Absence of gravitationally induced entanglement in certain semi-classical theories of gravity

This paper demonstrates that a specific class of semi-classical gravity models, including the Newton-Schrödinger and Bohmian analogues, fails to generate entanglement between massive systems, thereby distinguishing them from standard quantum gravity predictions in the context of proposed experimental tests.

The Gist

The Big Question: Is Gravity a Quantum Thing?

Imagine you have two heavy objects, like two tiny bowling balls. In the world of quantum physics, these balls can be in two places at once (a "superposition"). Scientists Bose, Marletto, and Vedral (the BMV team) proposed a clever experiment: if you let these two quantum balls interact only through gravity, will they become "entangled"?

Entanglement is a spooky connection where two particles act as a single unit, no matter how far apart they are. The BMV team argued: If gravity can make two things entangled, then gravity itself must be a quantum force, not a classical one.

However, some scientists (like Döner and Großardt) argued, "Wait a minute! Maybe gravity stays classical (like a smooth, continuous field) but can still create this spooky connection."

The Author's Argument: The "Separable" Wall

This paper, written by Ward Struyve, says: "No, that's not possible."

Struyve looks at a specific family of theories where gravity is treated as a classical force that acts on quantum particles. He argues that in these specific models, gravity acts like a personalized, non-communicating wall.

Here is the analogy:
Imagine two people, Alice and Bob, standing in separate rooms.

• The Standard Quantum View (Newtonian Gravity): Alice and Bob are connected by a single, shared rope. If Alice pulls, Bob feels it instantly. They are linked. This allows them to coordinate their actions perfectly (entanglement).
• The Semi-Classical Models (The ones Struyve analyzes): Alice and Bob are in rooms with their own private mirrors.
  • Alice looks in her mirror and sees a reflection of Bob.
  • Bob looks in his mirror and sees a reflection of Alice.
  • Crucially: Alice's mirror only shows her own idea of Bob, and Bob's mirror only shows his own idea of Alice. They are reacting to their own private reflections, not to each other directly.

Because they are reacting to their own separate reflections, they can never truly "sync up" or become entangled. Their movements remain independent, even though they are influenced by the idea of the other person.

The Three "Mirror" Models

Struyve examines three specific theories that use this "mirror" approach, and he proves they all fail to create entanglement:

1. The Newton-Schrödinger (NS) Model:

   • The Analogy: The "mirror" is made of a fuzzy cloud of probability. The gravity Alice feels depends on the average shape of Bob's fuzzy cloud.
   • The Result: Since the cloud is just a sum of possibilities, the gravity Alice feels is just a sum of separate forces. It doesn't link them together.

2. The Bohmian Analogue (NSB):

   • The Analogy: The "mirror" is made of a single, real point (like a tiny dot). The gravity Alice feels depends on exactly where Bob's dot is right now.
   • The Result: Even though the dot is real, Alice and Bob are still in separate rooms. Alice reacts to Bob's dot, and Bob reacts to Alice's dot, but they don't share a single quantum state.

3. The Döner and Großardt Model:

   • The Analogy: This was the model that claimed to break the rules. It was a mix of the two mirrors above.
   • The Result: Struyve shows that this model is actually just a math trick. It looks like it creates a connection, but if you look closely, it's still just two separate mirrors. The authors of that model made a calculation error by mixing up which "dot" was being used for which part of the calculation.

The "Additively Separable" Rule

The paper uses a fancy math term: "Additively Separable."

Think of it like a recipe.

• Entangling Gravity (Standard): The recipe is a smoothie. You blend Alice and Bob together. You can't separate them again.
• Non-Entangling Gravity (Semi-classical): The recipe is a salad. You have a bowl of Alice's lettuce and a bowl of Bob's tomatoes. You can mix them in a big bowl, but they are still just lettuce and tomatoes sitting next to each other. You can separate them back into their original bowls.

Struyve proves that in these semi-classical theories, gravity is always a "salad." It adds up the effects of Alice and Bob separately, so they never blend into a single quantum smoothie.

What Does This Mean for the Experiment?

The paper concludes that if the BMV experiment is performed:

• If the result shows entanglement (negative witness): It proves gravity is quantum (like the smoothie).
• If the result shows NO entanglement (positive witness): It suggests gravity might be classical (like the salad), specifically following one of the models Struyve analyzed.

The paper provides a way to tell the difference between the "smoothie" (standard quantum gravity) and the "salad" (these specific classical theories) by looking at a specific measurement called an "entanglement witness."

Summary

Ward Struyve's paper is a mathematical proof that certain ways of treating gravity as a classical force simply cannot create quantum entanglement. He shows that the model which claimed to do so was actually miscalculated. Therefore, if the upcoming experiment finds entanglement, it will be strong evidence that gravity is indeed a quantum force, and these specific classical theories are incorrect.

Technical Summary

1. Problem Statement

The central problem addressed is the reconciliation of quantum mechanics with gravity. A prominent proposal by Bose et al. (BMV) and Marletto and Vedral suggests a tabletop experiment to test if gravity can induce entanglement between two massive systems in spatial superposition. The prevailing argument is that if gravity can generate entanglement, it must be a quantum entity.

However, this claim has been challenged by authors (specifically Döner and Großardt) who proposed semi-classical models where gravity remains classical yet still generates entanglement. Struyve's paper aims to refute this specific counterexample and, more broadly, to demonstrate that a wide class of semi-classical gravity models (where gravity is treated as a classical potential in the Schrödinger equation) cannot generate entanglement.

