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Classical Self-Gravity Breaks Quantum State Tomography

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2026-07-09 00:49 UTC pith:IEUXXAPR

load-bearing objection Novel SN tomographic signatures (angle-set dependence, below-Heisenberg covariance) are real and cleanly derived within the CCSN framework; the framework itself is the load-bearing assumption. the 1 major comments →

arxiv 2607.06967 v1 pith:IEUXXAPR submitted 2026-07-08 quant-ph gr-qc

Tomography of a Macroscopic Quantum State influenced by Classical Self-Gravity

Wenjie Zhong , Yubao Liu , Yanbei Chen , Haixing Miao , Yiqiu Ma This is my paper
classification quant-ph gr-qc PACS 03.65.Ta03.65.Yz04.60.-m42.50.-p
keywords Schrödinger-Newton equationquantum state tomographysemi-classical gravityoptomechanicsHeisenberg uncertainty principlenonlinear quantum mechanicscontinuous measurementgravitational self-energy
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.

This paper studies what happens when you try to reconstruct the quantum state of a macroscopic mechanical oscillator using standard quantum-mechanical data analysis, but the oscillator's motion is actually influenced by its own classical gravitational field—the Schrödinger-Newton effect. The central object is the tomographic reconstruction map: the mathematical procedure that converts continuous optical measurement records into an inferred quantum-state covariance matrix. In standard quantum mechanics (or if gravity is quantized), this map is well-behaved: the reconstructed covariance is independent of which measurement angles you choose, and the inferred state always satisfies the Heisenberg uncertainty principle. The paper shows that when the underlying dynamics include classical self-gravity, the same reconstruction map acquires a state-dependent correction that depends on the conditional covariance trajectory during measurement. This correction makes the reconstructed covariance depend on the specific set of measurement angles chosen and can push the inferred covariance below the Heisenberg uncertainty bound or even make it non-positive-definite. The authors quantify the distinguishability between quantum-gravity and Schrödinger-Newton reconstructions using the Hellinger distance across measurement-strength and temperature parameter space, finding that the mismatch is strongest at low temperature and moderate measurement strength. They then generalize the result: any nonlinear quantum dynamics where the state being measured also controls the measurement process itself will produce analogous corrections to the standard tomographic reconstruction map, breaking the angle-set consistency that ordinary quantum tomography guarantees.

Core claim

The paper's central discovery is that the Schrödinger-Newton self-gravity correction to continuous optomechanical tomography, denoted sigma^2_SN, depends on the conditional covariance trajectory V_c(t) which is itself shaped by the chosen homodyne measurement angle. Because this correction is a functional of the same state parameters being reconstructed and varies with the measurement setting, applying the standard quantum-mechanical reconstruction map to Schrödinger-Newton-generated data produces three diagnostic signatures: (1) the reconstructed covariance matrix changes when different sets of tomography angles are used, unlike in standard quantum mechanics where the result is angle-set-in

What carries the argument

The Schrödinger-Newton self-gravity term in the center-of-mass Hamiltonian, M*omega_SN^2*(x_hat - <x_hat>)^2, sourced by the conditional mean position x_c = <psi_c|x_hat|psi_c> during continuous homodyne measurement. The conditional covariance matrix V_c(t) evolves via a Riccati equation at the SN-modified frequency omega_q = sqrt(omega_m^2 + omega_SN^2), and the SN tomographic correction sigma^2_SN[g1,g2] = -omega_SN^2 * integral(j2(t) * [hxx(t|theta)*j1(t) + hxp(t|theta)*j2(t)]) inherits the angle-dependence of this trajectory.

Load-bearing premise

The entire analysis depends on the Causal Conditional Schrödinger-Newton prescription, where the classical gravitational potential sources the conditional mean position x_c = <psi_c|x_hat|psi_c> rather than the unconditional expectation value. If a different prescription for how classical gravity couples to the conditional quantum state were correct, the angle-dependence and below-Heisenberg-bound signatures could change qualitatively or disappear.

What would settle it

If the conditional mean prescription for the Schrödinger-Newton potential is replaced by one where gravity sources the unconditional density, the tomographic correction sigma^2_SN would no longer depend on the conditional covariance trajectory in the same way, and the angle-set dependence and below-bound signatures could vanish.

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

If this is right

  • An experiment that reconstructs a macroscopic oscillator's quantum state at multiple angle sets and finds inconsistent covariance matrices would have evidence that the underlying dynamics are not purely standard quantum mechanics, potentially pointing to classical self-gravity or other nonlinear effects.
  • The below-Heisenberg-bound or non-positive-definite reconstructed covariance is a model-mismatch diagnostic: it does not mean the uncertainty principle is violated, but rather that the assumed linear reconstruction map is incompatible with the actual dynamics that generated the data.
  • Self-consistent tomography under Schrödinger-Newton dynamics cannot proceed by simply adding a correction term to the standard filter; it requires solving a nonlinear inverse problem where the unknown state parameters appear inside the reconstruction map itself.
  • The angle-set consistency test could serve as a theory-agnostic diagnostic for nonlinear quantum dynamics beyond the specific Schrödinger-Newton case: any theory where the measurement process depends on the state being inferred will break Radon consistency.

