Pith. sign in

REVIEW 3 major objections 5 minor 84 references

Detectability of post-Newtonian classical and quantum gravity via quantum clock interferometry

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A symmetric quantum clock interferometer around a rotating mass isolates the gravitomagnetic clock effect, giving a proper-time shift of $\Delta\tau = 16GJK/(c^4 w)$, but the predicted signal is too small to detect with laboratory-scale…

desk verdict Serious formal extension of quantum-clock interferometry to frame dragging, honest about its own undetectability, but with a factor-of-2 issue, a sign slip in an appendix, and an idealized symmetry cancellation that needs a caveat. read the letter →

arxiv 2506.15014 v6 pith:RJYJKQFA submitted 2025-06-17 quant-ph gr-qc

Eyuri Wakakuwa This is my paper · ORCID
classification quant-phgr-qc
keywords quantumclockinterferometryframedragginggravitomagneticeffectgravity-inducedentanglementequivalenceprinciplepost-Newtoniangravitypropertime
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 proposes a symmetric quantum clock interferometer around a rotating mass as a way to isolate the frame-dragging (gravitomagnetic) part of gravity from ordinary Newtonian gravity. Because the two arms are mirror images, Newtonian gravitational contributions cancel, while the rotation-induced metric term $g_{t\phi}$ produces a proper-time difference $\Delta\tau = 16GJK/(c^4 w)$ between the two paths; this phase difference shows up as a periodic amplitude modulation of the interference pattern. The same geometry, with the source mass rotating clockwise and counterclockwise in superposition, would generate gravity-induced entanglement whose amount is sensitive to whether the quantum equivalence principle holds, and a QEP violation in the GIE experiment alone would rule out models where entanglement is mediated by a spacetime geometry in a quantum superposition of classical configurations. The numerical message is sobering: with $w\sim1$ mm and dimensionless angular momentum $\ell$, the phase shift is of order $\ell\cdot10^{-60}$, so a detectable signal would need $\ell\sim10^{60}$, i.e. planetary-scale rotation. The author therefore presents the setup as a Gedankenexperiment and a starting point for future post-Newtonian quantum gravity probes.

What carries the argument

The central object is the symmetric two-arm quantum clock interferometer with a rotating axially symmetric source at its center. Its load-bearing feature is the mirror symmetry $\phi\to-\phi$: the two arms experience identical Schwarzschild contributions to first order, so Newtonian terms cancel, while the odd-parity frame-dragging component $h_{t\phi}$ contributes with opposite signs and survives. The calculation is carried by a path-integral propagator, derived from canonical quantization, in which the internal clock phase is governed by the proper time along each trajectory (Eq. (3)), together with the weak-field rotating-mass metric of Eq. (33). For the quantum equivalence principle extension, the machinery is a Routhian test theory with four internal Hamiltonians whose equality encodes the extended QEP, and for the entanglement experiment it is the preparation of the source in a superposition of opposite rotation directions plus the witness operator $W_\pm$.

What would settle it

A measurement of the left-port detection probability as a function of interferometer width $w$ around a source of known angular momentum $J$ would settle Eq. (48): a modulation with the predicted period in $1/w$ supports the claim, while a flat visibility at that period at the predicted scale would falsify it. Short of that, a numerical calculation of the Newtonian phase induced by a small fractional arm-length mismatch $\delta w/w$ around the symmetric configuration would determine whether the claimed isolation survives realistic asymmetries.

Watch

Extended reading notes

Core claim

The central claim is that a symmetric quantum clock interferometer is a direct post-Newtonian probe: the mirror-image arrangement cancels the static Newtonian terms, leaving only the frame-dragging contribution, and the proper-time difference between the two arms is $\Delta\tau = 16GJK/(c^4 w)$, with $K$ a dimensionless energy factor of order one. This proper-time difference appears in an interference pattern as both a phase shift and an amplitude modulation, so the visibility as a function of interferometer width $w$ is the observable signature. The paper further claims that preparing the rotating source in a superposition of opposite rotation directions produces gravity-induced entanglement, and that a generalized quantum equivalence principle can be tested through the presence or absence of amplitude modulation in the visibility and in the entanglement. It concludes from a numerical estimate that the effect is inaccessible in any realistic laboratory setting because the suppression by $c^{-4}$ is overwhelming.

Load-bearing premise

The two interferometer arms must be exact mirror images, with straight trajectories and symmetric placement about the rotating mass, so that Newtonian gravitational contributions cancel identically; any path asymmetry, deflection, or external potential gradient injects a Newtonian phase far larger than the gravitomagnetic signal.

