Amplification and generation bounds of gravity-induced entanglement in pulsed optomechanical systems

Alfred Li; Daisuke Miki; Yanbei Chen

Amplification and generation bounds of gravity-induced entanglement in pulsed optomechanical systems

T0 review · 1 major / 0 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read Gravity-induced entanglement in pulsed optomechanics requires gravitational coupling to exceed twice the thermal decoherence rate, and no input state lowers this threshold.

desk verdict The paper derives a thermal-noise threshold for gravity-induced entanglement that holds for Gaussian and Fock inputs but asserts it for arbitrary states without a general proof. read the letter →

arxiv 2605.26240 v1 pith:7B5W7AY3 submitted 2026-05-25 quant-ph gr-qc

Daisuke Miki , Alfred Li , Yanbei Chen This is my paper · ORCID

classification quant-phgr-qc

keywords gravity-inducedentanglementpulsedoptomechanicsthresholdthermaldecoherenceGaussianstatesFockbeam-splitterswap

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 studies two red-detuned pulsed optomechanical systems whose mechanical modes interact gravitationally. Each system uses rectangular pulses to first swap a nonclassical optical state onto its mechanical mode and then read the gravitationally induced entanglement back into the outgoing light. Squeezed or Fock inputs increase the amount of entanglement that appears in the optical outputs. The generation threshold itself, however, remains fixed by the inequality g_G greater than 2 gamma_m N_th. This bound is proven for two-mode Gaussian inputs and shown to hold as well for Fock inputs; imperfect detection further splits the parameter space into entanglement-annihilating and entanglement-breaking regimes governed solely by accumulated thermal decoherence.

What carries the argument

The beam-splitter state swap performed by the red-detuned optomechanical interaction under rectangular pulses, which transfers the effect of the gravitational coupling from the mechanical modes to the optical outputs.

What would settle it

An experiment that generates detectable entanglement with g_G less than or equal to 2 gamma_m N_th using either Gaussian or Fock inputs would falsify the claimed bound.

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Extended reading notes

Core claim

In two red-detuned pulsed optomechanical systems with gravitationally coupled masses, the optomechanical interaction realizes a beam-splitter state swap. Two rectangular pulses per system first imprint a nonclassical state on the mechanics and then read the gravitationally generated entanglement onto the outgoing optical fields. While squeezed or Fock inputs amplify the resulting entanglement, the threshold condition g_G > 2 gamma_m N_th cannot be lowered by any choice of input state; the bound is established for two-mode Gaussian inputs and remains valid for Fock-state inputs. Imperfect detection modifies the accessible regimes, which are ultimately set by thermal decoherence accumulated ov

Load-bearing premise

The optomechanical interaction must realize an ideal beam-splitter swap with rectangular pulses while thermal decoherence in the mechanical modes is the only competing noise.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core result is a lower bound g_G > 2 γ_m N_th on the gravitational coupling needed to generate entanglement between the output fields; the bound is independent of the input state and arises from competition with mechanical thermal decoherence. They show that squeezed or Fock inputs can increase the amount of entanglement above threshold, and they track how finite detection efficiency raises the effective threshold and carves out entanglement-breaking regimes set only by accumulated thermal noise.

The work is clearest on the Gaussian case, where the beam-splitter swap and covariance-matrix evolution let them prove the bound directly, and the Fock-state extension is checked explicitly. That part is solid and useful for experiment design. The model assumptions (rectangular pulses realizing a clean state swap, thermal decoherence as the sole competing process) are stated plainly.

The soft spot is the reach of the claim. The abstract says the threshold cannot be lowered by any input state, yet the proofs cover only two-mode Gaussians and Fock states. Nothing in the provided text supplies a general argument that input dependence drops out for every possible state; if a different non-Gaussian state alters the effective competition between g_G and thermal terms under the same interaction Hamiltonian, the bound could shift. The paper would be stronger with either a general proof or an explicit restriction to the cases shown.

This is for groups working on pulsed optomechanics and proposals for gravity-entanglement tests. A reader who needs concrete bounds on lab parameters will find the threshold and detection analysis helpful. It is worth sending to referees because the central claim is sharp and the derivations for the checked cases are reproducible, even though the generality step needs tightening.

