Can classical theories of gravity produce entanglement?

This paper refutes the claim that classical gravity can generate entanglement, demonstrating that the previously reported effect was an artifact of incorrectly discarding specific transition amplitudes which, when properly included, ensure that an initially unentangled state remains factorized.

The Gist

The Big Question: Can Gravity "Tangle" Things Without Being Quantum?

Imagine you have two dancers, Alice and Bob. They are standing on opposite sides of a stage. In the world of quantum physics, particles can exist in two places at once (a "superposition"). So, imagine Alice is dancing in two spots simultaneously (Left and Right), and Bob is also dancing in two spots simultaneously (Left and Right).

Recently, a team of scientists published a paper in Nature claiming something very exciting: Even if gravity is just a boring, classical force (like a smooth sheet of fabric), it can still make Alice and Bob "entangled."

In quantum mechanics, "entanglement" is like a magical invisible string. Once two things are entangled, what happens to one instantly affects the other, no matter how far apart they are. The Nature paper suggested that gravity alone could tie this string between Alice and Bob, proving that gravity might have a secret quantum side.

The New Paper's Verdict: "Wait, You Missed a Step!"

The authors of this new paper (Gundhi, Infantino, and Bassi) looked at that Nature claim and said, "Hold on. You made a calculation error."

They argue that the Nature team didn't look at the whole picture. They only looked at the "easy" parts of the math and ignored the "messy" parts. When you include the messy parts, the magic string disappears.

Here is the breakdown of their argument using a simple analogy:

1. The "Diagonal" Mistake (Looking at Only One Path)

Imagine you are trying to predict the future of our dancers.

• The Nature Team's Method: They only looked at the path where Alice stays in her "Left" spot and Bob stays in his "Left" spot. They calculated the gravity interaction for that specific path and found a weird result that looked like entanglement.
• The New Team's Critique: They say, "You can't just look at one path! In quantum mechanics, you have to add up every possible path Alice and Bob could take."

It's like trying to figure out the weather by only looking at the sky at noon. You might think it's sunny, but if you look at the whole day (morning, afternoon, evening), you realize it's actually raining. The Nature team ignored the "off-diagonal" paths (where Alice moves from Left to Right, or Bob moves from Right to Left).

2. The "Exchange" Confusion (The Identical Twin Problem)

The math gets tricky because the particles involved are identical. It's like having two identical twins, Alice and Alice-prime. If you swap them, the universe doesn't notice.

The Nature team found a term in the math called an "exchange term." This term looks like the twins swapped places. They thought this swapping was the "magic string" (entanglement) caused by gravity.

The new authors say: "No, that's just a bookkeeping error."
Because the twins are identical, the math must account for them swapping places to keep the rules of symmetry. When you do the math correctly and include all the paths (not just the ones where they stay put), the "swapping" terms cancel out or arrange themselves in a way that does not create a connection.

Think of it like this:

• The Nature Claim: "If I swap two identical coins, the table shakes!"
• The New Paper: "The table doesn't shake. The coins are identical, so swapping them is just a trick of the light. If you look at the whole table, nothing has actually changed."

The Final Conclusion

The new paper proves that if you have a fixed number of particles (no new particles popping into existence) and they interact only through a classical gravitational field:

1. No Entanglement: The particles will not become entangled. They remain independent.
2. The "Magic String" is a Mirage: The apparent entanglement found in the Nature paper was an illusion caused by throwing away half the math.
3. Gravity is Still a Mystery: This doesn't prove gravity is not quantum. It just proves that classical gravity cannot create entanglement in this specific setup. To get entanglement from gravity, you likely still need gravity to be a quantum force (involving virtual particles or particle creation), which is a much more complex scenario.

The Takeaway for Everyday Life

Imagine two people trying to communicate using a walkie-talkie.

• The Nature paper said: "If they stand in a field, the wind (gravity) will make their voices sync up perfectly, even if the wind isn't a radio signal."
• The new paper says: "No, you only listened to the wind blowing in one direction. If you listen to the wind blowing from all directions, you'll see their voices are still completely independent. The wind didn't sync them up; you just didn't listen to the whole storm."

In short: Classical gravity, on its own, cannot create the spooky "spooky action at a distance" (entanglement) between objects. The recent claim was based on an incomplete calculation.

Technical Summary

1. Problem Statement

The paper addresses a claim made in a recent Nature publication (Aziz and Howl, 2025) suggesting that two massive objects, each prepared in a spatial superposition, can become entangled solely through a classical gravitational interaction. The original authors argued that even if gravity is treated as a classical field (not a quantum mediator), the exchange of virtual particles (or equivalent interaction terms) between the objects generates quantum entanglement.

The authors of this rebuttal challenge this conclusion, arguing that the entanglement reported in the original study is an artifact of an inconsistent perturbative approximation (specifically, the "diagonal approximation") that discards necessary transition amplitudes. They aim to demonstrate that under the specific constraints of the original setup (fixed particle number, non-relativistic regime, classical potential), an initially factorized state remains factorized, and thus no entanglement is generated.

2. Methodology

The authors re-examine the mathematical framework used in the original study, employing both perturbative calculations and non-perturbative operator analysis.