Editorial extensions

If this is right

  • Any observed periodic amplitude modulation in this symmetric geometry, with period in $1/w$ set by Eq. (48), would be a direct signature of frame dragging acting on a quantum clock rather than of Newtonian gravity.
  • In the GIE version, the witness $W_\pm = 1 \pm V_{\Delta E,\Delta\tau}\sin(\bar{E}\Delta\tau/\hbar)$ exceeds 1 whenever frame dragging entangles the source and the path, so entanglement generation in this setup is tied to post-Newtonian effects.
  • If the quantum equivalence principle is violated, both the interference visibility and the generated entanglement acquire a $\theta$-dependent amplitude modulation; comparing the visibility and GIE experiments can distinguish a breakdown of the post-Newtonian QEP from a failure of superposed-geometry models of GIE.
  • At realistic laboratory scales the proper-time difference is far below detectability, so the frame-dragging regime is currently inaccessible to tabletop quantum experiments; the scheme functions as a conceptual template rather than a ready-to-build detector.

Reading between the lines

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

  • Editorial inference: the symmetry cancellation is an idealization; a practical experiment would need a tolerance analysis converting the $\ell\cdot10^{-60}$ requirement into bounds on relative arm-length asymmetry, external field gradients, and source placement, which the paper does not provide.
  • Editorial inference: because the same $h_{t\phi}$ term also couples to spin, a clock-based interferometer could in principle be compared with a spin-based interferometer to separate internal-energy coupling from magnetomechanical coupling, potentially trading the smallness of $\Delta E$ for stronger spin-dependent factors.
  • Editorial inference: the GIE conclusion is conditional on the three assumptions about superposed gravitational fields; a null result in the GIE experiment could alternatively be read as a failure of those assumptions rather than as evidence that superposed-geometry models are excluded.
  • Editorial inference: the $c^{-4}$ suppression is generic to gravitomagnetic coupling, so the same order-of-magnitude obstruction likely applies to any clock-based scheme that tries to sense the gravitomagnetic vector potential, not just to this particular geometry.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops a theoretical framework for quantum clock particles in stationary but nonstatic spacetimes, applies it to a quantum clock interferometer with a rotating source mass, and derives the gravitomagnetic proper-time difference Delta_tau = 16 G J K / (c^4 w) (Eq. 48). It then proposes a gravity-induced entanglement scheme based on a superposition of opposite rotation directions and an extension of the quantum equivalence principle involving a frame-dragging internal Hamiltonian H_f. The author concludes that the predicted phase shift is far below detectability for any realistic tabletop parameters, while emphasizing that the symmetry of the interferometer isolates the post-Newtonian contribution from Newtonian gravity.

Significance. If the derivation is correct, the paper provides a clean analytic treatment of a post-Newtonian clock effect in a quantum interferometer and extends the quantum equivalence principle framework to frame-dragging spacetimes. It also gives a concrete, albeit currently undetectable, signature that could in principle distinguish certain quantum-gravity models of gravity-induced entanglement. The paper is explicit and honest about the undetectability of the predicted effects, which is a strength: it marks a boundary for tabletop probes of post-Newtonian quantum gravity. The main formal derivation in Section IV is plausible, and no parameters are fitted to data; the undetectability scaling is robust to the details of the setup.

major comments (3)
  1. [Section V.A, Fig. 1, Eq. (45)] The central claim that the setup isolates frame dragging rests on the assumption that the two interferometer arms are exact mirror images, so that all Newtonian proper-time contributions cancel identically, and that each arm can be treated as a straight line. No tolerance analysis is provided for asymmetries in source placement, arm geometry, environmental potential gradients, or gravitational deflection of the trajectories. For a laboratory-scale source, the Newtonian phase contribution per arm can be many orders of magnitude larger than the gravitomagnetic phase, so even a tiny fractional asymmetry could swamp the signal. The final undetectability conclusion is robust, but the stronger claim that an observed signal would be purely post-Newtonian is not established by the present analysis. The manuscript should either provide an explicit error budget for the required symmetry, or explicitly restrict the isolation claim to an idealized Gedankenexperiment.
  2. [Appendix A, Eqs. (A36) and (A38)] There is a sign inconsistency in the g0i term of the path-integral derivation. Equation (A36) contains + sum_i g0i(bar q) (q'^i - q^i)/(c Delta t) in the exponent, while Eq. (A38) gives d tau/dt = 1 - v^2/2c^2 + Phi/c^2 - sum_i (g0i/c) dx^i/dt. The two expressions have opposite signs for the frame-dragging contribution. Since Eq. (A37) claims the exponent in (A36) equals Delta tau, and since this sign propagates to the Routhian in Eq. (71) and hence to the quantum equivalence principle predictions in Eqs. (78)-(87), the inconsistency needs to be resolved and the affected expressions recomputed.
  3. [Section VI.D, Table I] The inference in Table I, especially Case 3, that a QEP violation in the GIE experiment but not in the interferometric visibility experiment would 'rule out superposed geometry models,' depends on the specific test theory in Eq. (73) and on assumptions (i)-(iii) taken from Ref. [55]. These are stated assumptions rather than derived consequences, and the paper does not show that every superposed-geometry model of gravity-induced entanglement must satisfy the same relations. The conclusion should be framed as conditional on the validity of the adopted test theory and the Ref. [55] postulates, not as a model-independent exclusion.
minor comments (5)
  1. [Eq. (40)] Because Delta_tau is defined as half of tau(P1) - tau(P2), a sentence emphasizing this convention would help readers compare with the standard gravitomagnetic clock effect literature, where the full difference is often quoted.
  2. [Section IV and Appendix A] There are typographical slips: 'Boyer-Linquist' should be 'Boyer-Lindquist' and 'Schwalzschild' should be 'Schwarzschild'.
  3. [Eqs. (49), Figs. 3-6] The dimensionless variable w' is used in the figures without a definition in the captions; please state in each caption that w' = c^4 w / (16 G J K).
  4. [Section II.C, Ref. [55]] Ref. [55] is cited as an arXiv preprint although a journal version exists (Ref. [56]); citing the published version in the relevant narrative would be more appropriate.
  5. [Section VI.B, Eq. (78)] The approximation U(P1)^dagger U(P2) approx exp(H_f Delta_tau / i hbar) uses the same Delta_tau as in Eq. (48), which assumes exact cancellation of the H_N contribution between paths; this assumption should be stated explicitly at that point.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central proper-time formula is derived from the Kerr metric and a canonical path-integral framework, not assumed, and the quantum-equivalence-principle extensions are explicitly stated assumptions.