Where Pith is reading between the lines

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

  • The angle-set consistency test could be applied to other proposed nonlinear modifications of quantum mechanics—such as spontaneous collapse models or post-quantum theories of classical gravity—to check whether they produce analogous tomographic signatures without requiring knowledge of the specific nonlinear correction.
  • If the Schrödinger-Newton frequency omega_SN is extremely small relative to the mechanical frequency, the tomographic signatures vanish, which means the test is complementary to spectral-based SN tests: tomography probes the conditional dynamics during measurement rather than steady-state oscillation frequencies.
  • An adaptive or Bayesian tomography protocol that iteratively refines its estimate of the initial covariance and propagates the SN conditional equations for each candidate could in principle perform self-consistent SN tomography, but whether such a protocol is experimentally efficient at the required parameter regimes remains an open question.

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

1 major / 6 minor

Summary. This manuscript investigates how classical self-gravity, described by the Schrödinger-Newton (SN) theory in its Causal Conditional (CCSN) formulation, affects continuous quantum state tomography of a macroscopic mechanical oscillator. The authors derive the tomographic error functional in a Schrödinger-picture framework, showing that when a standard quantum-gravity (QG) optimized reconstruction map is applied to measurement data generated by SN dynamics, an additional state-dependent correction arises. This correction depends on the conditional covariance trajectory, which is itself angle-dependent, leading to three signatures: (1) the reconstructed covariance depends on the chosen set of tomography angles, (2) the inferred covariance can be driven outside the standard Gaussian-covariance domain (below the Heisenberg bound or even non-positive-definite), and (3) the QG-SN distinguishability, quantified by the Hellinger distance, exhibits nontrivial dependence on measurement strength and temperature. The paper also frames these results as a concrete instance of a general obstruction in tomography of nonlinear quantum mechanics.

Significance. The paper addresses a timely question in the quantum-gravity phenomenology program: how to operationally distinguish semi-classical gravity from quantum gravity using macroscopic optomechanical systems. The derivation from the stochastic master equation through the Riccati equation to the explicit SN correction (Eq. 44) is careful and internally consistent, with a clean algebraic identity (Eq. 66) demonstrating the angle-independence of the QG reconstruction. The generalization to nonlinear quantum mechanics in Sec. V, connecting the SN result to a Radon-consistency obstruction, is a conceptual contribution that extends the reach of the paper beyond the specific SN setting. The falsifiable predictions (Figs. 3, 5) and the honest assessment of experimental feasibility are commendable. The key physical insight—that the SN correction is not positive-semidefinite and therefore cannot be absorbed as ordinary added noise—is the central novel result.

major comments (1)
  1. Sec. II.B, Eqs. (10) and (21): The CCSN prescription, where the SN self-gravity potential sources from the conditional mean x_c = ⟨ψ_c|x̂|ψ_c⟩, is the load-bearing assumption of the paper. Under this prescription, the conditional mean evolves at ω_m (Eq. 10, drift A_m) while the conditional covariance evolves at ω_q (Eq. 21, drift A_q), producing the frequency split A_SN = A_q - A_m (Eq. 32) that generates the entire σ²_SN correction. The paper acknowledges (citing Refs. 47-48) that alternative thermal-noise prescriptions exist, but it does not discuss whether the CCSN prescription itself is the consensus or a contested choice, nor how the central results would change under an alternative semi-classical gravity coupling (e.g., one where the gravitational potential sources the unconditional density). Since the angle-dependence and below-Heisenberg-bound signatures are structurally tied to
minor comments (6)
  1. Sec. II.C, Eq. (46): The approximation sign is used without a clear statement of what is being approximated. The text mentions performing an integral by parts, but the conditions under which the remaining terms are negligible should be specified.
  2. Table I: The parameters (M = 0.2 kg, ω_m/2π = 4×10⁻³ Hz, Q = 10⁷) are extremely challenging. The paper acknowledges this, but a brief quantitative estimate of the required measurement strength relative to the SN frequency would help the reader assess the regime of validity.
  3. Fig. 2: The two angle sets Θ_A = {0, π/4, π/2} and Θ_B = {π/3, 2π/3, π} differ by a common shift of π/3. It would be instructive to also show a case where the angle sets are not related by a simple rotation, to demonstrate that the angle-dependence is not an artifact of a particular symmetry.
  4. Sec. V, Eqs. (74)-(76): The general NLQM formulation is schematic. A brief comment on whether the SN nonlinearity satisfies the specific conditions discussed by Weinberg (Ref. 60) and Gisin (Ref. 61) would help situate the result.
  5. References: Ref. [21] is cited as a 2026 arXiv preprint; please verify the date and completion status.
  6. Typo in Acknowledgments: 'Y. M. W. Z. and Y. L. is supported' should read 'are supported'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies the CCSN prescription as the load-bearing assumption of the paper and asks for a discussion of its status and of how the central results would change under alternative semi-classical gravity couplings. We agree this discussion should be added and explain below how we will revise the manuscript.

