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Inference of gravitational field superposition from quantum measurements

Experiments are beginning to probe the interaction of quantum particles with gravitational fields beyond the uniform-field regime. In standard quantum mechanics, the gravitational field in such experiments is written as a superposition state. We empirically demonstrate that alternative theories of gravity can avoid gravitational superposition states only by decoupling the gravitational field energy from the quantum particle's time evolution. Furthermore, such theories must specify a preferred quantum reference frame in which the equations of motion are valid. To the extent that these properties are theoretically implausible, recent experiments provide indirect evidence that gravity has quantum features. Proposed experiments with superposed gravitational sources would provide even stronger evidence that gravity is nonclassical.


I. INTRODUCTION
Understanding the fundamental nature of gravity is a significant open problem in theoretical physics. Unlike the other interactions, gravity lacks a satisfactory description as a quantum field, leaving open the possibility that gravity is fundamentally classical [1] or entropic [2] rather than quantum. On the other hand, several arguments [3][4][5][6][7][8][9][10][11] suggest that gravity ought to have quantum features.
For many years, it was believed that experiments could not contribute meaningfully to this discussion. Owing to the weakness of the gravitational interaction, it is not feasible to detect a single graviton in a gravitational-wave observatory or a particle detector [12]. Experimental evidence of the quantum nature of gravity is expected to appear at the energy scale corresponding to the Planck mass (10 19 GeV), which is far beyond the reach of collider experiments.
Although the quantization of gravity cannot be probed directly, it may be possible to obtain experimental evidence of gravity's fundamental nature through indirect means. Two proposals [7,8] ("BMV experiments") suggest searching for entanglement generation in a pair of matter-wave interferometers that interact gravitationally. Such low-energy tests might open the first observational window on the quantum description of the gravitational field, but the precise theoretical implication of these experiments is still an open question [9][10][11][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. However, it is widely believed that if gravity mediates entanglement between two quantum systems, then the gravitational field cannot be classical [28]. The experiments described in these proposals are challenging to implement because they require quantum control of large test masses (∼ 10 12 amu). Nevertheless, the idea of indirectly observing quantum-gravitational properties motivates an assessment of what we can infer from previous gravitational measurements in quantum systems.
Beginning with the COW experiment in 1975 [29], many matter-wave interferometers have demonstrated sensitivity to inertial forces. Until recently, these experiments operated in a regime where the gravitational field is approximately uniform at the length scale of the measuring device. According to the equivalence principle [30], all observable effects in such experiments arise from the relative acceleration between the matter waves and the phase reference, which is induced by the nongravitational forces that keep the laboratory in place on the surface of the Earth. Hence, in these experiments, the observed phase shift is caused by non-gravitational forces [31]. Matter-wave interferometers in the uniformfield regime test the equivalence principle [32,33] but do not provide quantum tests of any other gravitational properties [31]. Throughout this work, we will assume that the equivalence principle is valid to the accuracy of the experimental results we describe.
Genuine gravitational effects in quantum systems appear when the gravitational field is measurably nonuniform across the quantum state. Recent experiments have begun to investigate this scenario. Observations of phase shifts associated with gravitational tidal forces [34] and a gravitational Aharonov-Bohm effect [35] indicate that the trajectory [34] and action [35] of a matter wave in a nontrivial gravitational field are correctly predicted by standard quantum mechanics.
Taken by themselves, these results might appear to provide little information about gravity's fundamental nature. However, their implications are much stronger if one makes additional assumptions about the fundamental principles underlying the description of the experiment. In particular, we identify the following three principles: 1. Existence of gravitational fields: Any massive particle that is well-localized at a position x 0 sources a gravitational field g with functional form g(x − x 0 ) [36]. arXiv:2209.02214v1 [quant-ph] 6 Sep 2022 2. Field energy principle. The phase of an interferometer is a function of the energies of the fields that interact with the interfering particle.
3. Quantum relativity principle. The laws of physics take the same form in every reference frame, including the reference frames associated with quantum particles (quantum reference frames [37]).