full rationale

The paper's central quantitative result, Delta_tau = 16 G J K / (c^4 w) (Eq. 48), is obtained by substituting the weak-field Kerr metric (Eqs. 33-36) into a first-order perturbation formula for proper-time differences (Eqs. 15, 32) and integrating along straight-line trajectories (Eq. 45). No parameter is fitted to data, and the result is not an input to the derivation. The claimed cancellation of Newtonian contributions is a geometric symmetry assumption, not a circular reduction: the paper explicitly treats the two arms as mirror-image paths and approximates them as straight lines, and any failure of that symmetry is a validity concern, not a circularity. The quantum-equivalence-principle analysis in Section VI introduces a phenomenological Hamiltonian H_f (Eq. 73) and adopts the superposition-of-geometries postulates of Ref. [55] as stated assumptions; the derived visibility and entanglement modulations (Eqs. 84, 86, 89, 90) follow from those assumptions by explicit calculation and are not equivalent to the assumptions themselves. Citations to prior work are used for background, for the quantum-clock formalism, and for the superposition protocol, but the load-bearing derivation is carried out in this paper from the metric and the path integral. No fitted input is renamed as a prediction, and no load-bearing claim reduces to a self-citation. The analysis is therefore self-contained against external benchmarks; remaining weaknesses, such as the lack of a tolerance analysis for the exact-symmetry idealization and the extreme smallness of the predicted phase, are correctness and feasibility risks rather than circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The central detectability calculation is self-contained and parameter-free apart from experimental variables. The QEP section adds a phenomenological test theory with free parameters and an invented Hamiltonian H_f. The GIE part depends on borrowed assumptions about superposed gravitational fields and on a specific source preparation protocol.

free parameters (2)
  • theta (QEP violation angle)
    Introduced in Section VI B as the angle between eigenstates of H_N and H_f in the phenomenological test theory. It is a free model parameter, not fitted to data, and is used to illustrate how violations would modulate visibility and entanglement.
  • H_f eigenvalues E'_g, E'_e
    Chosen by hand in Section VI B to illustrate the QEP violation patterns (Figure 5 uses E'_g = E_g, E'_e = E_e, Delta_E' = Delta_E = Ebar/24). They parameterize the test theory without independent evidence.
assumptions (6)
  • domain assumption Weak gravity, stationary spacetime, slow motion, and low internal energy gap (Conditions (i)-(iv), Section III).
    These define the regime of the post-Newtonian expansion and the validity of the approximate propagator Eq. (1).
  • domain assumption Semiclassical path dominance: the path integral in Eq. (1) reduces to the classical trajectory, giving U(P) = exp(H_rest tau / i hbar).
    Invoked in Section III to turn the full path integral into a proper-time phase along a single classical path.
  • standard math Canonical quantization with Weyl ordering and the midpoint prescription is valid for the post-Newtonian Hamiltonian.
    Used in Appendix A to derive the short-time propagator; standard within the path-integral literature.
  • domain assumption Postulates from Ref. [55]: macroscopically distinguishable gravitational fields are assigned orthogonal states, each field obeys general relativity, the superposition principle applies to gravitational fields, and back action of the clock on the field is negligible.
    Adopted in Sections V B and VI C for the gravity-induced entanglement calculation; if false, the entanglement predictions are invalid.
  • ad hoc to paper The quantum equivalence principle test theory with a frame-dragging Hamiltonian H_f, requiring H_N = H_f and [H_N, H_f] = 0.
    Introduced in Section VI A as a phenomenological extension of Zych-Brukner. The Hamiltonian H_f has no independent derivation or evidence.
  • domain assumption The source mass can be prepared in a superposition of opposite rotation directions using the protocol of Ref. [48].
    Assumed in Section V B for the GIE protocol; the paper does not analyze experimental feasibility of this preparation.
invented entities (1)
  • Frame-dragging internal Hamiltonian H_f
    purpose: Models possible violations of the extended quantum equivalence principle by coupling the internal clock degrees of freedom to the g0i components of the metric.
    Introduced ad hoc in Section VI A. It produces an amplitude-modulation signature if QEP is violated, but the operator itself has no independent falsifiable handle outside the paper's phenomenological model.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Detectability of post-Newtonian classical and quantum gravity via quantum clock interferometry." pith.science (2026). https://pith.science/paper/RJYJKQFA