read point-by-point responses

  1. Referee: Sec. II.B, Eqs. (10) and (21): The CCSN prescription, where the SN self-gravity potential sources from the conditional mean x_c = ⟨ψ_c|x̂|ψ_c⟩, is the load-bearing assumption of the paper. Under this prescription, the conditional mean evolves at ω_m (Eq. 10, drift A_m) while the conditional covariance evolves at ω_q (Eq. 21, drift A_q), producing the frequency split A_SN = A_q - A_m (Eq. 32) that generates the entire σ²_SN correction. The paper acknowledges (citing Refs. 47-48) that alternative thermal-noise prescriptions exist, but it does not discuss whether the CCSN prescription itself is the consensus or a contested choice, nor how the central results would change under an alternative semi-classical gravity coupling (e.g., one where the gravitational potential sources the unconditional density). Since the angle-dependence and below-Heisenberg-bound signatures are structurally tied to

    Authors: The referee is correct that the CCSN prescription is the load-bearing assumption and that this point deserves explicit discussion. We will add a dedicated paragraph in Sec. II.B addressing the following points. revision: yes

  2. Referee: [Continued from above — the comment was cut off, but the thrust is: how do the central results change under alternative semi-classical gravity couplings, and is CCSN consensus or contested?]

    Authors: We address the two sub-questions in turn. (1) Status of CCSN: The CCSN prescription is not the unique formulation of semi-classical gravity; it is one of several prescriptions discussed in the literature (Refs. 20, 45–48). The key distinction is whether the gravitational source is the conditional state |ψ_c⟩⟨ψ_c| (as in CCSN) or the unconditional density matrix ρ = E[|ψ_c⟩⟨ψ_c|]. The CCSN choice is motivated by the requirement of causality: the gravitational field at time t should be sourced by the matter distribution at time t, not by a future ensemble average. This is the position taken in Refs. 20, 45–46 and further discussed in Ref. [46] (Miki et al., 2025). However, we agree that this is a contested choice, not a settled consensus, and the manuscript should say so explicitly. (2) How results change under alternative prescriptions: The frequency split A_SN = A_q − A_m arises specifically because, under CCSN, the conditional mean evolves at ω_m (the SN force vanishes on the mean trajectory) while the conditional covariance evolves at ω_q. Under a prescription where the gravitational potential sources the unconditional density, the situation is structurally different: the unconditional covariance would evolve at ω_q, but the conditional covariance evolution would not necessarily carry the same A_SN correction during the measurement process. In that case, the specific σ²_SN correction derived in Eq. (44) would take a different form, and the angle-dependence and below-Heisenberg-bound signatures may be weakened or absent. However, the general mechanism identified in Sec. V — that state-dependent nonlinear dynamics during readout introduces a model-dependent correction to the tomographic map — does not depend on the CCSN prescription specifically. Any semi-classical orhy revision: no

Circularity Check

0 steps flagged

No significant circularity: the SN tomographic correction is derived from the CCSN equations of motion without fitting to the target result; self-citations provide the model framework but are not load-bearing in a circular way.

full rationale

The paper's central result — that σ²_SN depends on tomography angles and can drive the QG-filtered reconstructed covariance outside the Gaussian domain — is derived from the CCSN equations of motion (Eq. 1, Eq. 10, Eq. 21) through a careful integration-by-parts calculation (Eqs. 28-44) without fitting any parameter to produce the target signature. The key structural ingredient, A_SN = A_q − A_m (Eq. 32), arises because under CCSN the SN force vanishes at the conditional mean, so the mean evolves at ω_m while the covariance evolves at ω_q. This is a physical consequence of the model choice, not a definition that presupposes the conclusion. The self-citations (Refs. 20, 45, 46, by overlapping authors) provide the CCSN framework, but the paper is transparent that this is a model prescription, explicitly acknowledges alternative prescriptions (Refs. 47-48), and states that excluding SN signatures 'would only rule out the SN description in the present setting.' The tomographic analysis (Eqs. 40-67), the Hellinger distance quantification (Eq. 73), and the general NLQM formulation (Sec. V) are new contributions. The parameters in Table I are set from physical considerations, not adjusted to produce below-Heisenberg-bound signatures. The one minor concern is that the CCSN prescription itself is load-bearing for the specific structure of σ²_SN, but this is a model-validity concern (correctness risk), not circularity — the paper does not claim CCSN is uniquely correct or derive it from itself. Score 1 reflects the presence of load-bearing self-citations for the framework, with the central derivation remaining independent.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