Referee Report

1 major / 0 minor

Summary. The paper investigates gravity-induced entanglement between the output optical fields of two red-detuned pulsed optomechanical systems whose masses interact gravitationally. Using rectangular pulses to realize beam-splitter state swaps, it shows that preparing input states in squeezed or Fock form amplifies the generated entanglement, but derives a threshold g_G > 2 γ_m N_th for entanglement generation that is set by competition with thermal decoherence and cannot be lowered by input-state choice. The bound is proven for two-mode Gaussian inputs and shown to hold for Fock inputs; the work also analyzes imperfect detection and identifies entanglement-annihilating and entanglement-breaking regimes independent of g_G.

Significance. If the central bound holds, the result supplies a concrete, input-state-independent limit on gravity-induced entanglement generation in pulsed optomechanics, together with explicit amplification mechanisms for nonclassical inputs. The explicit proofs for Gaussian and Fock cases, the identification of detection-modified regimes, and the parameter-free character of the threshold (arising directly from g_G versus thermal terms) are strengths that would guide experimental efforts to observe quantum gravity effects.

major comments (1)

[Abstract] Abstract: the claim that the threshold 'cannot be lowered by any choice of input state' is not supported by the proofs that are explicitly limited to two-mode Gaussian inputs and Fock-state inputs. If the manuscript contains no general argument showing that input-state dependence drops out for arbitrary states (e.g., via properties of the beam-splitter swap or thermal noise that are state-independent), the claim must be qualified to the cases actually proven; otherwise the central assertion about the threshold is over-stated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need to align the abstract claim with the scope of the proofs. We address the single major comment below and will make the corresponding revision.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the threshold 'cannot be lowered by any choice of input state' is not supported by the proofs that are explicitly limited to two-mode Gaussian inputs and Fock-state inputs. If the manuscript contains no general argument showing that input-state dependence drops out for arbitrary states (e.g., via properties of the beam-splitter swap or thermal noise that are state-independent), the claim must be qualified to the cases actually proven; otherwise the central assertion about the threshold is over-stated.

Authors: We agree that the abstract statement is stronger than the explicit proofs provided. The threshold g_G > 2 γ_m N_th originates from the additive thermal noise contributed by the mechanical baths during the interaction time; because this noise is independent of the input state and the beam-splitter swap merely transfers the gravitational phase accumulation to the optical outputs, the minimal coupling required to overcome the noise is expected to be state-independent. Nevertheless, we have only derived the bound rigorously for two-mode Gaussian states and verified it for Fock states. To correct the overstatement we will (i) revise the abstract to read that the threshold “cannot be lowered by the choice of Gaussian or Fock input states” and (ii) add a short paragraph after the Fock-state section explaining why the same bound is anticipated for general states on the basis of the state-independent thermal channel. These changes will be implemented in the revised manuscript. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; bound derived from model competition

full rationale

The central bound g_G > 2 γ_m N_th arises directly from the competition between gravitational coupling and thermal decoherence terms under the stated optomechanical beam-splitter model and rectangular-pulse assumptions. The paper explicitly proves the bound for two-mode Gaussian inputs and extends it to Fock inputs without reducing the result to a fitted parameter, self-citation chain, or definitional equivalence. No load-bearing self-citations, ansatz smuggling, or renaming of known results are indicated; the derivation remains independent of the target claim and self-contained against the model inputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Ledger inferred solely from abstract statements since full text unavailable; no explicit free parameters or invented entities named.

assumptions (2)

domain assumption Optomechanical interaction realizes a beam-splitter state swap between incident optical pulse and mechanical mode
Stated as the basis for the two-pulse protocol in the abstract.
domain assumption Thermal decoherence is the dominant competing process with gravitational coupling
The threshold g_G > 2 γ_m N_th is defined via this competition.

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Cite this review

Pith. "Pith review of Amplification and generation bounds of gravity-induced entanglement in pulsed optomechanical systems." pith.science (2026). https://pith.science/paper/7B5W7AY3