@misc{pith2026250615014,
  author       = {Pith},
  title        = {Pith review of: Detectability of post-Newtonian classical and quantum gravity via quantum clock interferometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RJYJKQFA}},
  note         = {Machine review of arXiv:2506.15014}
}
read the original abstract

Understanding physical phenomena at the intersection of quantum mechanics and general relativity remains a major challenge in modern physics. While various experimental approaches have been proposed to probe quantum systems in curved spacetime, most focus on the Newtonian regime, leaving post-Newtonian effects such as frame dragging largely unexplored. In this study, we propose and theoretically analyze an experimental scheme to investigate how post-Newtonian gravity affects quantum systems. We consider two setups: (i) a quantum clock interferometry setup designed to detect the gravitational field of a rotating mass, and (ii) a scheme exploring whether such effects could be used to generate gravity-induced entanglement. Due to the symmetry of the configuration, the proposed setup is insensitive to Newtonian gravitational contributions but remains sensitive to the frame-dragging effect. Furthermore, our scheme allows for testing whether the observed gravity-induced entanglement is consistent with the quantum equivalence principle. While the predicted effects appear too small to detect with current technology, our scheme offers a starting point for future experiments probing post-Newtonian quantum gravitational effects.

Figures

Figures reproduced from arXiv: 2506.15014 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic diagram of the proposed experiment. An [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The paths for which the proper time difference is [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The interference pattern predicted from Eqs. (57) [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The amount of entanglement predicted from Eqs. (64) [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The interference pattern predicted from Eqs. (84) and [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The amount of entanglement predicted from Eqs. (89) [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

84 extracted references · 52 canonical work pages

  1. [55]

    Higgins, A

    G. Higgins, A. Di Biagio, and M. Christodoulou, Truly relativistic gravity mediated entanglement protocol using superpositions of rotational energies, Physical Review D 110, L101901 (2024)

    work page 2024

  2. [1]

    Here, the source mass is assumed to be a particle that has an electric dipole moment represented by{|0⟩,|1⟩} and a large magnetic dipole moment

    Description of The Protocol For the preparation of the superposition state of rota- tional directions, we exploit the protocol proposed in [48]. Here, the source mass is assumed to be a particle that has an electric dipole moment represented by{|0⟩,|1⟩} and a large magnetic dipole moment. At the first step, the electric dipole moment is prepared in a supe...

  3. [2]

    Using (44), the inner product of the clock states is calculated to be ⟨ξ1|ξ2⟩=V ∆E,∆τ exp( ¯E∆τ/iℏ) (54) and ⟨η±|η±⟩= 1±V ∆E,∆τ cos( ¯E∆τ/ℏ),(55) ⟨η±|η∓⟩=±iV ∆E,∆τ sin( ¯E∆τ/ℏ),(56) where the visibility parameterV ∆E,∆τ is given by V∆E,∆τ := cos ∆E∆τ ℏ .(57) Thus, the detection probabilities are given by Pr(L′) = 1 2 1 +V ∆E,∆τ cos ¯E∆τ ℏ ,(58) Pr(R′) = 1...

  4. [3]

    Measurement Requirement We briefly analyze a measurement requirement for wit- nessing the generated entanglement. In order to wit- nesstripartiteentanglement amongS,PandC, mea- surements should be performed not only on the source mass and the path degree of freedom of the clock parti- cle, but also on the clock degree of freedom of it. This requires a dir...

    work page

  5. [4]

    In particular, we chooseθ 0 =π/2

    Then an electric field is applied, by which the orientationθof the particle becomes entan- gled so that the state is (|0⟩|θ=−θ 0⟩+|1⟩|θ=θ 0⟩)/ √ 2. In particular, we chooseθ 0 =π/2. Next, an alternating 11 magnetic field is applied so that the particle starts ro- tating around a given axis with frequencyω 0. The state will become (|0⟩|θ=−π/2,ω=ω 0⟩+|1⟩|θ=...