No new particles, forces, dimensions, or conserved quantities are introduced. The SN theory and CCSN prescription are from prior literature. The paper's contribution is the analysis of their consequences for tomography.

free parameters (4)
  • ω_SN/(2π) = 7.8×10⁻² Hz
    SN frequency for 0.2 kg osmium mirror, set by the material density and mass; not fitted to tomography data but chosen to make the effect visible in the parameter scan.
  • ω_m/(2π) = 4×10⁻³ Hz
    Bare mechanical frequency, chosen low to enhance relative SN contribution; acknowledged as experimentally challenging.
  • Q = 10⁷
    Mechanical quality factor, chosen to keep thermal occupation manageable; authors note this challenges experimental feasibility.
  • V₀ (initial squeezed state) = V_xx=0.2ℏ/2Mω_q, V_xp=ℏ/2, V_pp=10ℏMω_q/2
    Initial Gaussian squeezed state parameters chosen to satisfy the quadratic SN potential validity condition (Δx_c.m. ≪ Δx_zp) while making the SN effect visible.
axioms (4)
  • domain assumption The Causal Conditional Schrödinger-Newton (CCSN) prescription: the SN self-gravity potential sources the conditional mean x_c = ⟨ψ_c|x̂|ψ_c⟩ during measurement.
    Invoked in Sec. I (Eq. 1) and Sec. II.B (Eq. 10). This is the specific prescription from Refs. 20, 45-46; alternative prescriptions exist (Refs. 47-48). The entire SN correction σ²_SN depends on this choice.
  • domain assumption Classical thermal noise prescription: thermal Langevin force is treated as classical force noise affecting the quantum trajectory, while conditional covariance is only affected by quantum radiation pressure noise.
    Stated in Sec. II.B after Eq. 13. Refs. 47-48 note two different prescriptions; this choice affects the Riccati equation (Eq. 21) and hence the SN correction.
  • standard math High-temperature limit k_BT/ℏω_m ≫ 1 for the thermal bath.
    Used in Eq. 12-13 to justify the classical thermal noise correlation function. Valid for the parameter regime considered (mHz frequencies, mK temperatures give k_BT/ℏω_m ~ 10⁶).
  • domain assumption Quadratic SN Hamiltonian validity: Δx_c.m. ≪ Δx_zp (CoM uncertainty much smaller than atomic zero-point fluctuation).
    Stated in Sec. III.A; verified numerically (Δx_c.m./Δx_zp ~ 10⁻⁴). Required for the SN Hamiltonian (Eq. 1) to be a valid approximation.

pith-pipeline@v1.1.0-glm · 30487 in / 3432 out tokens · 406645 ms · 2026-07-09T00:49:02.427592+00:00 · methodology

0 comments
read the original abstract

Macroscopic optomechanical systems offer a promising testbed for distinguishing whether gravity acts as a quantum interaction or as a classical field. Schrodinger-Newton (SN) theory is the nonrelativistic limit of semi-classical gravity where quantum matter couples to classical gravity. Based on SN theory, this work investigates how classical self-gravity affects continuous quantum state tomography of a macroscopic mechanical oscillator monitored by variable-angle homodyne detection. In the Schrodinger-Newton (SN) theory, the measurement record arises from a different conditional test mass dynamics from that in quantum-gravity (QG)/standard quantum mechanics, consequently, applying the QG-optimised reconstruction map introduces an additional state-dependent contribution. We show that this contribution makes the reconstructed covariance depend on the chosen set of tomography angles and can drive the SN covariance--after QG filtering--outside the standard Gaussian-covariance domain set by the Heisenberg uncertainty principle. We quantify the resulting QG-SN distinguishability via the Hellinger distance and analyse its dependence on measurement strength and temperature. We then formulate the same issue in the broader setting of nonlinear quantum mechanics: when the system's conditional dynamics during the readout process depends on the state being inferred, the tomographic map acquires nonlinear, model-dependent corrections to the usual Radon or Gaussian reconstruction map.

Figures

Figures reproduced from arXiv: 2607.06967 by Haixing Miao, Wenjie Zhong, Yanbei Chen, Yiqiu Ma, Yubao Liu.

Figure 1
Figure 1. Figure 1: FIG. 1. Concept of tomography for a self-gravitating mechanical [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dependence of the reconstructed covariance on the choice of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Gaussian-covariance consistency of the SN reconstruction [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Tomographic error-matrix components at two different temperatures. Panel (a) shows the low-temperature [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Distinguishability of the QG and SN reconstructions in [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Low-temperature slice of the reconstruction at [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. High-temperature slice of the reconstruction at [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

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