The quantum relativity principle (QRP) encodes the relational nature of physical statements. This means that all physical quantities are expressed in terms of the relationship between physical systems, without appealing to an absolute background structure. In the case of two particles interacting gravitationally in the linearized regime, we show in Appendix 1 that the standard description of the gravitational field allows us to describe the interaction as sourced by either one of the two particles, with the other particle moving in this external potential. The QRP extends the relativity principle to the case in which the two particles are in a quantum superposition state relative to one another.
In this article, we analyze the gravitational Aharonov-Bohm experiment [35] to determine its implications for gravitational superposition [38]. We show that the standard quantum-mechanical description of this experiment is compatible with all three of the above principles and that in standard quantum mechanics, the gravitational field in this experiment is described as entangled with the superposed matter degrees of freedom. We present data demonstrating the insensitivity of the interferometer phase to the probability distribution of the test particleas predicted by standard quantum mechanics, but in tension with semiclassical expectations. We assess the theoretical reach of the experiment by comparison to future searches for gravitationally mediated entanglement. Finally, we argue that in order to deny the existence of a gravitational superposition state in this apparatus, a theory in which gravity is treated as a physical system must either reject principle (1) or reject both principles (2) and (3).
The remainder of the article is organized as follows: Section II presents a technical overview and simplified theoretical model of the gravitational Aharonov-Bohm experiment; Section III describes how the gravitational phase shift can be computed from the field energy; Section IV presents experimental data; Section V analyzes the experiment in a quantum reference frame [37] associated with the position of the interfering particle; Section VI compares the gravitational Aharonov-Bohm experiment to the proposed BMV experiments; and Section VII discusses the inference of a gravitational superposition state from (1)-(3) and explains how these assumptions are violated in alternative theories of gravity.

II. EXPERIMENTAL OVERVIEW AND SIMPLIFIED THEORETICAL MODEL
The idea of the gravitational Aharonov-Bohm experiment [35] is to observe the gravitational interaction between a quantum test particle and a classical source mass in the regime where the wave packet separation of the test particle is larger than the distance to the source mass. In this regime, the phase response of the interfering test particle becomes decoupled from the forces on the interferometer arms. In particular, one can observe a gravitational phase shift even in a configuration where the deflection-induced phase contribution vanishes, as demonstrated experimentally by independent measurements of the arm deflections [35]. For the purpose of this work, the significance of the experimental result is that it is consistent with the predictions of standard quantum mechanics for the behavior of a quantum particle in a nontrivial gravitational field. In the experiment [35] (Fig. 1A), an ultracold cloud of 87 Rb is used as the input of a single-source gradiometer [34] with baseline 24 cm and wave packet separation 25 cm. The upper interferometer in the gradiometer is sensitive to the tungsten source mass, while the lower interferometer mainly acts as a phase reference. The matter-wave beam splitters and mirrors are implemented by applying a sequence of laser pulses to the atoms, inducing sequential 2 k Bragg transitions. The population of each output port is detected by fluorescence imaging. Fig. 1B shows the spacetime trajectories of the source mass, interferometers, and detector in a freely falling reference frame. For a full description of the experimental apparatus and measurement protocol, see Ref. [35].
