# Quantum fluctuations of spacetime generate quantum entanglement between gravitationally polarizable subsystems

## Abstract

There should be quantum vacuum fluctuations of spacetime itself, if we accept that the basic quantum principles we are already familiar with apply as well to a quantum theory of gravity. In this paper, we study, in linearized quantum gravity, the quantum entanglement generation at the neighborhood of the initial time between two independent gravitationally polarizable two-level subsystems caused by fluctuating quantum vacuum gravitational fields in the framework of open quantum systems. A bath of fluctuating quantum vacuum gravitational fields serves as an environment that provides indirect interactions between the two gravitationally polarizable subsystems, which may lead to entanglement generation. We find that the entanglement generation is crucially dependent on the polarizations, i.e, they cannot get entangled in certain circumstances when the polarizations of the subsystems are different while they always can when the polarizations are the same. We also show that the presence of a boundary may render parallel aligned subsystems entangled which are otherwise unentangled in a free space. However, the presence of the boundary does not help in terms of entanglement generation if the two subsystems are vertically aligned.

## 1 Introduction

Recently, gravitational wave signals from black hole merging systems have been directly detected [1, 2, 3, 4], and this confirms the prediction based on Einstein’s general relativity over a hundred years ago [5]. Naturally, one may wonder what happens if gravitational waves are quantized. One of the consequences, if gravity is quantized, is the quantum fluctuations of spacetime itself. A direct result of spacetime fluctuations is the flight time fluctuations of a probe light signal from its source to a detector [6, 7, 8]. Another expected effect is the Casimir-like force which arises from the quadrupole moments induced by quantum gravitational vacuum fluctuations [9, 10, 11, 12, 13], in close analogy to the Casimir and the Casimir-Polder forces [14, 15].

In the present paper, we are concerned with yet another effect associated with quantum fluctuations of spacetime, that is, quantum entanglement generation by quantum fluctuations of spacetime. Quantum entanglement is crucial to our understanding of quantum theory, and it has many interesting applications in various novel technologies. However, a significant challenge to the realization of these quantum technologies with quantum entanglement as a key resource is the environmental noises that lead to the quantum to classical transition. In the quantum sense, one environment that no physical system can be isolated from is the vacuum that fluctuates all the time. On one hand, there are obviously vacuum fluctuations of matter fields, and on the other hand, there should also be quantum vacuum fluctuations of spacetime itself if we accept that the basic quantum principles we are already familiar with apply as well to a quantum theory of gravity. Currently, there are discussions with vacuum fluctuations of quantized matter fields as inevitable environmental noises that cause quantum decoherence [16].

Like matter fields, the fluctuations of gravitational fields, i.e. the fluctuations of spacetime itself, may also cause quantum decoherence. Since gravitation can not be screened unlike electromagnetism, gravitational decoherence is universal. Different models for gravitational decoherence have been proposed [17, 18, 19, 20, 21, 22, 23, 24, 25, 26], see Ref. [27] for a recent review. However, decoherence is not the only role the environment plays. In certain circumstances, the indirect interactions provided by the common bath may also create rather than destroy entanglement between the subsystems via spontaneous emission and photon exchange [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. Here the bath can be either in the vacuum state [28, 29, 30, 31, 32, 33, 34], or in a thermal state [35, 36, 37, 38]. In Ref. [39], a general discussion showed that two independent atoms in a common bath can be entangled during a Markovian, completely positive reduced dynamics. In Ref. [40], this approach has been applied in the study of the entanglement generation for two atoms immersed in a common bath of massless scalar fields, and found that entanglement generation can be manipulated by varying the bath temperature and the distance between the two atoms. This work was further generalized to the case in the presence of a reflecting boundary [41, 42], which showed that the presence of a boundary may offer more freedom in controlling entanglement generation.

In this paper, we study whether a common bath of fluctuating gravitational fields may generate entanglement between subsystems just as matter fields do. We consider a system composed of two gravitationally polarizable two-level subsystems in interaction with a bath of quantum gravitational fields in vacuum. We will show that in certain conditions, the two subsystems may get entangled due to the fluctuating gravitational fields. Like matter fields, the fluctuations of gravitational fields are also expected to be modified if a boundary is present. Although it is generally believed that gravitational waves can hardly be absorbed or reflected, there have been proposals that the interaction between gravitational waves and quantum fluids might be significantly enhanced compared with normal matter (see Ref. [45] for a review), and recently there have been interesting conjectures that superconducting films may act as mirrors for gravitational waves due to the so-called Heisenberg–Coulomb effect [46, 47]. Therefore, we are interested in how the conditions for entanglement generation due to quantum fluctuations of spacetime is modified if such gravitational boundaries exist.

