claim

Systems exhibiting effectively inertial dynamics in the short-time regime reproduce the same gravitational phase accumulation responsible for entanglement generation. Deviations from the free-fall phase enter at order (t/T)^2, where t is the interferometer timescale and T is the characteristic period of the constrained motion. Analysis of a representative mechanically constrained implementation using carbon nanotube pendula shows that the resulting correction to the entangling phase remains below 10^{-6} in experimentally relevant regimes, leading to a negligible modification of the interference visibility used to certify entanglement.

What carries the argument

The short-time inertial approximation for constrained motion, under which the gravitational phase matches the free-fall result up to higher-order corrections in (t/T).

If this is right

  • Gravity-induced entanglement protocols apply to mechanically constrained systems in addition to free-fall implementations.
  • The entangling phase and interference visibility remain essentially identical for nanotube pendula under realistic conditions.
  • Experimental requirements for realizing such protocols are relaxed by permitting constrained dynamics.
  • Any system whose short-time motion is effectively inertial can be used to generate the necessary gravitational entanglement.

Where Pith is reading between the lines

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

  • The same short-time approximation could apply to other constrained platforms such as electromagnetic traps provided the inertial condition holds.
  • Optimizing the ratio t/T offers a direct way to suppress residual phase errors in future designs.
  • This equivalence suggests that gravity-entanglement tests can be pursued in more compact or stable laboratory settings than pure free-fall schemes allow.

Load-bearing premise

The constrained motion remains effectively inertial over the interferometer timescale t, keeping the phase correction below 10^{-6} for the chosen parameters.

What would settle it

Measure the gravitational phase shift accumulated in a nanotube-pendulum interferometer and compare it to the free-fall prediction; a discrepancy larger than order (t/T)^2 would falsify the equivalence.

Figures

Figures reproduced from arXiv: 2605.00967 by Hollis Williams.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the proposed nanotube-based platform. view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic of two adjacent nanotube-based inter view at source ↗
read the original abstract

Tests of gravity-induced entanglement have been proposed as a route to probing the quantum nature of gravity, but existing schemes rely on free-fall interferometry of massive spatial superpositions, imposing severe experimental constraints. We show that systems exhibiting effectively inertial dynamics in the short-time regime reproduce the same gravitational phase accumulation responsible for entanglement generation. Deviations from the free-fall phase enter at order $(t/T)^2$, where $t$ is the interferometer timescale and $T$ is the characteristic period of the constrained motion. We analyse a representative mechanically constrained implementation using carbon nanotube pendula and show that the resulting correction to the entangling phase remains below $10^{-6}$ in experimentally relevant regimes, leading to a negligible modification of the interference visibility used to certify entanglement. These results demonstrate that gravity-induced entanglement protocols extend beyond free-fall implementations to a broader class of constrained dynamical systems, significantly relaxing the requirements for experimental realisation of the Bose-Marletto-Vedral protocol.

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

2 major / 2 minor

Summary. The manuscript claims that systems with constrained dynamics, exemplified by carbon nanotube pendula, can reproduce the gravitational phase accumulation of free-fall interferometry in the short-time regime (t << T), with deviations only at O((t/T)^2). For experimentally relevant parameters, this correction is below 10^{-6}, resulting in negligible change to the interference visibility used to certify gravity-induced entanglement in the Bose-Marletto-Vedral protocol. This extends the protocol to a broader class of mechanically constrained implementations.

