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What gravity mediated entanglement can really tell us about quantum gravity

We revisit the Bose-Marletto-Vedral (BMV) table-top experimental proposal -- which aims to witness quantum gravity using gravity mediated entanglement -- analyzing the role of locality in the experiment. We show how gravity mediated entanglement in the BMV experiment can be accounted for without appealing to the quantum nature of the gravitational field. We then carry out a fully quantum modelling of the interaction of matter and gravity and identify the modifications that a BMV-like experiment should have in order to truly reveal quantum features of the gravitational interaction.

Introduction.-A full formulation of quantum gravity is among the most coveted theories in physics. However, we lack experimental testbeds to probe gravity in the quantum regime. Recently, exciting proposals for tabletop experiments combining quantum mechanics and gravity have been put forward by Bose et. al [1] and Marletto and Vedral [2] (the so-called BMV experiment). These proposals have captured the eye of the community for their alleged potential to provide proof of the quantum nature of gravity.
The experiment consists of two particles, each prepared in a superposition of two different trajectories. The two particles interact only gravitationally. The rough idea is that the different superposed paths generate different gravitational fields, which can entangle the particles. Despite the inherent experimental challenges to isolate the particles while shielding them from decoherence, we are close to having technology that will allow the experiment in the near future [1][2][3][4][5][6]. Although it is exciting to have an experiment that can explore aspects of gravity when quantum theory is relevant, some possible objections to the fact that the BMV experiment can reveal the quantum nature of gravity exist in the literature [7][8][9][10][11][12]. This indicates that the conclusions that can be drawn from the experiment require careful analysis. For example, it has been claimed that if the gravitational field is capable of entangling the masses, then either we must abandon the principle of locality, or the gravitational field must be quantum [13,14]. In this letter we will present a series of arguments that this is not exactly the case. Instead, we will discuss that while the experiment, as proposed, can prove that gravity can establish a quantum channel between the particles, it cannot decide whether gravity has quantum degrees of freedom. However we will also show that it is in principle possible to modify the experiment so that it demonstrates the fundamental quantum behaviour of gravity.
It is well-known that the coupling of classical and quantum systems is theoretically inconsistent (see, e.g., [15,16]). This prevents theories where matter is quantum and gravity is classical from being fundamental. However, the theoretical argument alone cannot be used to claim that gravity is quantum (e.g., the interaction between matter and gravity might perhaps be described by a different framework outside quantum theory). Instead, for an experimental claim of quantumness, one requires observing specific markers of quantum behaviour. While different authors may disagree on what conditions are sufficient to prove that a system is quantum, there are some uncontroversial properties of a system that, if observed, would prove its quantumness. For example, observing Wigner negativity [17,18], or violations of Bell inequalities [18,19].
In this light, our goal is twofold. First, we will discuss that the current proposals of the BMV experiment cannot witness indisputably quantum behaviour of the gravitational field. That is, we will argue that the mere observation that two masses (in a quantum superposition of trajectories) get entangled through the gravitational interaction is not sufficient to experimentally witness quantum behaviour of gravity. Second, we will also identify under which regimes a BMV-like experiment could truly reveal quantum aspects of gravity.
Here we provide a comparison between two descriptions for the gravitational field in the BMV experiment. One of them is a relativistically local description that does not use any quantum feature of the gravitational field, and yet is able to predict the entanglement of the masses. The second description considers a fully quantum model for the degrees of freedom of (weak) gravity. We will highlight the differences in the predictions of the two models, emphasizing which experimentally observable features can be linked to genuinely quantum features of gravity. Finally, we will suggest modifications to the BMV experimental proposal in order to observe them.
Locality.-In order to discuss the interpretation of the experiment, it is important to distinguish between two fundamentally different notions of locality. The first notion comes from the description of spacetime and is deeply linked with causality. It states operations happen at events in spacetime, and do not affect other events which are causally disconnected from them. We will call this notion event locality [20]. The second notion of lo-cality comes from quantum mechanics, and states that operations that independently affect two quantum systems must be separable. We will call this notion system locality. The notion of system locality alone is agnostic about causal structure or any underlying notion of spacetime. Although these notions of locality are different there are particular frameworks in which we link the two. For example, in quantum field theory (QFT) the postulate of microcausality prevents operations in local systems from violating the notion of event locality prescribed by relativity.
The distinction between the two notions of locality is particularly important when talking about local operations and classical communication (LOCC). While the notion of system locality is operationally captured by the 'L' and the 'O', we often define what we mean by classical communication (the 'CC') in terms of event locality.
After all, what is classical communication if not sending information from one event to another?