We construct a theoretical model based on the gravitational Aharonov-Bohm experiment by implementing several simplifying assumptions. First, we ignore the back action of the test particles on the source mass trajectory, and we also ignore any perturbations of the source mass induced by its interaction with the environment. Second, we neglect the gravitational interaction between the source mass and the lower interferometer, allowing us to exclude the lower interferometer and the optical phase reference from the model. Third, we ignore the nongravitational acceleration of the detector, which does not affect the measurement outcomes. Fig. 1C shows the spacetime trajectories of the test mass, source mass, and detector in this model. A test particle of mass m ("particle A"), initialized in a Gaussian state [40], is used as the input to a Mach-Zehnder interferometer. The particle is placed in a superposition of momentum states by a beam splitter at time t = 0, which are reflected by a mirror at time t = T and interfered by a second beam splitter at time t = 2T . The central trajectories of the two interferometer arms are given by where is the reduced Planck's constant and k is the magnitude of the beam splitter wave vector. The details of the beam splitter and mirror implementation are unimportant for our analysis. During the interferometer, a source particle of mass M ("particle B"), initialized in a Gaussian state, is brought close to one interferometer arm and then removed. For concreteness, we will assume that the source particle follows a parabolic trajectory: A third particle ("particle D") with mass M D defines the reference frame. We assume that particle D is far enough away from particles A and B that its gravitational influence is negligible. In standard quantum mechanics and in accordance with our principle (1), particles A and B interact gravitationally with potential energy where G is Newton's gravitational constant andx i is the position operator of particle i [41]. In the time intervals (0, T ) and (T, 2T ), the system can be described by the Hamiltonian After the final beamsplitter, the two output ports spatially separate due to their momentum difference. When the measurement occurs, the two ports are displaced by d 1 and d 2 from the detector, respectively. The interferometer phase φ is defined by the probabilities that the test particle will be observed at distance d 1 or d 2 from the detector: where P i is the probability of observing particle A at distance d i from particle D. Note that the interferometer phase is defined entirely in terms of relative coordinates, as is necessary for an observable quantity. One can compare the interferometer phase with and without the source mass present to measure the phase shift ∆φ due to the source mass. For the purpose of calculating ∆φ, we can neglect the trajectory perturbations of particle A induced by U . The phase shift due to the gravitational interaction between A and B can be computed [43] as In the reference frame of particle D, this phase shift can be understood to arise from the different paths of particle A in the background classical gravitational field sourced by particle B. This means that the phase shift can be rewritten as where V(x i ) is the classical gravitational potential associated with particle B integrated over each trajectory x i of particle A in the interferometer. During the interferometer, the quantum state (in the reference frame of particle D) takes the form Here we have written the state of the gravitational field |α i G , where α i represents the field sourced by point particles of mass m and M at positions x i and x s , respectively. The necessity of including the gravitational field in the quantum state description is discussed in [9], and the explicit form of the gravity state is given in [44].
Note that in the reference frame of particle D, the gravitational field (which is sourced by both particles A and B) is entangled with the position of particle A and is in a spatial superposition with length scale kT /m. In a general-relativistic context, this state is described by writing a superposition of metrics [11].

III. PHASE SHIFT FROM FIELD ENERGY
The electrostatic interaction energy U EM of a particle with charge q in an electric potential V EM is given by From a field-theoretic perspective, it is well known [45,46] that this quantity is equivalent to the electric field energy E EM , given by where E is the electric field and the integral is taken over all space, with the divergent self-energy of the particle's electric field subtracted from the integrand. Notice that if the electrostatic interaction were due to a non-local potential, and not to a field, it would not be possible to associate the field energy E EM with it. An analogous relationship holds for the classical gravitational field. In the weak-gravity, non-relativistic limit, the interaction energy U G of a particle of mass m in a gravitational potential V G is given by This interaction energy is equivalent to the gravitational field energy E G , which is given by where G is Newton's constant, g is the gravitational field, and the integral is taken over all space. As in electromagnetism, the gravitational self-energy of the particle is subtracted out of the integrand. This definition of the gravitational field energy can be generalized to linearized gravity (or even to full general relativity when the spacetime is asymptotically flat), provided that one uses a coordinate system that asymptotically approaches Minkowski coordinates [46]. The usual approach in quantum mechanics is to account for the gravitational interaction energy by including the potential V G in the Hamiltonian. Alternatively, we can take the perspective that the energy resides in the gravitational field. This amounts to replacing V G with E G in the Hamiltonian, where |g| 2 is promoted to an operator that acts on the gravitational field's quantum state. The ability of the gravitational field to carry energy has been experimentally confirmed by the observation of gravitational radiation [47].