Significance. Should the short-time approximation and the smallness of the phase correction be rigorously established, this work would meaningfully broaden the experimental landscape for probing quantum gravity by removing the stringent requirement for free-fall conditions. The concrete analysis of nanotube pendula adds practical value by identifying a feasible platform. The leading-order equivalence to free-fall phase accumulation is a conceptual strength if the expansion is shown to hold without additional assumptions.

major comments (2)
  1. [§3 (constrained dynamics and phase derivation)] The short-time expansion must be derived explicitly from the constrained equations of motion to confirm that the gravitational phase matches the free-fall case at leading order. Any unaccounted O(t) term arising from the constraint force would directly impact the entangling phase and undermine the equivalence claim.
  2. [§4 (nanotube pendula implementation)] The assertion that the phase correction is below 10^{-6} for relevant regimes requires the specific values of the characteristic period T (determined by nanotube parameters) and interferometer time t, along with the computation showing (t/T)^2 < 10^{-6}. Without these details, the bound on visibility modification cannot be verified.
minor comments (2)
  1. [Abstract] The abstract refers to 'the resulting correction to the entangling phase' without defining the phase explicitly; a brief mention of the phase expression would improve clarity.
  2. [Notation] The characteristic period T is introduced but its precise definition in terms of the pendulum parameters should be stated early to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below and have revised the manuscript to provide the requested clarifications and explicit details.

read point-by-point responses

  1. Referee: [§3 (constrained dynamics and phase derivation)] The short-time expansion must be derived explicitly from the constrained equations of motion to confirm that the gravitational phase matches the free-fall case at leading order. Any unaccounted O(t) term arising from the constraint force would directly impact the entangling phase and undermine the equivalence claim.

    Authors: We agree that an explicit derivation from the constrained equations of motion is necessary to substantiate the leading-order equivalence. In the revised manuscript we have expanded Section 3 with a step-by-step derivation starting from the constrained Lagrangian and the resulting equations of motion. The expansion demonstrates that the gravitational phase accumulation reproduces the free-fall result at order t, with the constraint forces contributing no O(t) term to the entangling phase; the first deviations appear at O((t/T)^2). revision: yes

  2. Referee: [§4 (nanotube pendula implementation)] The assertion that the phase correction is below 10^{-6} for relevant regimes requires the specific values of the characteristic period T (determined by nanotube parameters) and interferometer time t, along with the computation showing (t/T)^2 < 10^{-6}. Without these details, the bound on visibility modification cannot be verified.

    Authors: We thank the referee for highlighting the need for explicit numerical values. The revised Section 4 now includes the specific nanotube parameters that determine the characteristic period T, the interferometer time t employed in the Bose-Marletto-Vedral protocol, and the direct computation establishing that (t/T)^2 remains below 10^{-6} for these experimentally relevant values, confirming the negligible effect on interference visibility. revision: yes

Circularity Check

0 steps flagged

Short-time expansion of constrained dynamics is derived independently from equations of motion

full rationale

The paper starts from the constrained equations of motion for systems such as nanotube pendula and performs a short-time expansion to show that the gravitational phase matches the free-fall case to leading order, with corrections entering only at O((t/T)^2). This is a direct perturbative calculation from the dynamics rather than a redefinition, fit, or self-citation chain. No load-bearing step reduces to a prior result by the same authors or to a fitted parameter renamed as a prediction. The bound <10^{-6} follows from evaluating the expansion with stated mechanical parameters, keeping the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the short-time inertial approximation for constrained motion and on the specific dynamical model of carbon-nanotube pendula; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Constrained motion is effectively inertial over the short interferometer timescale t
    Invoked to equate the gravitational phase to the free-fall case; appears in the statement that deviations enter only at (t/T)^2.

pith-pipeline@v0.9.0 · 5447 in / 1159 out tokens · 43874 ms · 2026-05-09T19:39:27.142619+00:00 · methodology

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    To probe gravity-induced entanglement, we consider two such interferometers placed in close proximity (shown in Fig

    within a mechanically constrained platform which provides enhanced stability and control. To probe gravity-induced entanglement, we consider two such interferometers placed in close proximity (shown in Fig. 2) [1, 2]. Each system evolves in a spatial su- perposition and the resulting branch-dependent gravita- tional interaction gives rise to an entangling...

    work page

  2. [2]