  6. [5]

    Exactly the same procedure in the reverse order will take the state back to the initial state

    Taking the state of the gravitational field into account, this process realizes the transformation |0⟩+|1⟩√ 2 |θ= 0,ω= 0⟩|g 0⟩→ 1√ 2 (|0⟩|↑⟩|g↑⟩+|1⟩|↓⟩|g ↓⟩)≡ |⇑⟩+|⇓⟩√ 2 ,(60) whereg 0,g ↑ andg ↓ are the states of the gravitational field generated by the source mass that is at rest, rotating clockwise or counterclockwise, respectively. Exactly the same pr...

    work page

  7. [6]

    Note that Ψ′ fin is a pure state onS,PandC

    Evaluation of Entanglement Let us evaluate the amount of entanglement of the state Ψ′ fin. Note that Ψ′ fin is a pure state onS,PandC. Thus, the entanglement between the source mass and the clock particle is quantified by the entanglement entropy 0.02 0.03 0.04 0.05 0.06 0.07 0.2 0.4 0.6 0.8 1.0 a b c ℰ Δ𝐸=0Δ𝐸=𝐸%24⁄,𝑆|𝑃𝐶Δ𝐸=𝐸%24⁄,𝑆|𝑃 𝑤′ FIG. 4. The amount ...

    work page

  8. [7]

    M. Zych, I. Pikovski, F. Costa, and ˇC. Brukner, General relativistic effects in quantum interference of “clocks”, inJournal of Physics: Conference Series, Vol. 723 (IOP Publishing, 2016) p. 012044

    work page 2016

Show all 84 references
  1. [8]

    S. Bose, I. Fuentes, A. A. Geraci, S. M. Khan, S. Qvar- fort, M. Rademacher, M. Rashid, M. Toroˇ s, H. Ulbricht, and C. C. Wanjura, Massive quantum systems as inter- faces of quantum mechanics and gravity, Reviews of Mod- ern Physics97, 015003 (2025)

    2025

  2. [9]

    M. Zych, F. Costa, I. Pikovski, and ˇC. Brukner, Quantum interferometric visibility as a witness of general relativis- tic proper time, Nature Communications2, 505 (2011)

    2011

  3. [10]

    Loriani, A

    S. Loriani, A. Friedrich, C. Ufrecht, F. Di Pumpo, S. Kleinert, S. Abend, N. Gaaloul, C. Meiners, C. Schu- bert, D. Tell,et al., Interference of clocks: A quantum twin paradox, Science advances5, eaax8966 (2019)

    2019

  4. [11]

    Roura, Gravitational redshift in quantum-clock inter- ferometry, Physical Review X10, 021014 (2020)

    A. Roura, Gravitational redshift in quantum-clock inter- ferometry, Physical Review X10, 021014 (2020)

    2020

  5. [12]

    Arndt and C

    M. Arndt and C. Brand, Interference of atomic clocks, Science349, 1168 (2015)

    2015

  6. [13]

    which path

    Y. Margalit, Z. Zhou, S. Machluf, D. Rohrlich, Y. Japha, and R. Folman, A self-interfering clock as a “which path” witness, Science349, 1205 (2015)

    2015

  7. [14]

    Mashhoon, F

    B. Mashhoon, F. W. Hehl, and D. S. Theiss, On the grav- itational effects of rotating masses: The thirring-lense pa- pers, General relativity and gravitation16, 711 (1984)

    1984

  8. [15]

    Castro Ruiz, F

    E. Castro Ruiz, F. Giacomini, and ˇC. Brukner, Entan- glement of quantum clocks through gravity, Proceedings of the National Academy of Sciences114, E2303 (2017)

    2017

  9. [16]

    A. R. Smith and M. Ahmadi, Quantum clocks observe classical and quantum time dilation, Nature Communi- cations11, 5360 (2020)

    2020

  10. [17]

    Paige, A

    A. Paige, A. Plato, and M. Kim, Classical and nonclas- sical time dilation for quantum clocks, Physical Review Letters124, 160602 (2020)

    2020

  11. [18]

    Di Pumpo, C

    F. Di Pumpo, C. Ufrecht, A. Friedrich, E. Giese, W. P. Schleich, and W. G. Unruh, Gravitational redshift tests with atomic clocks and atom interferometers, PRX Quan- tum2, 040333 (2021)

    2021

  12. [19]

    Lense and H

    J. Lense and H. Thirring, ¨Uber den einfluß der eigenro- tation der zentralk¨ orper auf die bewegung der planeten und monde nach der einsteinschen gravitationstheorie, Physikalische zeitschrift19, 156 (1918). 16

    1918

  13. [20]

    Thirring, ¨Uber die wirkung rotierender ferner massen in der einsteinschen gravitationstheorie., Physikalische Zeitschrift19, 33 (1918)

    H. Thirring, ¨Uber die wirkung rotierender ferner massen in der einsteinschen gravitationstheorie., Physikalische Zeitschrift19, 33 (1918)

    1918

  14. [21]

    P. K. Schwartz and D. Giulini, Post-newtonian hamilto- nian description of an atom in a weak gravitational field, Physical Review A100, 052116 (2019)

    2019

  15. [22]

    M. Aspelmeyer, When zeh meets feynman: how to avoid the appearance of a classical world in gravity experi- ments, inFrom Quantum to Classical: Essays in Honour of H.-Dieter Zeh(Springer, 2022) pp. 85–95

    2022

  16. [23]

    gravitomagnetic

    (see [24–26] for general frameworks). Therefore, if measurable entanglement between the two particles is ob- served, it would serve as evidence that gravity itself must be inherently quantum in nature. The entanglement gen- erated in such a setup is referred to asgravity-induc...