As discussed in Section 2, the interferometer phase shift of a particle of mass m evolving in a perturbing gravitational potential V G is approximately given by where x 1 (t) and x 2 (t) are the trajectories of the interferometer arms. Equivalently, the phase shift can be written as where E i (t) is E G evaluated on the quantum state of the gravitational field associated with trajectory x i (t). Standard quantum mechanics thus satisfies the field energy principle. Crucially, the gravitational field energy cannot give rise to the experimentally observed relative phase in this way unless the gravitational field is in a superposition state that is entangled with the interferometer arms.
As in the case of electromagnetism, the identification of a gravitational energy (and hence the field energy principle) is deeply related with gravity being a field, rather than just a non-local potential. Note that the gravitational energy difference between interferometer arms cannot be observed when the gravitational field is approximately uniform at the experimental length scale [29]. In that case, the phase shift arises from the relative acceleration between the interfering particles and the phase reference [48], which is induced by non-gravitational forces.

IV. INDEPENDENCE OF PHASE SHIFT FROM PROBABILITY DISTRIBUTION
We next consider whether the field energy principle can be satisfied in alternative gravitational theories that lack gravitational superposition states. In semiclassical gravitational theories, the gravitational field is typically sourced by the expectation value of the stress-energy tensor, which implies that the field is always in a classical state [6]. Since the gravitational field in such theories is not entangled with the interferometer trajectories, Eq. (15) cannot hold. It is possible that the interferometer phase shift could still be a function of the gravitational field energy, but only if the phase shift depends on the probability distribution of the interfering particle.
We present data from the experiment described in Ref. [35] to demonstrate the insensitivity of the gravitational phase shift to the test particle's probability distribution. Fig. 2 shows the phase shift induced by the tungsten source mass as a function of the probability on the upper interferometer arm. An asymmetric probability distribution is created by altering the Rabi frequency of the initial beam splitter pulse. Systematic effects, e.g. from ac-Stark shifts, are suppressed by comparing the interferometer phase with and without the source mass installed; each point in Fig. 2 represents the phase shift between these two configurations. Imaging-related systematic errors are further reduced by reversing the direction of the horizontal detection fringe [32]. Within the experimental resolution, there is no statistically significant correlation between the test particle's center of mass and the interferometer phase shift (slope −0.07 ± 0.05, p = 0.39). This result is correctly predicted by quantum mechanics, in which the time evolution depends linearly on the wavefunction. However, the apparent independence of phase shift from center-of-mass position is surprising from the perspective of semiclassical gravitational theories, where the gravitational field energy depends on the test particle's probability distribution. Such theories cannot express the phase shift in terms of the gravitational field energy while remaining consistent with experiment.

V. THE RELATIVITY OF SUPERPOSITION
In classical mechanics, every particle is associated with a reference frame-a coordinate system in which that particle is located at the origin. With appropriate coordinate transformations, any physical interaction can be described in the reference frame of any particle. This property also holds in quantum mechanics, where the reference frame associated with a quantum particle is called a quantum reference frame (QRF) [37]. The core of the formulation introduced in Ref. [37] is the generalization of a reference frame transformation to a quantum superposition of reference frame transformationsŜ. Such a transformation allows one to associate a reference frame with a particle that is in a quantum superposition of positions from the perspective of the initial reference frame. QRFs have been shown to be relevant for quantum systems in a gravitational field or in a superposition thereof [49][50][51][52][53][54][55].
