How Gravity Entangles Quantum Particles: Lessons from the Schrödinger–Newton Model ✦
The Schrödinger–Newton equation describes the quasi-classical evolution of massive quantum systems with self-gravity. A two-particle model is proposed, separating the contributions of local nonlinear self-localization and the non-separable Newtonian pair potential. It is analytically shown that nonlinear self-interaction preserves the Schmidt spectrum, and direct entanglement generation is due solely to the pair potential. Numerical simulation in regularized one-dimensional geometry demonstrates a pronounced dependence of entanglement generation on the initial spatial configuration and mass ratio. Strongly localized self-bound wavepackets barely increase entanglement during scattering, whereas spatial delocalization and kinetic dispersion broaden the interaction region, enhancing the pair potential's ability to generate entanglement and excite higher modes. For dispersive Gaussian initial states, mass asymmetry leads to fragmentation of the lighter particle, causing Wigner negativity and rapid entanglement growth. Stationary Schrödinger–Newton profiles significantly suppress this effect, allowing the contribution from the pure pair potential to be isolated.
Context
The question of whether classical gravity can generate quantum entanglement lies at the heart of modern searches for a quantum theory of gravity. If the gravitational field acts as a classical mediator, it should not increase the Schmidt number of separable states. However, the Schrödinger–Newton model, which combines quantum evolution with a Newtonian self-gravitational potential, offers a different perspective. Its two-particle generalization allows us to separate the effects of nonlinear self-localization and mutual attraction, clarifying which mechanism is responsible for the emergence of quantum correlations.
Methods
A phenomenological two-particle equation was used for analysis, where self-gravitational potentials depend on marginal densities, and the direct pair interaction is given by a non-separable operator. Numerical simulation was performed in a one-dimensional periodic system with a softened Coulomb kernel; the evolution was calculated using a split-step method with spectral accuracy. Four initial configurations were considered: localized product, delocalized product, and two superposition “cat” states. For each, the von Neumann entropy and Wigner functions in phase space were computed.
Results
It was found that at zero pair interaction, no entanglement arises, confirming the isospectrality of self-interaction. For stationary self-gravitating profiles, entanglement growth is minimal (S ≈ 0.19 after collisions), while dispersing Gaussian packets lead to significant generation (S ≈ 0.87). In delocalized and superposition states, entropy reaches S ≈ 1.5–1.7, indicating the excitation of higher modes beyond the initial two-level subspace. Under mass asymmetry, the lighter particle undergoes “gravitational shattering,” sharply enhancing decoherence and quantum information capacity.
Implications
The work demonstrates that any semiclassical model of gravity that combines wave function collapse à la Diósi–Penrose with Newtonian interaction must explicitly specify whether mutual attraction is a separability-preserving mean field or a non-separable pair operator. This has direct relevance to testing the quantum nature of gravity through experiments with levitated optomechanical systems.
Future development
Future research will include two- and three-dimensional geometries with non-zero impact parameters, as well as the addition of external disorder to explore connections with Anderson localization. Full-scale 3D modeling is necessary to assess whether gravitational entanglement can be isolated in real experiments.
Impact
The results will impact quantum information technologies and fundamental physics, especially the development of quantum gravity tests with massive particles.
Next steps
The next step is to move to realistic three-dimensional simulations and include additional effects such as external fields and noise, in order to quantitatively evaluate the feasibility of observing gravitationally induced entanglement in the laboratory.
Key open problems
The study is directly connected to the measurement problem in quantum mechanics and the question of quantizing the gravitational field. If gravity can entangle particles, it would be an argument for its quantum essence, which has profound implications for unifying general relativity and quantum theory.
🎯 The Schrödinger–Newton equation was originally proposed as a model of spontaneous wave function collapse, explaining the transition from the quantum to the classical world, and today it is also used to describe the dynamics of dark matter and boson stars.
Key numbers
- maximum entropy (Gaussian packets, localized product): 0.87
- maximum entropy (stationary profiles, localized product): 0.19
- entropy for delocalized product: 1.58
- entropy for anti-correlated superposition: 1.67
- mass ratio in asymmetric collisions: 4:1