• System Setup: Two massive objects (1 and 2), each composed of $N$ Klein-Gordon particles, are prepared in spatial superpositions of locations $\{L, R\}$. The initial state is factorized:
  $$|\Psi(0)\rangle = \frac{1}{2} (|N\rangle_{1L} + |N\rangle_{1R}) \otimes (|N\rangle_{2L} + |N\rangle_{2R})$$
  The objects interact via a classical gravitational potential $\Phi(x)$.

• Perturbative Analysis (4th Order):

  • The authors calculate the time-evolved coefficients $\beta_{ij}(t)$ using the full Dyson series expansion of the evolution operator $\hat{U}(t)$.
  • They specifically analyze the exchange terms (involving two propagators connecting the two objects) at the 4th order of perturbation theory.
  • Critique of Original Approach: They identify that the original authors used a "diagonal approximation," calculating only terms where the initial and final branches match ($i=m, j=k$). The authors argue this is inconsistent because, in the specific geometry considered (where the distance between branches $d_{RL}$ is much smaller than other distances), the off-diagonal terms ($i \neq m$ or $j \neq k$) are actually dominant.
  • Correction: By including all terms in the summation $\sum_{m,k}$, they show that the exchange terms factorize into a product of single-particle amplitudes.

• Non-Perturbative Proof:

  • The authors provide a rigorous proof using the formalism of second quantization for a complex Klein-Gordon field.
  • They demonstrate that if the Hamiltonian conserves particle number (no pair creation) and the interaction is linear in the field operators (coupling to an external classical potential), the time evolution operator maps a symmetrized tensor product (factorized state) to another symmetrized tensor product.
  • They utilize the property that the interaction Hamiltonian is the second quantization of a one-particle operator, ensuring that the many-body state remains a product state (up to symmetrization).

• First Quantization Verification:

  • To rule out the possibility that the result is an artifact of the field-theoretic "virtual particle" language, they re-derive the results using non-relativistic first quantization for identical and distinguishable particles.
  • They show that for identical particles, the "exchange terms" arise purely from the symmetrization of the wavefunction, not from an entangling interaction. For distinguishable particles, these terms vanish entirely, and the state remains strictly factorized.

3. Key Contributions

1. Identification of Approximation Error: The paper pinpoints the specific mathematical error in the original Nature paper: the inconsistent truncation of the perturbative series. By keeping only diagonal terms, the original authors broke the factorization structure of the amplitude, artificially creating entanglement.
2. Restoration of Factorization: By consistently including off-diagonal terms (which are dominant in the relevant geometric limit), the authors show that the 4th-order exchange coefficients $\beta_{ij}^{(4)}$ take the form $a_{1i} b_{2j}$, proving the state remains factorized.
3. General Proof of No Entanglement: The authors prove generally that for any system of particles interacting with a classical external potential where particle number is conserved, an initially factorized state cannot become entangled.
4. Clarification of "Virtual Particles": They demonstrate that the mathematical terms resembling "virtual particle exchange" in the perturbative expansion are merely artifacts of the symmetrization requirement for identical particles and do not imply a quantum mediator capable of generating entanglement.

4. Results

• Perturbative Result: When the full expression for the transition amplitudes is used (Eq. 3 in the text), the 4th-order exchange contribution to the state is:
  $$|\Psi(t)\rangle^{(4)}|_{2c}^{ex} \propto \left( \sum_k V_{ik} |N\rangle_{1i} \right) \otimes \left( \sum_m V_{mj} |N\rangle_{2j} \right)$$
  This is a product state. The apparent non-factorizable term $(V_{ij})^2$ found in the original paper is an artifact of ignoring the dominant off-diagonal contributions.
• Non-Perturbative Result: The time evolution operator $\hat{U}(t)$ acting on a Hartree state (symmetrized product state) yields another Hartree state. Therefore, no entanglement is generated in the non-relativistic regime with fixed particle number.
• Distinguishable vs. Identical: For distinguishable particles, the exchange terms vanish completely, and the state remains factorized. For identical particles, the exchange terms exist but preserve the product structure due to symmetrization.

5. Significance

• Resolution of a Controversy: The paper refutes the claim that classical gravity can generate entanglement in the specific non-relativistic, fixed-particle-number regime. This reinforces the standard wisdom that entanglement generation requires a quantum mediator or relativistic effects (like particle creation).
• Clarification of Quantum Gravity Tests: The result has implications for proposed experiments (such as the Bose-Marletto-Vedral proposal) aiming to test the quantum nature of gravity via entanglement. The authors clarify that if such an experiment observes entanglement, it cannot be explained by a classical gravitational field interacting with fixed-mass objects; the field itself must be quantized or relativistic effects (pair creation) must be involved.
• Methodological Rigor: The paper serves as a cautionary example regarding perturbative expansions in quantum mechanics, highlighting the danger of inconsistent truncations (keeping only diagonal terms) when off-diagonal terms may be dominant due to geometric constraints.

In conclusion, the authors demonstrate that classical theories of gravity cannot produce entanglement between two massive objects in a spatial superposition if the number of particles is fixed and relativistic effects are neglected. The entanglement reported in the original study was a mathematical artifact of an incomplete calculation.