  17. [24]

    Marletto and V

    C. Marletto and V. Vedral, Quantum-information meth- ods for quantum gravity laboratory-based tests, Reviews of Modern Physics97, 015006 (2025)

    2025

  18. [25]

    Zych and ˇC

    M. Zych and ˇC. Brukner, Quantum formulation of the einstein equivalence principle, Nature Physics14, 1027 (2018)

    2018

  19. [26]

    Roura, Quantum probe of space-time curvature, Sci- ence375, 142 (2022)

    A. Roura, Quantum probe of space-time curvature, Sci- ence375, 142 (2022)

    2022

  20. [27]

    Pikovski, M

    I. Pikovski, M. Zych, F. Costa, and ˇC. Brukner, Universal decoherence due to gravitational time dilation, Nature Physics11, 668 (2015)

    2015

  21. [28]

    Zych,Quantum systems under gravitational time di- lation(Springer, 2017)

    M. Zych,Quantum systems under gravitational time di- lation(Springer, 2017)

    2017

  22. [29]

    M. L. Basso and J. Maziero, The effect of stationary ax- isymmetric spacetimes in interferometric visibility, Gen- eral Relativity and Gravitation53, 70 (2021)

    2021

  23. [30]

    S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroˇ s, M. Paternostro, A. A. Geraci, P. F. Barker, M. Kim, and G. Milburn, Spin entanglement witness for quantum gravity, Physical Review Letters119, 240401 (2017)

    2017

  24. [31]

    Krisnanda, M

    T. Krisnanda, M. Zuppardo, M. Paternostro, and T. Pa- terek, Revealing nonclassicality of inaccessible objects, Physical Review Letters119, 120402 (2017)

    2017

  25. [32]

    L. Lami, J. S. Pedernales, and M. B. Plenio, Testing the quantumness of gravity without entanglement, Physical Review X14, 021022 (2024)

    2024

  26. [33]

    T. D. Galley, F. Giacomini, and J. H. Selby, A no-go theorem on the nature of the gravitational field beyond quantum theory, Quantum6, 779 (2022)

    2022

  27. [34]

    M. J. Hall and M. Reginatto, On two recent proposals for witnessing nonclassical gravity, Journal of Physics A: Mathematical and Theoretical51, 085303 (2018)

    2018

  28. [35]

    Anastopoulos, M

    C. Anastopoulos, M. Lagouvardos, and K. Savvidou, Gravitational effects in macroscopic quantum systems: a first-principles analysis, Classical and Quantum Gravity 38, 155012 (2021)

    2021

  29. [36]

    Fragkos, M

    V. Fragkos, M. Kopp, and I. Pikovski, On inference of quantization from gravitationally induced entanglement, AVS Quantum Science4(2022)

    2022

  30. [37]

    Y. Ma, T. Guff, G. W. Morley, I. Pikovski, and M. Kim, Limits on inference of gravitational entanglement, Phys- ical Review Research4, 013024 (2022)

    2022

  31. [38]

    Mart´ ın-Mart´ ınez and T

    E. Mart´ ın-Mart´ ınez and T. R. Perche, What gravity me- diated entanglement can really tell us about quantum gravity, Physical Review D108, L101702 (2023)

    2023

  32. [39]

    Christodoulou, A

    M. Christodoulou, A. Di Biagio, R. Howl, and C. Rov- elli, Gravity entanglement, quantum reference systems, degrees of freedom, Classical and Quantum Gravity40, 047001 (2023)

    2023

  33. [40]

    Huggett, N

    N. Huggett, N. Linnemann, and M. D. Schneider,Quan- tum gravity in a laboratory?(Cambridge University Press, 2023)

    2023

  34. [41]

    Aziz and R

    J. Aziz and R. Howl, Classical theories of gravity produce entanglement, Nature646, 813 (2025)

    2025

  35. [42]

    Marletto and V

    C. Marletto and V. Vedral, Gravitationally induced en- tanglement between two massive particles is sufficient evidence of quantum effects in gravity, Physical Review Letters119, 240402 (2017)

    2017

  36. [43]

    Krisnanda, G

    T. Krisnanda, G. Y. Tham, M. Paternostro, and T. Pa- terek, Observable quantum entanglement due to gravity, npj Quantum Information6, 12 (2020)

    2020

  37. [44]

    S. Bose, A. Mazumdar, M. Schut, and M. Toroˇ s, Mech- anism for the quantum natured gravitons to entangle masses, Physical Review D105, 106028 (2022)

