Collapse-based models for gravity do not violate the entanglement-based witness of non-classicality

This paper defends the validity of entanglement-based witnesses for non-classicality by demonstrating that collapse-based models of gravity, such as the Diósi-Penrose model, inherently violate the principle of locality and therefore cannot genuinely generate entanglement through purely local means.

The Gist

The Big Question: Is Gravity "Quantum" or "Classical"?

Imagine you are trying to figure out if a mysterious force (gravity) is made of "quantum stuff" (weird, fuzzy, connected particles) or "classical stuff" (solid, predictable, separate objects).

Scientists recently proposed a clever test called the "Entanglement Witness." Here is the rule:

If you have two separate objects (like two heavy balls) and you let them interact only through a middleman (gravity), and they end up becoming "entangled" (a spooky, quantum connection where they instantly know what the other is doing), then the middleman must be quantum.

Think of it like this: If two people in different rooms start finishing each other's sentences perfectly without speaking, there must be a secret, high-tech phone line connecting them. If the connection was just a regular, old-fashioned letter (classical), they couldn't do that.

The Controversy: The "Cheat Code" Argument

Recently, some researchers claimed they found a "cheat code." They said, "Wait! There is a model of gravity called the Diósi-Penrose (DP) model that is supposed to be classical (not quantum), but it can still make those two balls become entangled."

If this were true, it would break the test. It would mean you could have a classical gravity that creates quantum connections, proving that the "Entanglement Witness" is useless.

The Paper's Verdict: The Cheat Code is Actually a Secret Quantum Device

The authors of this paper (Feng, Vedral, and Marletto) say: "No, the test still works. The DP model is not actually a classical cheat code."

Here is why, using an analogy:

1. The Setup: Two Dancers and a Mirror

Imagine two dancers (the masses) in separate rooms. They are supposed to dance only by looking at a giant mirror (gravity) in the middle.

• The Rule: If the mirror is just a normal piece of glass (classical), the dancers can't coordinate a complex, synchronized dance (entanglement).
• The Claim: The DP model says, "We have a special mirror that is just glass, but it can still make them dance together."

2. The Investigation: What's Inside the Mirror?

The authors looked closely at how the DP model works. They realized that for this "special mirror" to make the dancers sync up, it isn't just a passive piece of glass.

Inside the mirror, there are hidden, invisible cameras (called "hidden detectors") constantly watching the dancers.

• These cameras aren't just recording; they are quantum cameras.
• When the cameras see a dancer move, they don't just send a signal; they instantly "collapse" the reality of the dancer's position and share that information with the other camera instantly, no matter how far apart they are.

3. The "Non-Local" Secret

The paper argues that these hidden cameras are non-local.

• Local: If I tap a table here, you feel it there only after a wave travels through the wood.
• Non-Local: If I tap the table here, you feel it there instantly, as if the wood is connected by magic.

The DP model relies on these cameras being connected by this "magic" (non-locality). Because the cameras are quantum and connected instantly, they are the ones creating the entanglement between the dancers, not the gravity itself.

The Conclusion: The Witness is Safe

The authors conclude that the DP model is not a purely classical theory. Even though it claims to describe "classical gravity," it secretly sneaks in quantum features (the hidden, non-local cameras) to make the math work.

• If the gravity was truly classical: It would be like a silent, passive mirror. It could never make the dancers entangle.
• Since the dancers do entangle in the DP model: It proves the "mirror" (gravity) must have a hidden quantum side.

In short: The "Entanglement Witness" is still valid. If we see gravity entangle two masses in a lab, it is 100% proof that gravity is quantum. The DP model doesn't break the rules; it just breaks its own definition of being "classical" by secretly using quantum tricks.

Why This Matters

This paper defends the upcoming experiments (like the Bose-Marletto-Vedral experiment) that aim to prove gravity is quantum. It tells scientists: "Don't worry about the DP model trying to confuse us. If the experiment works, gravity is definitely quantum, and the DP model's 'classical' explanation is actually a disguise for a quantum one."

Technical Summary

1. Problem Statement

The paper addresses a critical debate regarding the General Witness Theorem (GWT) for non-classicality. The GWT posits that if a system $M$ (e.g., the gravitational field) can mediate entanglement between two spatially separated quantum systems ($A$ and $B$) via local means only, then $M$ must be non-classical (i.e., possess non-commuting variables).

Recently, claims emerged (specifically by Trillo and Navascués) suggesting that collapse-based models of classical gravity, such as the Diósi-Penrose (DP) model, can predict gravitationally induced entanglement (GIE) between quantum masses. If true, this would imply that observing GIE is insufficient to prove gravity is fundamentally quantum, thereby invalidating the GWT as a test for quantum gravity.

The authors aim to vindicate the GWT by demonstrating that the DP model does not generate entanglement through classical gravity, but rather through mechanisms that violate the assumption of locality or introduce hidden quantum degrees of freedom.