    Bose et al., Spin Entanglement Witness for Quantum Gravity, Phys

    S. Bose et al., Spin Entanglement Witness for Quantum Gravity, Phys. Rev. Lett.119, 240401 (2017)

    work page 2017

  3. [3]

    Marletto and V

    C. Marletto and V. Vedral, Gravitationally Induced En- tanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity, Phys. Rev. Lett. 119, 240402 (2017)

    work page 2017

  4. [4]

    Carney et al., Tabletop experiments for quantum gravity: a user’s manual, Class

    D. Carney et al., Tabletop experiments for quantum gravity: a user’s manual, Class. Quantum Grav.36(3), 034001 (2019)

    work page 2019

  5. [5]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cambridge University Press, 2000)

    work page 2000

  6. [6]

    Horodecki et al., Quantum entanglement, Rev

    R. Horodecki et al., Quantum entanglement, Rev. Mod. Phys.81, 865 (2009)

    work page 2009

  7. [7]

    Nguyen and F

    H.C. Nguyen and F. Bernards, Entanglement dynamics of two mesoscopic objects with gravitational interaction, 5 Eur. Phys. J. D,74(4), 69 (2020)

    work page 2020

  8. [8]

    Chevalier et al., Witnessing the nonclassical nature of gravity in the presence of unknown interactions, Phys

    H. Chevalier et al., Witnessing the nonclassical nature of gravity in the presence of unknown interactions, Phys. Rev. A102(2), 022428 (2020)

    work page 2020

  9. [9]

    van de Kamp et al., Quantum gravity witness via entanglement of masses: Casimir screening, Phys

    T.W. van de Kamp et al., Quantum gravity witness via entanglement of masses: Casimir screening, Phys. Rev. A,102(6), 062807 (2020)

    work page 2020

  10. [10]

    Schut et al., Improving resilience of quantum-gravity- induced entanglement of masses to decoherence us- ing three superpositions, Phys

    M. Schut et al., Improving resilience of quantum-gravity- induced entanglement of masses to decoherence us- ing three superpositions, Phys. Rev. A105(3), 032411 (2022)

    work page 2022

  11. [11]

    Schut el al., Relaxation of experimental parameters in a quantum-gravity-induced entanglement of masses pro- tocol using electromagnetic screening, Phys

    M. Schut el al., Relaxation of experimental parameters in a quantum-gravity-induced entanglement of masses pro- tocol using electromagnetic screening, Phys. Rev. Res., 5(4), 043170 (2023)

    work page 2023

  12. [12]

    Ghosal et al., Distribution of quantum gravity induced entanglement in many-body systems, J

    P. Ghosal et al., Distribution of quantum gravity induced entanglement in many-body systems, J. Phys. A: Math. Theo.57(44), 445302 (2024)

    work page 2024

  13. [13]

    Liu et al., Multiqubit entanglement due to quantum gravity, Phys

    S. Liu et al., Multiqubit entanglement due to quantum gravity, Phys. Lett. A493, 129273 (2024)

    work page 2024

  14. [14]

    Machluf et al., Coherent Stern–Gerlach momentum splitting on an atom chip, Nat

    S. Machluf et al., Coherent Stern–Gerlach momentum splitting on an atom chip, Nat. Commun.4, 2424 (2013)

    work page 2013

  15. [15]

    Margalit et al., Analysis of a high-stability Stern–Gerlach spatial fringe interferometer, New J

    Y. Margalit et al., Analysis of a high-stability Stern–Gerlach spatial fringe interferometer, New J. Phys. 21, 073040 (2019)

    work page 2019

  16. [16]

    Margalit et al., Realization of a complete Stern- Gerlach interferometer: Toward a test of quantum grav- ity

    Y. Margalit et al., Realization of a complete Stern- Gerlach interferometer: Toward a test of quantum grav- ity. Sci. Adv.7(22), eabg2879 (2021)

    work page 2021

  17. [17]

    Scala et al., Matter-Wave Interferometry of a Levi- tated Thermal Nano-Oscillator Induced and Probed by a Spin, Phys