    2022

  38. [45]

    Yant and M

    J. Yant and M. Blencowe, Gravitationally induced en- tanglement in a harmonic trap, Physical Review D107, 106018 (2023)

    2023

  39. [46]

    Carney, H

    D. Carney, H. M¨ uller, and J. M. Taylor, Using an atom interferometer to infer gravitational entanglement gener- ation, PRX Quantum2, 030330 (2021)

    2021

  40. [47]

    R. J. Marshman, A. Mazumdar, and S. Bose, Locality and entanglement in table-top testing of the quantum na- ture of linearized gravity, Physical Review A101, 052110 (2020)

    2020

  41. [48]

    Toroˇ s, T

    M. Toroˇ s, T. W. Van De Kamp, R. J. Marshman, M. Kim, A. Mazumdar, and S. Bose, Relative acceler- ation noise mitigation for nanocrystal matter-wave inter- ferometry: Applications to entangling masses via quan- tum gravity, Physical Review Research3, 023178 (2021)

    2021

  42. [49]

    Kent and D

    A. Kent and D. Pital´ ua-Garc´ ıa, Testing the nonclassi- cality of spacetime: What can we learn from bell–bose et al.-marletto-vedral experiments?, Phys. Rev. D104, 126030 (2021)

    2021

  43. [50]

    R. Howl, N. Cooper, and L. Hackerm¨ uller, Gravitationally-induced entanglement in cold atoms, arXiv preprint arXiv:2304.00734 (2023)

    2023 arXiv

  44. [51]

    Fujita, Y

    T. Fujita, Y. Kaku, A. Matumura, and Y. Michimura, Inverted oscillators for testing gravity-induced quantum entanglement, arXiv preprint arXiv:2308.14552 (2023)

    2023 arXiv

  45. [52]

    Bengyat, A

    O. Bengyat, A. Di Biagio, M. Aspelmeyer, and M. Christodoulou, Gravity-mediated entanglement be- tween oscillators as quantum superposition of geometries, Physical Review D110, 056046 (2024)

    2024

  46. [53]

    Y. Kaku, A. Matsumura, and T. Fujita, Sudden decoher- ence by resonant particle excitation for testing gravity- induced entanglement, arXiv preprint arXiv:2501.18147 (2025)

    2025

  47. [54]

    Chen and F

    L.-Q. Chen and F. Giacomini, Quantum effects in gravity beyond the newton potential from a delocalised quantum source, arXiv preprint arXiv:2402.10288 (2024)

    2024 arXiv

  48. [56]

    T. B. Lanta˜ no, L. Petruzziello, S. F. Huelga, and M. B. Plenio, Low-energy test of quantum gravity via angular momentum entanglement, arXiv preprint arXiv:2409.01364 (2024)

    2024 arXiv

  49. [57]

    Christodoulou, A

    M. Christodoulou, A. Di Biagio, M. Aspelmeyer, ˇC. Brukner, C. Rovelli, and R. Howl, Locally mediated entanglement in linearized quantum gravity, Physical Re- view Letters130, 100202 (2023)

    2023

  50. [58]

    L.-Q. Chen, F. Giacomini, and C. Rovelli, Quantum states of fields for quantum split sources, Quantum7, 17 958 (2023)

    2023

  51. [59]

    C. M. Will,Theory and experiment in gravitational physics(Cambridge university press, 2018)

    2018

  52. [60]

    C. M. Will, The confrontation between general relativ- ity and experiment, Living Reviews in Relativity17, 1 (2014)

    2014

  53. [61]

    G. Rosi, G. D’Amico, L. Cacciapuoti, F. Sorrentino, M. Prevedelli, M. Zych, ˇC. Brukner, and G. M. Tino, Quantum test of the equivalence principle for atoms in coherent superposition of internal energy states, Nature Communications8, 15529 (2017)

    2017

  54. [62]

    Giacomini and ˇC

    F. Giacomini and ˇC. Brukner, Einstein’s equiv- alence principle for superpositions of gravitational fields and quantum reference frames, arXiv preprint arXiv:2012.13754 (2020)

    2020 arXiv

  55. [63]

    Giacomini and ˇC

    F. Giacomini and ˇC. Brukner, Quantum superposition of spacetimes obeys einstein’s equivalence principle, AVS Quantum Science4(2022)

    2022

  56. [64]

    Giacomini, Spacetime quantum reference frames and superpositions of proper times, Quantum5, 508 (2021)

    F. Giacomini, Spacetime quantum reference frames and superpositions of proper times, Quantum5, 508 (2021)

    2021

  57. [65]

    Giacomini, E

    F. Giacomini, E. Castro-Ruiz, and ˇC. Brukner, Quantum mechanics and the covariance of physical laws in quan- tum reference frames, Nature Communications10, 494 (2019)

    2019

  58. [66]

    S. Bose, A. Mazumdar, M. Schut, and M. Toroˇ s, En- tanglement witness for the weak equivalence principle, Entropy25, 448 (2023)