## 2 The basic formalism

*P*denotes the principal value. The coefficient matrix \(H_{ij}^{(\alpha \beta )}\) and \(C_{ij}^{(\alpha \beta )}\) can then be expressed as

## 3 The condition for entanglement generation

*T*denotes the matrix transposition, and the three-dimensional vectors \(\mu _{i}\), \(\nu _{i}\) are \(\mu _{i}=\nu _{i}=\{1,-i,0\}\). For brevity, we make the following substitution in Eq. (21), \(A^{(11)}\rightarrow A_1\), \(A^{(22)}\rightarrow A_2\), and \(A^{(12)}, A^{(21)}\rightarrow A_3\), and do the same to the coefficients \(B^{(\alpha \beta )}\). Plugging the new expressions of \(C_{ij}^{(\alpha \beta )}\) into Eq. (27), leads to

### 3.1 Entanglement generation in the free Minkowski spacetime

*L*denotes the separation between the two subsystems, as shown in Fig. 1. The explicit expressions of the coefficients \(A_i\) and \(B_{i}\) in Eq. (28) are given in appendix A, see Eqs. (A4)–(A6). Since \(A_{1}=B_{1}\) and \(A_{2}=B_{2}\), it is clear that the criterion for entanglement generation (28) becomes \(A_3^2>0\). That is, entanglement can be generated as long as \(A_{3}\), which is dependent on \(q_{ij}^{(\alpha )}\) and \(\omega L\), is nonzero.

When the polarizations of the two subsystems are the same, i.e., \(q^{(1)}_{ij}=q^{(2)}_{ij}=q_{ij}\), there are no solutions to \(F=K=0\) (see Eqs. (A5)–(A6)) for any given *L*, i.e., irrespective of the polarization of the two subsystems, the entanglement between the two subsystems with the same polarization can always be generated for a given separation. Therefore, when the polarizations of the two subsystems are the same, they can always get entangled if the separation is finite, except for a series of special values of \(\omega L\) which are the zero points of \(A_3\). Similar conclusions have been drawn in the case of two independent atoms coupled with massless scalar (matter) fields [40]. This suggests that there is no difference between spacetime fluctuations and matter fields fluctuations when the entanglement generation is concerned with two subsystems of the same gravitational polarization.

*x*axis,

*y*axis and

*z*axis, and the off-diagonal components \(q_{12}\), \(q_{13}\) and \(q_{23}\) respectively represent the contributions to the total mass quadrupole moment from the mass distributed in the

*xoy*,

*xoz*and

*yoz*plane. Therefore, when the mass distributions of the two subsystems induced by quantum gravitational vacuum fluctuations satisfy the condition (29), the two subsystems remain disentangled.

*xoy*plane, i.e. \(q^{(1)}_{3i}=q^{(2)}_{3i}=0\). In this case, we have

### 3.2 Entanglement generation for two gravitationally polarizable subsystems aligned parallel to the boundary

*xoz*plane, and the two subsystems separated from each other by a distance

*L*is placed along the

*z*direction at a distance of

*y*, see Fig. 3 . The coefficients \(A_1,~A_2,~A_3\) and \(B_1,~B_2,~B_3\) can be calculated from Eqs. (23) and (25) with the Fourier transform of Eq. (A7), and the results are lengthy, so we do not give them explicitly here. As before, we will examine when \(A'_{3}=0\) to determine whether entanglement can be generated or not, since \(A'_i=B'_i~(i=1,2,3)\) still holds in the present circumstances. In this case, the conditions that entanglement cannot be generated for any

*L*are solved as follows

*xoy*plane, \(q^{(1)}_{3i}=q^{(2)}_{3i}=0\). In this case, we have

*L*and

*y*, the corresponding coefficients satisfy \(F_{1}=K_{1}=M_{1}=N_{1}=0\), and then we have

### 3.3 Entanglement creation for two gravitationally polarizable subsystems aligned vertical to the boundary

*z*=0 in the

*xoy*plane and the two subsystems are placed on the

*z*-axis with a separation

*L*, the distance from the boundary to the nearer subsystem being

*z*, see Fig. 4.

Following the same procedures, we find that the conditions that entanglement generation cannot happen for any given *L* and *z* are the same with that obtained in the case without a boundary, see Eq. (29). That is, when the quadrupole moments of the two subsystems satisfy certain conditions such that entanglement cannot be generated in the free space, it cannot be generated in the presence of a boundary placed vertically to the alignment of the subsystems either.

## 4 Summary

In this paper, we have investigated the entanglement generation at the neighborhood of the initial time between two independent gravitationally polarizable two-level subsystems in interaction with a bath of fluctuating quantum gravitational fields in vacuum both with and without a boundary. The partial transposition criterion has been applied to determine whether entanglement can be generated or not at the beginning of evolution. In the free space case, when the polarizations of the two subsystems are the same, the two subsystems can always get entangled as long as the separation is finite. This is similar to what happens when the fluctuations of scalar (matter) fields are considered. When the polarizations of the two subsystems are different, they cannot get entangled in certain circumstances, whatever the separation is. This is in sharp contrast with the case of massless scalar (matter) fields. In the presence of a boundary, we find that in some of the cases where entanglement cannot be generated in a free space, the presence of a boundary placed parallel to the alignment of the subsystems may render the subsystems entangled, but this does not happen for a boundary placed vertically.

## Notes

### Acknowledgements

This work was supported in part by the NSFC under Grants No. 11435006, No. 11690034, and No. 11805063.

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