@misc{pith2026260526240,
  author       = {Pith},
  title        = {Pith review of: Amplification and generation bounds of gravity-induced entanglement in pulsed optomechanical systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7B5W7AY3}},
  note         = {Machine review of arXiv:2605.26240}
}

abstract

We investigate gravity-induced entanglement between the output optical fields of two red-detuned pulsed optomechanical systems with their masses coupled by mutual gravitational interaction. For each individual system, the optomechanical interaction realizes a beam-splitter state swap between an incident optical pulse and its mechanical mode. Using two rectangular pulses for each system -- the first to imprint a nonclassical state on the mechanical modes and the second to read the gravitationally generated entanglement back onto the outgoing light -- we show that the amount of entanglement can be amplified by preparing the input in a squeezed or Fock state. However, the threshold for entanglement generation is set by the competition between the gravitational coupling and thermal decoherence, $g_G>2\gamma_m N_{\rm th}$, and cannot be lowered by any choice of input state. We prove this bound for two-mode Gaussian inputs and show that it continues to hold for Fock-state inputs. We further analyze how imperfect detection modifies the threshold and identify the entanglement-annihilating and entanglement-breaking regimes, which are set by the thermal decoherence accumulated over the interaction time, independent of the gravitational coupling.

Figures

Figures reproduced from arXiv: 2605.26240 by the authors.

**Figure 2.** FIG. 2. Behavior of entanglement negativity with the thermal noise. We fix the gravitational coupling [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. Entanglement negativity for the squeezed input with [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Behavior of entanglement negativity with measurement loss. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. Entanglement region in the ( [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. E [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

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Reference graph

Works this paper leans on

67 extracted references · 5 canonical work pages

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This proves that Eq

In the regime γmtG ≪g GtG ≪1, this inequality impliesg G ≤2γ mNth. This proves that Eq. (33) is sufficient for separability preservation. Moreover, forΩ 2 ≥0, the conditionΩ≥0 is also neces- sary. We show the details of the proof in Appendix C. In the regimeγ mtG ≪g GtG ≪1, we haveΩ 2 ≥0. Hence, the con- dition 2γmNth ≥g G is necessary and sufficient cond...
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ˆD(1) B (αAC∗ 12 −α BC∗ 22) ×e iα∗ An11 A +iαAn11† A eiα∗ An12 B +iαAn12† B e−iα∗ Bn21 A −iαBn21† A e−iα∗ Bn22 B −iαBn22† B .(B7) Hence, the characteristic function is obtained by χout(αA, αB)=exp " −1 2 αAα∗ B(C∗ 11C21 +C ∗ 12C22)+ 1 2 α∗ AαB(C11C∗ 21 +C 12C∗ 22) # ×tr[ρ (1) in ˆD(1) A (−αAC∗ 11 +α BC∗
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ˆD(1) B (αAC∗ 12 −α BC∗ 22)] ×tr ρtheiα∗ An11 A +iαAn11† A eiα∗ An12 B +iαAn12† B e−iα∗ Bn21 A −iαBn21† A e−iα∗ Bn22 B −iαBn22† B .(B8) Using Eq. (18), the characteristic function can be decomposed as the product of a pure evolution part and a thermal part as χout(αA, αB)=χ inχth,(B9) where each part is χin =tr[ρ (1) in ˆD(1) A (−αAC∗ 11 +α BC∗
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Since the thermal noise is Gaussian, the commutation relation of the thermal fluctuation is a c-number

ˆD(1) B (αAC∗ 12 −α BC∗ 22)] (B10) χth =tr ρtheiα∗ An11 A +iαAn11† A e−iα∗ Bn21 A −iαBn21† A eiα∗ An12 B +iαAn12† B e−iα∗ Bn22 B −iαBn22† B (B11) Let us compute the thermal partχ th. Since the thermal noise is Gaussian, the commutation relation of the thermal fluctuation is a c-number. Using Wick’s theorem, we have χth =e 1 2 [iα∗ An11 A +iαAn11† A ,−iα∗ ...
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We first consider the lossless case

Lossless case Here, we derive the separability-preservation condition without expanding in the gravitational interaction angleθ≡g GtG. We first consider the lossless case. As discussed in Sec. IV A, the PPT condition for the output covariance matrix can be written as ˜V=V in +Σ PT +Ω≥0, whereΩis defined in Eq. (37). Since any initially separable two-mode ...
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Therefore,Ω≥0 is equivalent toΩ 1 ≥ q Ω2 2 + Ω2
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We next discuss when this sufficient condition is also necessary

Expanding only inγ mtG ≪1, we obtain λmin(Ω)≃ −2|sin[g GtG]|+4γ mtGNth.(C1) Thus, to this order, the sufficient condition for separability preservation is 2γmtGNth ≥ |sin(g GtG)|.(C2) This reduces to 2γmNth ≥g G when the additional short-interaction approximationg GtG ≪1 is used. We next discuss when this sufficient condition is also necessary. We do not ...
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For finiteg GtG, however,Ω 2 is not necessarily positive

This implies that there exists an initially separable squeezed product state for which ˜V̸≥0 and the conditionΩ≥0 is not only sufficient but also necessary for separability preservation. For finiteg GtG, however,Ω 2 is not necessarily positive. Therefore, outside the regimeΩ 2 ≥0, the conditionΩ≥0 should be regarded as a sufficient condition within the pr...

