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arxiv: 2605.06577 · v1 · submitted 2026-05-07 · 🪐 quant-ph · gr-qc

Entanglement generation in a two-body Schr\"odinger--Newton model

Marcin P{\l}odzie\'n , Julia Os\k{e}ka-Lenart , Maciej Lewenstein , Micha{\l} Eckstein This is my paper

Pith reviewed 2026-05-08 11:00 UTC · model grok-4.3

classification 🪐 quant-ph gr-qc
keywords Schrödinger-Newton equationquantum entanglementself-gravitating systemstwo-body modelNewtonian pair potentialnumerical simulationsSchmidt spectrum
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The pith

The nonseparable Newtonian pair potential generates entanglement in a two-body Schrödinger-Newton model while the nonlinear self-field preserves the Schmidt spectrum.

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

This paper introduces a two-body version of the Schrödinger-Newton equation that isolates the local nonlinear self-gravity from the mutual gravitational interaction. It shows analytically that only the nonseparable pair potential produces entanglement between the particles. Numerical work in one dimension then reveals that the amount of entanglement generated depends strongly on how spread out the initial wave packets are and on the ratio of the two masses. Localized, self-bound states barely entangle during scattering, while delocalized or dispersive states entangle more readily, especially when the masses are unequal.

Core claim

In the two-body Schrödinger-Newton model, the nonlinear self-field term preserves the Schmidt spectrum of the joint wave function, so it generates no entanglement. Entanglement is produced solely by the nonseparable Newtonian pair potential. In regularized one-dimensional simulations, highly localized stationary SN profiles experience minimal entanglement growth, whereas dispersive Gaussian states show mass-asymmetric shattering of the lighter particle, Wigner negativity, and rapid entanglement increase.

What carries the argument

Separation of the local nonlinear self-field from the nonseparable pair potential in the two-body Schrödinger-Newton equation.

If this is right

  • Stationary self-localized wave packets minimize entanglement growth during gravitational scattering.
  • Mass asymmetry combined with initial dispersion produces rapid entanglement growth and Wigner negativity in the lighter particle.
  • Spatial delocalization broadens the interaction region, excites higher spatial modes, and amplifies the entangling effect of the pair potential.
  • Stationary SN profiles isolate the bare contribution of the pair potential to entanglement generation.

Where Pith is reading between the lines

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

  • If the one-dimensional results carry over to three dimensions, laboratory tests of gravitational entanglement could prioritize control of initial wave-packet delocalization.
  • The mass ratio acts as a tunable knob that can enhance or suppress entanglement in proposed massive-particle gravity experiments.
  • Self-gravitational localization may act as a natural mechanism to limit unwanted entanglement in quantum systems interacting gravitationally.

Load-bearing premise

The semiclassical Schrödinger-Newton framework remains valid for the two particles, together with the specific regularization and reduction to one spatial dimension.

What would settle it

A full three-dimensional simulation of the two-body Schrödinger-Newton equation without regularization in which entanglement growth vanishes when the pair potential is made artificially separable would falsify the claim that the nonseparable pair potential alone drives entanglement.

Figures

Figures reproduced from arXiv: 2605.06577 by Julia Os\k{e}ka-Lenart, Maciej Lewenstein, Marcin P{\l}odzie\'n, Micha{\l} Eckstein.

Figure 1
Figure 1. Figure 1: Collision dynamics for the localized product state view at source ↗
Figure 2
Figure 2. Figure 2: Collision dynamics for the localized product state view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of the delocalized product state view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the correlated superposition state view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of the anticorrelated superposition state view at source ↗
Figure 6
Figure 6. Figure 6: Entanglement and Schmidt spectrum dynamics comparing Gaussian wavepackets (solid view at source ↗
Figure 7
Figure 7. Figure 7: Mass-asymmetric localized-product collision with view at source ↗
Figure 8
Figure 8. Figure 8: Gravitational shattering and entanglement generation in mass-asymmetric collisions. (a,c) view at source ↗
read the original abstract

The Schr\"odinger--Newton (SN) equation provides a semiclassical framework for the evolution of self-gravitating of massive quantum systems. We propose a two-body Schr\"odinger--Newton model that separates local nonlinear self-localization from the nonseparable Newtonian pair potential. Analytically, we show that the nonlinear self-field preserves the Schmidt spectrum, whereas direct entanglement generation arises from the nonseparable pair potential. Using numerical simulations in a regularized one-dimensional geometry, we find that entanglement generation depends sensitively on the initial spatial configuration and on the mass ratio. Highly localized, self-bound wavepackets experience minimal entanglement growth during scattering. Spatial delocalization and kinetic dispersion broaden the interaction region, amplifying the entangling power of the pair potential and exciting higher-order spatial modes. For dispersive Gaussian initial states, mass asymmetry shatters the lighter particle, producing Wigner negativity and rapid entanglement growth, whereas stationary SN profiles strongly suppress this effect. Stationary SN profiles isolate the bare pair-potential contribution; dispersive Gaussian initial states inflate it.

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 paper introduces a two-body Schrödinger-Newton model that analytically separates the local nonlinear self-localization term (which preserves the Schmidt spectrum of the two-particle state) from the nonseparable Newtonian pair potential (which generates entanglement). Numerical simulations in a regularized one-dimensional geometry then show that entanglement growth is highly sensitive to the initial spatial support and the mass ratio: stationary self-bound SN profiles exhibit minimal entanglement, while dispersive Gaussian states with mass asymmetry produce rapid entanglement increase, Wigner negativity, and shattering of the lighter particle.