In Section 2, our simplified model of the gravitational Aharonov-Bohm experiment was described in a QRF associated with the detector particle D, which we may conceptualize as the "laboratory" QRF. We now use a QRF transformation to describe the same model in a QRF associated with the test particle (particle A). Specifically, we transform to the QRF centered on the position of particle A. The position operators of the other particles correspond, in this QRF, to the displacements between those particles and the trajectory of particle A. This transformation is accomplished bŷ HereP AD is the parity-swap operator, defined aŝ The position operatorsŷ B ,ŷ D and momentum operatorŝ π B ,π D in particle A's QRF are given bŷ The unperturbed trajectories are where x i (t) and x s (t) were defined respectively in Eq. BD that includes the gravitational interaction with particle B is given bŷ The definition of the interferometer phase, given in Eq. (5), is unchanged by the QRF transformation. Fig. 3 illustrates the experiment in particle A's reference frame. When the gravitational interaction is included, the quantum state in the reference frame of particle A takes the form where β i represents the gravitational field sourced by point particles of mass m and M at positions 0 and x s − x i , respectively. Note that in this reference frame, particles B and D are in an entangled superposition state, The gravitational interaction between test particle A and source particle B induces a phase shift. (B): Same experiment in the QRF of unperturbed particle A. In this QRF, the phase shift of particle A arises because particle A evolves in the gravitational field superposition sourced by particle B. and the gravitational field is also in an entangled superposition state with length scale kT /m. We now need to interpret ∆φ, the phase shift of particle A due to the gravitational interaction, in the QRF associated with particle A: In this case, particle A is located at the origin [56], so ∆φ cannot be explained in terms of the different paths taken by particle A. Instead, particle B is in a quantum superposition of two semiclassical states, and hence the phase is due to the gravitational field being in a quantum superposition at the location of particle A (see Eq. (24)). From this perspective, the phase can be obtained as where V i , i = 1, 2 are the two different gravitational potentials associated with the two different positions of particle B, integrated over the interferometer time. Quantitatively, this result agrees with the calculation in particle D's reference frame (as expected, since ∆φ is an observable). Comparing these two descriptions of the system allows us to draw several conclusions. First, since the reference frames of particles A and D can both be used to calculate the observable ∆φ, the experiment provides no reason to prefer one reference frame over the other. Second, the expression for ∆φ takes the same form in both reference frames. This occurs because the phase shift depends only on the relative configuration of the masses interacting gravitationally, namely the relative displacement between particles A and B. This is a manifestation of the relational character of general relativity, which has been widely confirmed for classical sources. The invariance of physical laws under coordinate transformations (general covariance) is familiar from general relativity, where the laws of physics are valid regardless of the choice of a specific reference frame (relativity principle). This experiment exhibits a quantum version of the relativity principle, where the physical laws are valid in any QRF.
In general, QRF transformations demonstrate that superposition and entanglement are relative properties [37]. In the gravitational Aharonov-Bohm experiment, the relative superposition between the test particle and the source mass gives rise to a gravitational field superposition in every QRF. Note that in previous experiments [29] where the gravitational field is approximately uniform over the length scale of the apparatus, the QRF transformation in Eq. (16) leaves the gravitational field of the source mass in a state that is not resolvably entangled with the test mass trajectory. As implied by the equivalence principle, local experiments cannot detect a gravitational field, even if the gravitational field is generated by a superposed source [57] (see Appendix 2).

VI. COMPARISON TO THE PROPOSED BMV EXPERIMENTS
By assuming that the quantum relativity principle is valid, the results of future experiments to detect gravita- tionally mediated entanglement generation [7,8] can be inferred from gravitational Aharonov-Bohm-like experiments. This follows from the fact that QRF transformations can map entangled states into single-particle superposed states and vice versa. Hence, BMV experiments can be related to measurements of single-particle superpositions by QRF transformations. Fig. 4A illustrates a generic BMV experiment, where two massive particles become entangled due to their gravitational interaction. Fig. 4B shows the same experiment in the QRF of one of the particles. In this QRF, one of the massive particles is in a well-localized state, and the other is in a superposition state. The signature of entanglement generation (namely, correlations between the probabilities of finding the system in each of its four output states) appears identically in both QRFs.
According to the QRP, the experiment in Fig. 4A is equivalent to the experiment in Fig. 4B performed in the laboratory reference frame. In other words, if the QRP is valid, then the result of a measurement like the one in Ref. [35] (but with the test particle superposed into four arms, rather than two) determines whether or not a BMV experiment would detect entanglement.