Many works in the literature which argue that the BMV experiment can be used to probe the quantum nature of gravity use an argument based on LOCC. In essence, the argument goes as follows: LOCC does not increase the entanglement between quantum systems, thus, if the masses interact only gravitationally and get entangled, the gravitational field which mediates the interaction is going beyond 'CC', hence the field cannot be classical [1,2,13,14]. However, this argument assumes a relationship between event and system locality in order to reach its conclusion: it assumes that, in order to get them entangled, the field has to couple system-locally to the masses. This assumption can be reworded as follows: mass A couples to the field and then the field carries quantum information to mass B, or otherwise we would have action-at-a-distance. However, the assumption that the gravitational interaction is implemented through system-local operations is not based on first principles. It is reasonable to demand that gravity must satisfy event locality to prevent action-at-a-distance, but it is not unthinkable to consider that it may not necessarily be system-local since this notion is just operational rather than fundamental.
We know that if we use QFT to describe gravity, the interaction will be both event-local and system-local. However, assuming this relationship between the two notions of locality to interpret the results of the experiment implicitly assumes a-pirori that the system is described by a framework like QFT. This is not satisfactory if our objective is to prove the quantum nature of a relativistic theory. For instance, the fact that a classical Coulomb potential can entangle two charges is not why we believe that the electromagnetic field is quantum: q r is clearly not a quantum field. The electromagnetic field has only been proved to be quantum when QFT was required to model experiments which no classical theory could account for.
Local classical fields can entangle.-We first set up the local interaction between the masses by coupling them via the gravitational field in the linear regime. Consider the metric where η µν = diag(−1, 1, 1, 1) and h µν is a metric perturbation with units of energy. We choose these conventions so that the field propagators do not pick up factors of G.
We write the solution of the linearized Einstein's equations for a source T µν as where G µν R α β (x, x ) denotes a retarded Green's function for the linearized Einstein's equations and dV is the invariant spacetime volume element.
We now present a description of the BMV experiment which does not rely on any quantum property of gravity and is compatible with relativistic locality. Yet, we will show that this description is able to predict the expected results of the experiment. Consider two pointlike particles labelled by i ∈ {1, 2} with masses m 1 and m 2 , whose centres of mass are quantum and can undergo two possible trajectories each, z Ri (t) and z Li (t). We associate each possible trajectory to states |R i and |L i (see Fig. 1). Each possible path combination |R 1 R 2 , |R 1 L 2 , |L 1 R 2 , |L 1 L 2 is associated with a different classical gravitational interaction between the masses. In this sense, this description is a quantum-controlled classical model for the gravitational field, where gravity has no quantum degrees of freedom. The total interaction Hamiltonian can be written aŝ where Φ p1p2 (t) denotes the retarded propagation of the classical gravitational field between the two particles undergoing trajectories z p1 (t) and z p2 (t), i.e., where denotes the gravitational field generated by particle i undergoing path p i and is the stress-energy tensor corresponding to each path z pi (t) for the particles with four-velocities u µ pi (t). This approach is the relativistic (and thus causal) unapproximated version of the interaction Gm 1 m 2 /|x 1 −x 2 |, which does not rely on any quantum property of the gravitational field whatsoever. It also does not involve any superposition of gravitational fields since there is no Hilbert space for gravity: a quantum-controlled classical field is certainly not itself a quantum field. This is in tension with the wording used in e.g., [3,21,22]. The interaction (3) defines a quantum channel between the particles that respects event locality. However establishing a quantum channel cannot be taken as proof that gravity has quantum degrees of freedom: it is merely a consequence of having quantum sources and not of any assumptions about the field 1 . Indeed, this model does not assume that the gravitational field has degrees of freedom of its own that carry information between the particles (i.e. gravity is not an active mediator). We use the term "classical" to refer to (3) because it associates to each state of the particles the classical field sourced by each particle undergoing each path. The name "classical" is given in contrast to the model presented later where the field has local quantum degrees of freedom and acts as a mediator for the interaction.
Since the gravitational interaction is implemented by a classical field,Ĥ I (t) (Eq. (3)) commutes with itself at different times. The time-evolution operator iŝ 1 Notice that in the nomenclature of quantum information an interaction between two systems of the form (3) is called quantum because it establishes a quantum channel. However in this manuscript we argue that establishing a quantum channel is not a valid proof that the gravitational interaction has a Hilbert space structure of any kind or any local quantum degrees of freedom. A proof that gravity is quantum requires, in our opinion, to see an experimental phenomenon that cannot be explained unless there are local degrees of freedom for the gravitational field, as we argue below. = p1∈{L1,R1} p2∈{L2,R2} e 2πiG∆p 1 p 2 |p 1 p 2 p 1 p 2 | , where ∆ p1p2 is a double integral in spacetime of the retarded plus advanced propagator contracted with the stress-energy tensor of the sources corresponding to each path: Importantly, the time evolution generated by the classical field interaction is not local in the operational quantum sense (that is,Û I =Û 1 ⊗Û 2 is not system-local), although the interaction is intrinsically local in the relativistic sense and thus satisfies event locality. This means that time evolution implemented byÛ I can create entanglement, even though the gravitational field is classical and the interaction is event-local in this description. The notion of event locality is the one that comes from first principles and captures no-action-at-a-distance: the classical field only interacts with the particles locally at each instant of time, due to the (relativistically causal) retarded propagation.