    2023

  59. [67]

    Landau and E

    L. Landau and E. Lifshitz,Mechanics: Volume 1, Course of theoretical physics (Butterworth-Heinemann, 1982)

    1982

  60. [68]

    M. E. Peskin,An Introduction to quantum field theory (CRC press, 2018)

    2018

  61. [69]

    Srednicki,Quantum field theory(Cambridge univer- sity press, 2007)

    M. Srednicki,Quantum field theory(Cambridge univer- sity press, 2007)

    2007

  62. [70]

    J. M. Cohen and B. Mashhoon, Standard clocks, inter- ferometry, and gravitomagnetism, Physics Letters A181, 353 (1993)

    1993

  63. [71]

    Ciufolini and J

    I. Ciufolini and J. A. Wheeler,Gravitation and inertia (Princeton university press, 1995)

    1995

  64. [72]

    Landau and E

    L. Landau and E. Lifshitz,The Classical Theory of Fields: Volume 2, Course of theoretical physics (Elsevier Science, 1975)

    1975

  65. [73]

    Pikovski, M

    I. Pikovski, M. Zych, F. Costa, and ˇC. Brukner, Time di- lation in quantum systems and decoherence, New Journal of Physics19, 025011 (2017)

    2017

  66. [74]

    Ufrecht, F

    C. Ufrecht, F. Di Pumpo, A. Friedrich, A. Roura, C. Schubert, D. Schlippert, E. M. Rasel, W. P. Schleich, and E. Giese, Atom-interferometric test of the universal- ity of gravitational redshift and free fall, Physical Review Research2, 043240 (2020)

    2020

  67. [75]

    C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schu- macher, Concentrating partial entanglement by local op- erations, Physical Review A53, 2046 (1996)

    1996

  68. [76]

    W. K. Wootters, Entanglement of formation of an arbi- trary state of two qubits, Physical Review Letters80, 2245 (1998)

    1998

  69. [77]

    Chevalier, A

    H. Chevalier, A. Paige, and M. Kim, Witnessing the non- classical nature of gravity in the presence of unknown interactions, Physical Review A102, 022428 (2020)

    2020

  70. [78]

    T. Guff, N. Boulle, and I. Pikovski, Optimal fidelity wit- nesses for gravitational entanglement, Physical Review A 105, 022444 (2022)

    2022

  71. [79]

    Kashiwa, Y

    T. Kashiwa, Y. Ohnuki, and M. Suzuki,Path integral methods(Clarendon Press, 1997). Appendix A: Derivation of the path integral formula In this section, we provide a detailed proof that the time evolution of a quantum clock particle is described by a path-integral formula as ⟨ξ...

    1997

  72. [80]

    invariant in the coordinate time) but nonstatic (i.e

    Calculation of the Spacetime Metric We first establish relations among metric components for stationary (i.e. invariant in the coordinate time) but nonstatic (i.e. it has non-zerog 0i =g i0 components) spacetimes, which will be used to express the Hamiltonian in canonical form...

  73. [81]

    Consider a classical particle with the rest massm

    Derivation of An Approximate Classical Hamiltonian We next derive the classical Hamiltonian consistent with the post-Newtonian expansion under the above metric. Consider a classical particle with the rest massm. For the moment, we assume that it has no internal degree of freed...

  74. [82]

    20 Note that (ˆpi,ˆqi) and ˆHrest act on different Hilbert spaces, so that [ ˆHrest,ˆpi] = [ ˆHrest,ˆqi] = 0

    Derivation of A Quantum Hamiltonian The classical Hamiltonian (A21) is then quantized by replacing the canonical variables (pi,qi) with operators (ˆpi,ˆqi) satisfying the canonical commutation relation [ˆpi,ˆqj] =iℏδ ij, and the rest energyE rest by the rest Hamiltonian ˆHrest...

  75. [83]

    Calculation of The Short-Time T ransition Amplitude DefineU(∆t) = exp( ˆHtot∆t/iℏ). Based on (A26), the short-time transition amplitude is calculated as follows: ⟨α′,q′|U(∆t)|α,q⟩ =⟨α′,q′| ˆI+ ˆHtot∆t iℏ ! |α,q⟩+O(∆t 2) (A27) =⟨α′,q′|α,q⟩+ ∆t iℏ⟨α′,q′| ˆHtot|α,q⟩+O(∆t 2) (A28)...

  76. [84]

    Z Dq p γ(q) exp Eα iℏ Z fin ini dτ !# ⟨α|ξini⟩(A49) =⟨ξ fin|

    Derivation of The Propagator Take an arbitraryN∈N, let (q ini,t ini) = (q0,t 0), (qN,tN) = (qN,tN), ∆t= (t fin−t ini)/Nandt k+1−tk = ∆t. We take the initial state and the final state of the internal degree of freedom to be eigenstates of the rest Hamiltonian, 22 αini =α 0 andα...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.