Significance. If the central separation and numerical trends hold, the work clarifies how gravitational self-interaction contributes to entanglement in multi-particle quantum systems within the semiclassical SN framework. The analytical spectrum-preservation argument is a clear strength, as is the explicit demonstration that stationary SN profiles isolate the bare pair-potential contribution while dispersive states amplify it. These results could inform future studies of gravity-induced decoherence or entanglement, though their generality is currently limited by the 1D setting.

major comments (2)
  1. [Numerical simulations] Numerical simulations section: the regularization procedure for the 1D Newtonian kernel is not fully specified (no explicit value or functional form for the regularization parameter, no convergence tests with respect to grid size or cutoff, and no error bars on entanglement measures or Wigner negativity). These omissions are load-bearing for the claims that entanglement depends sensitively on initial configuration and mass ratio.
  2. [Numerical simulations] Numerical simulations section: the reduction to a regularized 1D geometry replaces the 3D 1/r potential with a linear or logarithmic kernel and restricts the spatial-mode basis, qualitatively altering the interaction range and higher-mode excitation. The manuscript does not discuss whether the reported phenomena (minimal entanglement for stationary profiles, rapid growth and Wigner negativity for asymmetric dispersive Gaussians) survive in 3D or are artifacts of the dimensionality reduction.
minor comments (2)
  1. The value chosen for the regularization parameter and all other simulation parameters (grid spacing, time step, initial widths, mass ratios) should be stated explicitly so that the numerics can be reproduced.
  2. A brief comparison of the 1D regularized potential to the 3D case, even if qualitative, would help readers assess the robustness of the entanglement-generation mechanism.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for recognizing the value of the analytical separation between the local self-localization term and the nonseparable pair potential. We address the two major comments on the numerical simulations below and will revise the manuscript to improve clarity and rigor in that section.

read point-by-point responses

  1. Referee: [Numerical simulations] Numerical simulations section: the regularization procedure for the 1D Newtonian kernel is not fully specified (no explicit value or functional form for the regularization parameter, no convergence tests with respect to grid size or cutoff, and no error bars on entanglement measures or Wigner negativity). These omissions are load-bearing for the claims that entanglement depends sensitively on initial configuration and mass ratio.

    Authors: We agree that the regularization details must be stated explicitly to support the reported sensitivity of entanglement growth to initial conditions and mass ratio. In the revised manuscript we will specify the functional form of the regularized 1D kernel, the numerical value of the regularization length scale used throughout the simulations, the results of convergence tests with respect to grid spacing and cutoff, and error bars (or uncertainty estimates) on the entanglement entropy and Wigner negativity. These additions will be placed in the Numerical simulations section and will not alter the qualitative trends already presented. revision: yes

  2. Referee: [Numerical simulations] Numerical simulations section: the reduction to a regularized 1D geometry replaces the 3D 1/r potential with a linear or logarithmic kernel and restricts the spatial-mode basis, qualitatively altering the interaction range and higher-mode excitation. The manuscript does not discuss whether the reported phenomena (minimal entanglement for stationary profiles, rapid growth and Wigner negativity for asymmetric dispersive Gaussians) survive in 3D or are artifacts of the dimensionality reduction.

    Authors: We acknowledge that the 1D reduction changes the interaction kernel and restricts the mode structure relative to three dimensions. The analytical result that the local nonlinear self-field leaves the Schmidt spectrum invariant holds in any dimension and is independent of the numerical implementation. We will add a dedicated paragraph in the revised manuscript that (i) states the rationale for the 1D regularized model as a controlled setting in which the pair-potential contribution can be isolated, (ii) notes that quantitative entanglement rates will differ in 3D, and (iii) argues that the qualitative dependence on initial delocalization and mass asymmetry should persist because it originates from the non-separability of the pair potential rather than from the specific form of the kernel. A full 3D numerical study lies beyond the scope of the present work. revision: partial

Circularity Check

0 steps flagged

No circularity: analytical separation and direct numerical evolution are self-contained

full rationale

The paper's central claims rest on an explicit separation of the two-body Schrödinger-Newton equation into a local nonlinear self-field term and a nonseparable pair potential, followed by an analytical demonstration that the former preserves the Schmidt spectrum while the latter generates entanglement, plus direct numerical integration of the regularized 1D dynamics. No step defines a quantity in terms of another that is then treated as a prediction, no parameters are fitted to subsets of results and repurposed, and no load-bearing uniqueness or ansatz is imported via self-citation. The derivation chain is independent of its inputs and does not reduce by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on the background validity of the semiclassical SN equation and on an ad-hoc separation into self and pair terms; a regularization parameter is introduced for the 1D numerics but its value is not reported. No new physical entities are postulated.

free parameters (1)
  • regularization parameter
    Introduced to handle singularities in the 1D geometry; value and sensitivity not specified in abstract.
axioms (2)
  • domain assumption Semiclassical Schrödinger-Newton equation is an adequate description for the masses and regimes considered
    Taken as the starting framework without further justification in the abstract.
  • ad hoc to paper The local nonlinear self-localization term can be cleanly separated from the nonseparable Newtonian pair potential
    Proposed by the authors to isolate entanglement sources.

pith-pipeline@v0.9.0 · 5500 in / 1407 out tokens · 35759 ms · 2026-05-08T11:00:10.186336+00:00 · methodology

discussion (0)

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Reference graph

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