The QRP thus represents the key assumption that links the gravitational Aharonov-Bohm experiment and proposed BMV experiments. The value of performing a BMV experiment is that it would provide the first evidence for or against the QRP in the gravitational sector. Alternatively, the QRP could be tested (and semiclassical gravitational theories constrained) by means of singleparticle superposition experiments. Fig. 5 illustrates the type of measurement that needs to be performed to validate the QRP: a detection of the gravitational influence of a localized lower-mass particle on a higher-mass particle in a superposition state. Since the QRP can be tested with measurements on single particles, the direct observation of gravitationally mediated entanglement genera-tion may ultimately not be necessary to constrain semiclassical gravity. The key experimental challenge is simply to create a spatial superposition of a massive particle and measure its gravitational interactions with sufficient precision.

VII. RESULTS AND DISCUSSION
As outlined in Section 1, the standard quantummechanical description of the gravitational Aharonov-Bohm experiment is compatible with three general principles: (1) the existence of gravitational fields, (2) the field energy principle, and (3) the quantum relativity principle. If we assume that certain subsets of these principles are valid, then this experiment demonstrates a gravitational superposition state even in the context of potential alternatives to standard quantum mechanics.
First, assume that (1) and (2) are valid. For a contradiction, suppose that a theory describes the gravitational field in Ref. [35] as a classical state. Then in order to be consistent with classical mechanics, the gravitational field energy must be a function of the test particle's probability distribution. On the other hand, in order to be consistent with the experimental results in Section 4 of this work, the theory must predict that the interferometer's phase shift is independent of the test particle's probability distribution. The phase shift is thus independent of the gravitational field energy, and the theory violates (2).
The same conclusion holds if we instead assume that (1) and (3) are valid. For each of the gravitating objects (test mass and source mass), there exists a quantum reference frame in which the object is well-localized at some position x i . In that reference frame, by (1) and (3), the object sources a gravitational field g i (x − x i ). We can then write the full gravitational field in a common reference frame by means of quantum reference frame transformations. Since there is a relative superposition between the test mass and source mass, there is a relative superposition between their gravitational fields as well, i.e. the full gravitational field is in a superposition state. Moreover, the gravitational field superposition has an observable consequence in the experiment because gravity induces the phase shift in the interferometer, so the experiment demonstrates the existence of the superposition state.
To deny that the gravitational field is in a superposition state in the experiment of Ref. [35], theories of gravity in which (1) is valid must reject both (2) and (3) [60]. As an example, consider the Schrödinger-Newton theory [6], which treats the gravitational field as a fundamentally classical entity that is sourced by the probability distribution of matter. The equation of motion for this theory contains a gravitational self-interaction term that prevents the creation of large spatial superpositions of mas- In this QRF, the test particle is at rest, while the source mass and detector are in an entangled superposition state. (C): Hypothetical experiment in which the source mass is placed in superposition relative to the test particle and detector. The relative quantum state of the test particle and source mass is the same in both experiments. By the quantum relativity principle, Experiment 1 has the same measurement outcome as Experiment 2. Performing an experiment like the one depicted in (C) would test the QRP.
sive particles, where the critical length scale depends sensitively on the mass. In the laboratory reference frame, the Schrödinger-Newton equation predicts that the gravitational field during the interferometer is in a classical state of the form where A i is the wavefunction amplitude of the test mass on trajectory i. The energy of the gravitational field is therefore The predicted (and observed) independence of the interferometer phase shift from A 1 and A 2 indicates that the phase shift is not a function of E S-N G , violating (2). Furthermore, the Schrödinger-Newton theory implicitly contains a preferred quantum reference frame. Although the Schrödinger-Newton equation evaluated in the laboratory frame appears to be consistent with the experimental results of Ref. [35], in the quantum reference frame of the test mass, the Schrödinger-Newton equation inconsistently predicts the collapse of the source mass superposition. The theory thus violates (3) [58]. A similar consideration can be also made in the case of Penrose's spontaneous state reduction [53], where the equivalence of all quantum reference frames can be stated in terms of a generalisation of the Einstein Equivalence Principle [50].