The specific BMV setup considers both particles to be in a superposition of the two paths, so that the initial state of the system is corresponding to the density operator The final density operator after the interaction can then be written aŝ The entanglement between the two particles can be evaluated through the negativity 2 of the stateρ c , which reads We use negativity instead of entanglement entropy to compare this classical calculation with the quantum description for the gravitational field below. When the field is quantum there will be particle-field entanglement and the state of the particles will not be pure. Entanglement entropy is therefore not a valid entanglement measure. Negativity is always a faithful entanglement quantifier for bipartite two-level systems [23].
With the typical choice of paths for the BMV experiment the above quantity is non-zero. Therefore, it is possible to model entanglement creation via an event-local classical interaction. This description does not involve any quantum property of gravity and yet it is able to predict the expected gravity-mediated entanglement generation in the BMV experiment without violating the fundamental notion of event locality. This disqualifies the mere presence of entanglement in the masses-system from being evidence of quantum gravity. A quantum description of the experiment.-We now describe the BMV experiment when the masses are coupled to linearized quantum gravity. Linearized quantum gravity is a weak field limit of the gravitational field which can be quantized [24,25]. This weak field limit description should be a prediction of any theory of quantum gravity. The quantum description of the experiment involves two interactions of each of the particles with the quantum gravitational field (i.e., event-local and systemlocal within the framework of QFT). It is prescribed by the interaction Hamiltonian densitŷ whereĥ µν (x) is the field corresponding to linearized quantum gravitational perturbations, and T µν pi (x) is given by (6). The interaction Hamiltonian is given bŷ . (15) Unlike in the classical description previously discussed, this interaction Hamiltonian does not commute with itself at different times. The unitary time evolution operator is then given byÛ I = T exp −i dtĤ(t) , where T exp denotes the time-ordered exponential. We analyze this through perturbation theory. These calculations are standard (see Appendix B).
To obtain the final state of the two-particle system we assume that the field is initially in the vacuum state. After the interaction, one must trace out the gravitational degrees of freedom. This results in a mixed state for the two masses, in contrast to the classical case: here the particles become entangled with the gravitational field, which has true quantum degrees of freedom. In order to simplify the result, we assume that all trajectories are related by rotations and translations in space, so that the local vacuum effect in each trajectory is the same. Under this assumption, the leading order negativity is where L is a local noise term associated to the particle's interaction with the gravitational field vacuum (see Ap-pendix C), (17) and G µνα β (x, x ) = 0| T ĥ µν (x)ĥ α β (x ) |0 is the Feynman propagator of the linearized quantum gravitational field. It can be written as where ∆ µνα β (x, x ) is the state-independent part of the Feynman propagator, which is already present in a classical theory, and given by Eq. (8) and H µνα β (x, x ) = 0| {ĥ µν (x),ĥ α β (x )} |0 is the Hadamard distribution, which corresponds to the state-dependent part of the propagator and is not present in the classical case, thus showcasing the genuinely quantum nature of the field in this interaction.
Comparing Eqs. (18) and (16) with (13), we see that, apart from the vacuum noise L that appears in the quantum case the entanglement acquired in the quantum case is larger than the one obtained in the classical description. In the regime of long interaction times, the noise term L is insignificant compared to the effect of the propagators, so that it can be neglected. The contribution to the negativity due to the imaginary part of the Feynman propagator can be associated with entanglement mediated by communication via the gravitational field [26], while the real part of the propagator is associated with the entanglement extracted from the vacuum state of the gravitational field (in relativistic quantum information this phenomenon is known as entanglement harvesting [26][27][28]). In this sense, the real part of the propagator is the one which captures the entanglement associated to the quantum degrees of freedom of the gravitational field. Violating Bell inequalities with this entanglement would then be proof of the quantum behaviour of gravity. In many setups where the paths are causally connected, the contribution of the real part of the propagator is negligible compared to its imaginary counterpart. In particular, the proposed implementations of the BMV experiment use masses with smallest separation of the order of L ∼ 10 −6 m and interaction times of the order of T ∼ 1s. For these parameters we find that the imaginary part contribution of the propagator is 10 14 times larger than its real part, casting doubts on whether quantum effects can be acknowledged in this regime.