    M. Scala et al., Matter-Wave Interferometry of a Levi- tated Thermal Nano-Oscillator Induced and Probed by a Spin, Phys. Rev. Lett.111, 180403 (2013)

    work page 2013

  18. [18]

    Wan et al., Free Nano-Object Ramsey Interferometry for Large Quantum Superpositions, Phys

    C. Wan et al., Free Nano-Object Ramsey Interferometry for Large Quantum Superpositions, Phys. Rev. Lett.117, 143003 (2016)

    work page 2016

  19. [19]

    Pedernales, G.W

    J.S. Pedernales, G.W. Morley and M.B. Plenio, Motional Dynamical Decoupling for Interferometry with Macro- scopic Particles, Phys. Rev. Lett.125, 023602 (2020)

    work page 2020

  20. [20]

    B.D. Wood, S. Bose and G.W. Morley, Spin dynamical decoupling for generating macroscopic superpositions of a free-falling nanodiamond, Phys. Rev. A105, 012824 (2022)

    work page 2022

  21. [21]

    Zhang et al., Growth of Half-Meter Long Carbon Nan- otubes Based on Schulz–Flory Distribution, ACS Nano 7(7), 6156-6161 (2013)

    R. Zhang et al., Growth of Half-Meter Long Carbon Nan- otubes Based on Schulz–Flory Distribution, ACS Nano 7(7), 6156-6161 (2013)

    work page 2013

  22. [22]

    Kumar and M.B

    S.P. Kumar and M.B. Plenio, On quantum gravity tests with composite particles, Nat. Commun.11, 3900 (2020)

    work page 2020

  23. [23]

    Nanofabricated torsion pendulums for tabletop gravity experiments,

    J. Manley, Nanofabricated torsion pendulums for table- top gravity experiments, arXiv:2601.11366v1 (2026)

    work page arXiv 2026

  24. [24]

    Kryhin and V

    S. Kryhin and V. Sudhir, Distinguishable Consequence of Classical Gravity on Quantum Matter, Phys. Rev. Lett. 134, 061501 (2025)

    work page 2025

  25. [25]

    G. Cao, X. Chen and J.W. Kysar, Apparent thermal con- traction of single-walled carbon nanotubes, Phys. Rev. B 72, 235404 (2005)

    work page 2005

  26. [26]

    G. Cao, X. Chen and J.W. Kysar, Thermal vibration and apparent thermal contraction of single-walled car- bon nanotubes, J. Mech. Phys. Solids54(6), 1206-1236 (2006)

    work page 2006

  27. [27]

    Wu et al., Structural Design and Fabrication of Mul- tifunctional Nanocarbon Materials for Extreme Environ- mental Applications, Adv

    S. Wu et al., Structural Design and Fabrication of Mul- tifunctional Nanocarbon Materials for Extreme Environ- mental Applications, Adv. Mater.34, 2201046 (2022)

    work page 2022

  28. [28]

    Zhao et al., Super-elasticity of three-dimensionally cross-linked graphene materials all the way to deep cryo- genic temperatures, Sci

    K. Zhao et al., Super-elasticity of three-dimensionally cross-linked graphene materials all the way to deep cryo- genic temperatures, Sci. Adv.5, eaav2589 (2019)

    work page 2019

  29. [29]

    Gonzalez-Ballestero et al., Levitodynamics: Levita- tion and control of microscopic objects in vacuum, Sci- ence374, eabg3027 (2021)

    C. Gonzalez-Ballestero et al., Levitodynamics: Levita- tion and control of microscopic objects in vacuum, Sci- ence374, eabg3027 (2021)

    work page 2021

  30. [30]

    Lialys et al., Optical trapping of sub-millimeter sized particles and microorganisms, Sci

    L. Lialys et al., Optical trapping of sub-millimeter sized particles and microorganisms, Sci. Rep.13, 8615 (2023)

    work page 2023