2. Methodology

The authors employ a rigorous analysis of the underlying physics of collapse-based models, specifically the DP model, using Continuous Measurement Theory and Stochastic Master Equations (SME).

• Model Formulation: They review the DP model as a continuous monitoring of mass density by "hidden detectors" with finite spatial resolution $\sigma$. The dynamics are governed by a Stochastic Master Equation (SME) that includes a monitoring process and a back-action on the gravitational field.
• Decomposition of Dynamics: The authors decompose the evolution of the quantum state into two distinct processes:
  1. Monitoring Process: The evolution of the quantum matter density matrix due to the hidden detectors (governed by the SME).
  2. Back-action Process: The sequential evolution where the quantum matter interacts with the stochastic gravitational potential generated by the monitoring signal.
• Analytical Derivation: They derive the time-evolution of the density matrix for a two-mass system in a spatial superposition (similar to the Bose-Marletto-Vedral or BMV experiment setup). They explicitly calculate the negativity (an entanglement measure) of the system at first order in time $\Delta t$.
• Locality Check: They analyze the commutation relations and the structure of the interaction terms to determine if the entanglement generation is local or non-local.

3. Key Contributions

The paper makes three primary technical contributions:

1. Identification of the Source of Entanglement: The authors prove that in the DP model, the entanglement between masses is not induced by the classical gravitational field. Instead, it arises from the hidden monitoring process (the "jump" term in the SME).
2. Demonstration of Non-Locality: They show that the correlator $\gamma_{rs}$ in the DP model (which links the hidden detectors at different positions $r$ and $s$) is non-local (specifically proportional to $1/|r-s|$). This non-locality violates the "local means" assumption required by the GWT.
3. Reclassification of the DP Model: The paper argues that the DP model, while claiming to describe "classical gravity," implicitly relies on quantum-like hidden degrees of freedom (entangled hidden detectors) to function. Therefore, it is not a purely classical theory in the sense required to falsify the GWT.

4. Key Results

• The Interaction is Local but Non-Entangling: The interaction term $e^{-i\hat{V}_G t}$ induced by the classical gravitational potential is a local unitary operation. Since the mass density operators for distinct particles commute ($[\hat{\varrho}_\sigma(r), \hat{\varrho}_\sigma(s)] = 0$ for $r \neq s$), a purely classical gravitational field cannot generate entanglement between the masses.
• Entanglement via Hidden Detectors: The authors derive the density matrix evolution (Eq. 12) and show that the negativity (Eq. 13) becomes non-zero at first order in $\Delta t$. This entanglement is generated solely by the monitoring term in the SME:
  $$\frac{d\hat{\rho}}{dt} \bigg|_{\text{monitoring}} \propto -\int dr ds \frac{G}{|r-s|} [\hat{\varrho}_\sigma(r), [\hat{\varrho}_\sigma(s), \hat{\rho}]]$$
  The kernel $1/|r-s|$ represents a non-local correlation between the hidden detectors.
• Violation of Locality: The generation of entanglement in the DP model relies on the hidden detectors at location $r$ and $s$ being correlated. If these detectors are part of the gravitational field, the field possesses a hidden quantum degree of freedom. If they are external, they mediate information non-locally. In either case, the principle of locality (a prerequisite for the GWT) is violated.
• Mathematical Proof: The calculated negativity $N(\rho^{T_2}(\Delta t))$ is strictly positive, confirming that entanglement is generated. However, the authors emphasize that this is a result of the non-local map inherent in the collapse model, not a classical gravitational interaction.

5. Significance

• Validation of the GWT: The paper conclusively defends the General Witness Theorem. It establishes that observing gravitationally induced entanglement in a local setup remains a sufficient condition to conclude that the mediator (gravity) is non-classical.
• Refutation of Counter-Claims: It directly refutes recent claims that the DP model serves as a counter-example to the quantum nature of gravity. The authors argue that the DP model is effectively a hybrid theory containing non-classical elements (hidden quantum degrees of freedom) and is not a valid "classical gravity" model in the context of the GWT.
• Implications for Experiments: The results support the validity of proposed experiments (like the BMV experiment) to test the quantum nature of gravity. If such experiments observe entanglement, it rules out not only standard classical gravity but also specific collapse-based models like DP, as those models rely on non-local mechanisms that would be distinguishable or are fundamentally incompatible with the "classical" label in a local framework.
• Theoretical Clarity: The work clarifies the distinction between "classical gravity" and "stochastic gravity models with hidden variables," suggesting that any model capable of generating entanglement via local means must inherently possess non-classical features.

In summary, the authors demonstrate that the Diósi-Penrose model does not invalidate the entanglement witness because the entanglement it predicts stems from non-local hidden quantum variables, not from a classical gravitational field. Thus, the GWT remains a robust tool for testing the quantum nature of gravity.