However, it is possible to adapt the BMV experiment to acknowledge the quantum nature of the gravitational field: if it could be implemented for times of the order of the light-crossing time of the separation between paths, then the real part of the Feynman propagator would be larger than its imaginary part. Only if the experiment can be adapted to implement these short interaction times (or can be sensitive to changes in negativity of the order of 10 −14 with the current proposed parameters [1]) one can claim to witness this genuine quantum behaviour of the gravitational field.
What can the BMV experiment tell us about quantum gravity?-The BMV experiment can be described treating the gravitational field classically or quantumly. Although these descriptions yield different predictions (therefore the experiment does have the potential to acknowledge quantum gravitational effects), they can only be experimentally distinguished in regimes where the duration of the experiment is of the order of the light crossing time between the paths.
We want to address previous claims in the literature about the implications of the BMV experiment on the quantum nature of gravity. One possible description for the BMV experiment is to assign a Hilbert space to the gravitational field [2][3][4]14]. However, we stress that while a quantum description for gravity is sufficient to account for the experimental results, it is by no means necessary. This obstructs absolute claims about the quantum nature of the physical systems described in the experiment-one requires to not be able to provide a classical description in order to claim non-classicallity of a theory. In fact, whether a system is quantum or not is not decided by whether it can mediate entanglement between two quantum sources, but by whether or not it admits a statistical description in a classical phase space [17,18]. This cannot be decided by finding entanglement in the BMV experiment.
On the other hand, it would be unfair to say that the BMV experiment-even in the regimes considered in [1,2]-gives no information about the interplay of gravity and quantum mechanics. If the gravitational field proves to be able to entangle the particles, we experimentally confirm that gravity mediates a quantum channel between the masses. In fact, if the BMV experiment confirms that gravity can mediate entanglement, it would rule out many classical descriptions for gravity coupled to quantum matter (such as some of the examples in [29]). However, as we discussed, this is different from proving that gravity itself has quantum degrees of freedom.
Conclusions-We discussed the implications of the BMV experiment for quantum gravity. If the experiment is implemented for long times, classical gravity mediates entanglement between the masses. This can give us information about how gravity couple to quantum systems, but may not be argued as proof of quantum behaviour of the gravitational field itself. If on the other hand the experiment can be performed in times of the order of the light-crossing time between the paths, then the entanglement between the particles would be significantly affected by the entanglement previously present in the gravitational field. This could be used to experimentally assess whether gravity admits a quantum field theoretic description or not. In order to adapt the previously proposed experiment so that the entanglement of the masses is due to the quantum nature of the field, either the spatial separation between the trajectories would have to be increased, or the interaction time would have to be decreased to guarantee spacelike separation. Alternatively, one could also see quantum behaviour with enough sensitivity in the experiment to distinguish between the classical result (Eq. (13)) and the quantum one (Eq. (16)). It is even possible to find regimes where the quantum prediction results in less entanglement than the classical model predicts. This is due to the fact that for short interaction times, the local vacuum noise term L can dominate the sum in Eq. (16). An effect that is well known within entanglement harvesting [26][27][28]30].
While the physical relevance of the BMV experiment is unquestionable, we argued that the experiment as prescribed is not enough to witness true quantum behaviour of the gravitational field, unless taken to regimes where a local classical model cannot predict the entanglement acquired by the masses. While we do not want to subtract from the many merits of the BMV proposal, it is important to qualify the range of implications of its experimental realization. Only once we measure the genuine quantum behaviour of gravity in a laboratory we will be able to claim that we have empirical evidence that the gravitational field is quantum.
where T αβ denotes the stress-energy tensor of the source. For the case of a pointlike particle undergoing a trajectory z 1 (t) with four-velocity u µ 1 (t), it reads so that we obtain Now, let t r be the retarded time such that t − t r − |x − z(t r )| = 0, so that wherer 1 (t) = (x − z 1 (t))/|x − z 1 (t)| and we obtain The interaction Hamiltonian of a particle labelled 2 with the gravitational potential sourced by particle 1 will then be In the quantum case, the final state of the particles to leading order can be written asρ + (δρ c + δρ q + δρ l ), where where H p1p2 denotes the integrated Hadamard function along each pair of paths for particles 1 and 2, while L v and L i are noise terms due to the interaction of each particle with the vacuum of the field. These are explicitly given by where, due to choice of paths, the indices i can be 1 or 2, and still yield the same result due to the fact that the paths are related to translations and rotations, which are symmetries of the quantum field theory. Then, the L v term is a local noise associated to each path, while the L i term represents an interference term associated with each particle undergoing the superposition of paths. Also notice that due to the fact that the propagator decreases with distance, we have L i ≤ L v , with equality holding only if the paths 1 and 2 are identical. Moreover, these vacuum noise terms decay fast with the interaction time, so that they are negligible for long interaction times. Then we can interpret δρ l as a local vacuum contribution and δρ q can be seen as the additional correlation contribution due to the quantum nature of the field.