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Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.

[1] [1]

This proves that Eq

In the regime γmtG ≪g GtG ≪1, this inequality impliesg G ≤2γ mNth. This proves that Eq. (33) is sufficient for separability preservation. Moreover, forΩ 2 ≥0, the conditionΩ≥0 is also neces- sary. We show the details of the proof in Appendix C. In the regimeγ mtG ≪g GtG ≪1, we haveΩ 2 ≥0. Hence, the con- dition 2γmNth ≥g G is necessary and sufficient cond...

[2] [2]

ˆD(1) B (αAC∗ 12 −α BC∗ 22) ×e iα∗ An11 A +iαAn11† A eiα∗ An12 B +iαAn12† B e−iα∗ Bn21 A −iαBn21† A e−iα∗ Bn22 B −iαBn22† B .(B7) Hence, the characteristic function is obtained by χout(αA, αB)=exp " −1 2 αAα∗ B(C∗ 11C21 +C ∗ 12C22)+ 1 2 α∗ AαB(C11C∗ 21 +C 12C∗ 22) # ×tr[ρ (1) in ˆD(1) A (−αAC∗ 11 +α BC∗

[3] [3]

ˆD(1) B (αAC∗ 12 −α BC∗ 22)] ×tr ρtheiα∗ An11 A +iαAn11† A eiα∗ An12 B +iαAn12† B e−iα∗ Bn21 A −iαBn21† A e−iα∗ Bn22 B −iαBn22† B .(B8) Using Eq. (18), the characteristic function can be decomposed as the product of a pure evolution part and a thermal part as χout(αA, αB)=χ inχth,(B9) where each part is χin =tr[ρ (1) in ˆD(1) A (−αAC∗ 11 +α BC∗

[4] [4]

Since the thermal noise is Gaussian, the commutation relation of the thermal fluctuation is a c-number

ˆD(1) B (αAC∗ 12 −α BC∗ 22)] (B10) χth =tr ρtheiα∗ An11 A +iαAn11† A e−iα∗ Bn21 A −iαBn21† A eiα∗ An12 B +iαAn12† B e−iα∗ Bn22 B −iαBn22† B (B11) Let us compute the thermal partχ th. Since the thermal noise is Gaussian, the commutation relation of the thermal fluctuation is a c-number. Using Wick’s theorem, we have χth =e 1 2 [iα∗ An11 A +iαAn11† A ,−iα∗ ...

[5] [5]

We first consider the lossless case

Lossless case Here, we derive the separability-preservation condition without expanding in the gravitational interaction angleθ≡g GtG. We first consider the lossless case. As discussed in Sec. IV A, the PPT condition for the output covariance matrix can be written as ˜V=V in +Σ PT +Ω≥0, whereΩis defined in Eq. (37). Since any initially separable two-mode ...

[6] [6]

Therefore,Ω≥0 is equivalent toΩ 1 ≥ q Ω2 2 + Ω2

[7] [7]

We next discuss when this sufficient condition is also necessary

Expanding only inγ mtG ≪1, we obtain λmin(Ω)≃ −2|sin[g GtG]|+4γ mtGNth.(C1) Thus, to this order, the sufficient condition for separability preservation is 2γmtGNth ≥ |sin(g GtG)|.(C2) This reduces to 2γmNth ≥g G when the additional short-interaction approximationg GtG ≪1 is used. We next discuss when this sufficient condition is also necessary. We do not ...

[8] [8]

For finiteg GtG, however,Ω 2 is not necessarily positive

This implies that there exists an initially separable squeezed product state for which ˜V̸≥0 and the conditionΩ≥0 is not only sufficient but also necessary for separability preservation. For finiteg GtG, however,Ω 2 is not necessarily positive. Therefore, outside the regimeΩ 2 ≥0, the conditionΩ≥0 should be regarded as a sufficient condition within the pr...