In conclusion, if one makes certain general assumptions about the existence and properties of gravitational fields, then the existence of a gravitational superposition state has already been demonstrated experimentally. Future experiments will seek to obviate the need for these assumptions, providing even more stringent tests of gravity's fundamental nature. Either the observation of grav-itational entanglement generation in a BMV experiment, or a sufficiently precise gravitational measurement using a superposed massive particle, would demonstrate the existence of a nonclassical state of the gravitational field even without assuming principle (2)   In this section, we demonstrate the reciprocity of the electromagnetic and the linearized gravitational actions. We also show that the actions depend only on the relative position of the source particles. These properties provide a more general justification for the quantum relativity principle.
Let us first review the electromagnetic case. The free action of the electromagnetic field is and A µ (x) is the vector potential. In addition to the free action, in the presence of a source term there is the matter-field interaction, which can be written as Here, J µ (x) is the current density associated with the source. In the presence of two sources with total current where each A µ i with i = 1, 2 solves the wave equation for the corresponding current J µ i (x). In perturbation theory, the interferometer phase shift can be calculated from the interaction term S int , which is completely symmetric under exchange of the two sources 1 and 2. This can be easily seen by making use of the where K µ ρ is a differential operator corresponding to the wave equation, which is obtained via the equations of motion of the free theory. A particular solution of the wave equation is obtained as A µ (x) = d 4 yG µ ν (x − y)J ν (y). Removing the self-interaction, we can rewrite Notice that this expression is valid in general, regardless of the relative magnitude of the charges sourcing the field. This shows that the phase can be equivalently calculated by considering the field sourced by system 1 coupled to the source term of system 2, or vice versa. The same reasoning applies to the weak gravitational field. From the weak gravity action (see, e.g. Ref. [61]) where T µν is the stress-energy tensor, it is possible to cast the equations of motion in the form Explicitly, one finds where = ∂ ρ ∂ ρ and the curly brackets indicate the symmetrization of the indices. A particular solution of Eq. (33) is where K µν αβ G αβ ρσ (x − y) = δ µ {ρ δ ν σ} δ (4) (x − y). In the linearized regime of gravity, the principle of linear superposition holds, so that in the presence of two sources of the gravitational field T µν (x) = T µν 1 (x) + T µν 2 (x) the full metric is the sum of the two independent solutions h µν 1 (x) and h µν 2 (x) corresponding respectively to the stress energy tensors T µν 1 (x) and T µν 2 (x). We then remove the self-interaction analogously to the electromagnetic case, and we obtain From the symmetry of the action, it is now clear that it is equivalent to consider the gravitational field sourced by system 1, namely h µν 1 (x) = d 4 y G µν αβ (x − y)T αβ 1 (y), coupled to system 2 via its stress-energy tensor T µν 2 (x) or, vice versa, the gravitational field sourced by system 2, namely h µν 2 (x) = d 4 y G µν αβ (x − y)T αβ 2 (y), coupled to system 1 via T µν 1 (x). Notice that the Green's function G µν αβ (x − y) is symmetric in its indices and is the same in both cases, as it is determined by the free theory. Furthermore, the Green's function depends only on the position difference between systems 1 and 2.
