[Paper Review] Entanglement of quantum fields via classical gravity

Marcel Reginatto, Michael J. W. Hall|arXiv (Cornell University)|Sep 13, 2018

Quantum Mechanics and Applications21 references3 citations

TL;DR

This paper demonstrates that quantum fields can become entangled through interaction with a classical gravitational field alone, using the configuration ensemble formalism to model classical-quantum coupling. Explicit relativistic calculations in spherical symmetry show entanglement generation from initially unentangled states, implying that observed entanglement cannot definitively prove gravity's nonclassical nature without additional assumptions about the hybrid model.

ABSTRACT

We consider the coupling of quantum fields to classical gravity in the formalism of ensembles on configuration space, a model that allows a consistent formulation of interacting classical and quantum systems. Explicit calculations show that there are solutions for which two quantum fields are in an entangled state, even though their interaction occurs solely via a common classical gravitational field, and that such entangled solutions can evolve from initially unentangled ones. These results support the observation of a previous paper that an observed generation of entanglement would not provide a definitive test of the nonclassical nature of gravity.

Motivation & Objective

• To investigate whether entanglement between quantum fields can emerge via a common classical gravitational field, without direct interaction.
• To assess whether observed entanglement in proposed gravity experiments could rule out classical gravity models.
• To extend previous non-relativistic results to a fully relativistic setting using the configuration ensemble formalism.
• To examine the viability of hybrid classical-quantum models as approximations toward a full quantum gravity theory.
• To explore the dynamical evolution of entanglement from initially separable states in a gravitational background.

Proposed method

• Employing the configuration ensemble formalism to describe coupled classical and quantum systems, using probability density P and conjugate field S on configuration space.
• Formulating a hybrid Hamiltonian that couples two quantized scalar fields to a classical gravitational field via the Einstein equations in spherical symmetry.
• Deriving an effective non-linear functional Schrödinger equation (Eq. 70) for the quantum fields, including a non-local correction term Δ arising from gravitational degrees of freedom.
• Using a weak-field approximation and introducing gravitational time to define time evolution in the curved background.
• Applying perturbative methods to solve the evolution equation and analyze entanglement generation from initial product states.
• Analyzing the functional dependence of the action S_B on fields φ₁, φ₂, R, and Λ to assess whether mixed terms in Δ can induce entanglement.

Experimental results

Research questions

• RQ1 Can two quantum fields become entangled through interaction with a classical gravitational field alone, without direct coupling?
• RQ2 Does the configuration ensemble formalism allow for entanglement generation in a fully relativistic setting with classical gravity?
• RQ3 Can entangled states evolve from initially unentangled states via gravitational interaction in this hybrid model?
• RQ4 Is the observed generation of entanglement sufficient to rule out classical gravity, or can it be explained by consistent hybrid models?
• RQ5 How does the non-linear correction term Δ in the effective Schrödinger equation affect the entanglement structure of the quantum fields?

Key findings

• Explicit calculations in spherical gravity show that two quantum scalar fields can evolve into an entangled state even when their only interaction is via a common classical gravitational field.
• The non-linear correction term Δ in the effective functional Schrödinger equation (Eq. 70) contains mixed derivatives with respect to φ₁ and φ₂, which can generate entanglement from initially separable states.
• Entanglement is generated even when the initial state is a product state, as the functional form of S_B evolves to include cross-terms under time evolution governed by the hybrid Hamiltonian.
• The configuration ensemble model supports entanglement generation in a relativistic setting, indicating that such entanglement is not exclusive to fully quantum gravity models.
• The results imply that observing entanglement in gravity-mediated experiments cannot definitively rule out classical gravity unless the underlying hybrid model is independently constrained.
• The formalism remains consistent under alternative canonical formulations of gravity, such as Ashtekar’s, suggesting robustness of the entanglement mechanism.