2. Methodology

The author analyzes a class of semi-classical models defined by the following framework:

• System Description: Systems are described by a wave function $\psi_t(x)$ and a configuration $Q_t$ (e.g., particle positions).
• Dynamics: The wave function evolves according to a non-linear Schrödinger equation:
  $$i\hbar\partial_t\psi_t(x) = \left[ -\sum \frac{\hbar^2}{2m_i}\nabla_i^2 + V(x, t; \psi_t, Q_t) \right] \psi_t(x)$$
  where the potential $V$ may depend on the wave function or the configuration.
• Key Criterion: The paper investigates the additive separability of the potential. A potential is additively separable if it can be written as a sum of single-particle potentials:
  $$V(x, t) = \sum_{i=1}^N V_i(x_i, t)$$
• Models Examined:
  1. Newton-Schrödinger (NS) Model: Gravity sourced by the mass density of the wave function ($|\psi|^2$).
  2. Newton-Schrödinger-Bohm (NSB) Model: Gravity sourced by actual point-particle positions (Bohmian trajectories).
  3. GRW-type Analogue: Gravity sourced by "flashes" (collapse events).
  4. Döner-Großardt Model: An interpolation between NS and NSB models.
  5. Standard Newtonian Potential: The standard quantum treatment where the potential depends on the positions of all particles simultaneously (non-separable).

The author applies these models to the specific experimental setup proposed by BMV, calculating the time-evolved state and evaluating an entanglement witness (proposed by Chevalier et al.) to determine if entanglement is generated.

3. Key Contributions and Theoretical Proofs

The paper provides two primary theoretical proofs regarding entanglement generation:

• Proof of Non-Entanglement for Separable Potentials:
  The author proves that if the potential $V$ is additively separable, the dynamics cannot generate entanglement from an initially separable state.

  • Logic: If the potential is separable, the Hamiltonian can be decomposed into a sum of single-particle Hamiltonians that commute with each other (in the context of a fixed background configuration or wave function). The time-evolution operator factorizes into a product of single-particle unitary operators. Consequently, a product state $\psi_0 = \prod \psi_i$ remains a product state $\psi_t = \prod \psi_i(t)$ at all times.
  • Implication: This holds even if the potential depends on $\psi$ or $Q$, provided the dependence maintains the additive separable structure.

• Refutation of the Döner-Großardt Counterexample:
  The paper specifically addresses the model by Döner and Großardt, which claimed to generate entanglement. Struyve demonstrates that their model relies on a calculation error:

  • Döner and Großardt incorrectly assumed a single wave function evolves with phase factors calculated using different particle configurations for each term in the superposition.
  • Struyve shows that in a Bohmian-like context (or any model with definite trajectories), the system evolves into a statistical mixture of four distinct separable states (corresponding to the four possible trajectory combinations).
  • Since the ensemble is a mixture of separable states, the system as a whole is not entangled.

4. Results: The BMV Experiment Analysis

The author applies the theoretical framework to the BMV experimental setup involving two masses in spatial superposition ($|L\rangle$ and $|R\rangle$).

• Standard Newtonian Potential ($V_N$):

  • The potential is not additively separable (it depends on the distance between particle 1 and particle 2).
  • Result: The state evolves into an entangled superposition.
  • Witness ($W_N$): The entanglement witness $W$ becomes negative for specific times, indicating entanglement.

• Semi-Classical Models (NS, NSB, Döner-Großardt):

  • The potentials are additively separable (each particle feels a potential sourced by the other, but the interaction term splits into single-particle terms).
  • Result: The state remains separable (or a mixture of separable states).
  • Witness ($W_{NS}, W_{NSB}$): The entanglement witness remains non-negative ($W \geq 0$) for all times.

• Quantitative Comparison:
  Using parameters from the original BMV proposal ($d=450\mu m$, $\delta=250\mu m$, $m=10^{-14}kg$), the paper plots the witness $W$ over time.

  • $V_N$ shows oscillations dipping below zero (entanglement).
  • $V_{NS}$ and $V_{NSB}$ remain strictly positive (no entanglement).

5. Significance and Implications

• Validation of the "Quantum Gravity" Argument: The paper strengthens the argument that observing gravitationally induced entanglement in the BMV experiment would indeed be evidence for the quantum nature of gravity. It rules out a specific class of "loophole" semi-classical theories that were thought to mimic quantum behavior.
• Experimental Discrimination: The paper demonstrates that the BMV experiment is not just a test for "quantum vs. classical" in a binary sense, but can specifically distinguish between the standard Newtonian potential and semi-classical models (like NS or NSB) based on the behavior of the entanglement witness.
• Clarification of Semi-Classical Dynamics: It clarifies that while semi-classical models with separable potentials allow for non-local dependencies (where particle A's evolution depends on particle B's position), this dependence does not suffice to create quantum entanglement.
• Correction of Literature: It corrects the erroneous claim by Döner and Großardt, ensuring that future theoretical debates regarding the quantum nature of gravity are built on accurate mathematical foundations.

In conclusion, Struyve establishes that additive separability is a sufficient condition for the absence of entanglement generation in semi-classical gravity models. Therefore, any semi-classical theory relying on such potentials will fail to reproduce the entanglement predicted by standard quantum mechanics with a Newtonian potential, making the BMV experiment a viable discriminator.