For two localized particles, we can choose a coordinate system that is centered on one of the particles, say in an initial position x 0 . It is easy to see that this choice of coordinates leaves the action invariant by changing the coordinates to x = x + x 0 , y = y + x 0 in Eq. (36). Then, when one of the particles is in a quantum superposition of two states that are well-localized in the position basis, it can be shown [44] that the gravitational field can be obtained (in the linearized regime) as the quantum superposition of the ground state configurations of the field associated with each state. Thanks to the fact that the action of Eq. (36) is invariant for the single transformation, we conclude that the difference of the action along the two paths is also invariant under a change of QRF. In the case of the gravitational Aharonov-Bohm experiment [35], which can be described in the Newtonian limit, this means that it is equivalent to use the laboratory description in which particle B is localized while particle A is in a quantum superposition of trajectories, or the QRF associated with the position of particle A, in which particle B is in a quantum superposition of trajectories [62]. According to the equivalence principle, gravity cannot be observed locally [63]. As applied to classical gravitational sources, the equivalence principle has been tested to high precision with classical [64] and quantum [32] test masses. The QRP motivates an extension of the equivalence principle to the case where a gravitational source is in a quantum superposition state [50,52]. We may state the equivalence principle for quantum sources as follows: no local experiment can detect the gravitational field produced by a superposed gravitational source. . Accelerometer (test particle A and reference D) in the gravitational field of a quantum source (B). The distance d between A and D is small enough that the accelerometer can be considered a local system. According to the equivalence principle, even though the trajectories of A and D become entangled with the trajectories of B, the distance between A and D is unaffected. Thus, the accelerometer cannot detect the gravitational field from B. Fig. 6 depicts a thought experiment that attempts to observe the gravitational field from a quantum source by using an accelerometer. The accelerometer consists of a test particle (which may be classical or quantum) and a reference; the accelerometer measures the relative acceleration between the test particle and the reference. As the system evolves in time, the gravitational field entangles the trajectories of the test particle and the reference with the trajectory of the gravitational source. However, as long as the distance between the test particle and reference is sufficiently small that gravitational tidal effects can be ignored, there is no relative acceleration between the test particle and the reference on either branch of the accelerometer's wave function. This follows from the equivalence principle, which states that the test particle and reference fall identically in a gravitational field, regardless of their masses or compositions.
As long as the accelerometer's time evolution is linear, no information can be transmitted from one branch of its wave function to the other, and therefore the accelerometer cannot detect the gravitational field from the quantum source. If particles A and D are replaced by clocks, an analogous argument shows that there is no observable redshift due to particle B (see Ref. [52] for an explicit derivation of this result in quantum reference frames). The equivalence principle for quantum sources is thus satisfied in standard quantum mechanics. Furthermore, any theory of gravity that incorporates the (classical) equivalence principle as well as linear time evolution also includes the equivalence principle for quantum sources.

APPENDIX 3: ENTANGLEMENT IN THE GRAVITATIONAL AHARONOV-BOHM EXPERIMENT
Since BMV experiments are usually discussed in terms of gravitationally mediated entanglement generation, one might wonder how the BMV result can be inferred from the gravitational Aharonov-Bohm experiment without mentioning entanglement. In fact, if the QRP is valid, there is gravitational back action and entanglement generation in the experiment of Ref. [35] (see Fig. 7). Specifically, the gravitational interaction between the source mass and the upper interferometer arm entangles their trajectories, causing phase shifts and deflections of the source mass trajectories during the interferometer. As pointed out by Vaidman in the context of the electromagnetic Aharonov-Bohm effect, the phase shift of the test particle due to the interaction is equivalent to the phase shift of the source [65]. At a particular launch height, the net deflections of the source mass and test mass vanish at the time of each beamsplitter pulse.
It is not possible to directly observe the back action on the source mass during the interferometer, even in principle, because the trajectory perturbation due to the back action is much smaller than the quantum uncertainty in the source mass position. For instance, even if the quantum uncertainty in the velocity of the source mass in Ref. [35] was as large as 1 mm/s, the quantum uncertainty in its position would still be at least 10 −32 m, which is larger than the maximum gravitational deflection of the source mass. Nevertheless, assuming that the QRP is valid, a self-consistent description of the experiment requires the entanglement generation and associated back action to The dashed (dotted) trajectory of particle A is entangled with the dashed (dotted) trajectory of particle B. Diagram is not to scale. In Ref. [35], the gravitational deflection of particle A is independently measured. Owing to the quantum uncertainty in particle B's position, the back action on particle B's trajectory cannot be observed. be included.