Field-induced entanglement in spatially superposed objects

The full picture of quantum gravity Feynmann1995 ; Aharony2000 ; Kiefer2006 ; Woodard2009 , which unifies general relativity and quantum mechanics, is still unclear. This is attributed to the lack of theoretical and experimental approaches to connect gravitational and quantum phenomena. However, with the recent development of various quantum technologies Doherty2013 ; Aspelmeyer2014 ; Matsumoto2019 ; Catano-Lopez2020 , there have been attempts to clarify quantum natures of gravity (for example, see Carney2018 and the references therein, or the recent works Marletto2018 ; Carlesso2019 ; Krisnanda2020 ; vandeKamp2020 ; Grobardt2020 ; Qvafort2020 ; Guerreiro2020 ; Kanno2020 ; Hall2018 ; Belenchia2018 ; Howl2019 ; Marshman2020 ; Anastopoulos2020 ; Balushi2018 ; Miao2020 ; Matsumura2020 ; Nguyen2020 ; Miki2020 ; Tilly2021 ). In such works, the quantum gravity induced entanglement of masses (QGEM) proposal Bose2017 ; Marletto2017 ; Marshman2020 has been attracting attentions. In the proposal, the authors considered two objects each in a superposition of two trajectories and assumed the Newtonian potential between them. The gravitational interactions generate the entanglement between the two objects. The detection of gravity-induced entanglement can be a witness of quantum nature of gravity.

The interesting point in the QGEM proposal is that two spatially superposed objects can probe quantum entanglement induced by fields. This is analogous to entanglement harvesting protocols Reznik2003 ; Lin2010 ; Salton2015 ; Pozas-Kerstjens2015 ; Pozas-Kerstjens2016 ; Simidzija2018a ; Stritzelberger2021 ; Henderson2020 ; Foo2021 by the Unruh-DeWitt detector. The Unruh-DeWitt detector is constructed by a particle with internal degrees of freedom, which locally interacts with a quantum field. In this model, the source of entanglement is the quantum field. In particular, it is known that the spacelike entanglement of a vacuum state induces the entanglement between the two spatially separated detectors (for example, see Reznik2003 ).

In this paper, we investigate how two superposed objects are capable to probe the entanglement of quantum fields. We assume that the fields have only dynamical degrees of freedom and any constraint equations are not imposed on the whole Hibert space of the objects and the fields. We consider the superposed objects which do not interact with each other and whose currents locally couple with the fields. By evaluating the currents along the objects’ trajectory, we compute the time evolution of the total system. For the case where the objects are spatially separated, we show that the state of trajectories remain disentangled if the microcausality condition holds for the quantum fields. In other words, such quantum fields cannot be mediators of spacelike entanglement for superposed trajectories of the objects. Our analysis also presents possible approaches and extensions of the objects’ model to verify the spacelike entanglement of fields; use of the internal degrees of freedom and extended model with multi-objects or multi-trajectories.

This paper is organized as follows. In Sec. II, the QGEM proposal to test quantum gravity and its thoretical approach are reviewed. In Sec. III, we introduce the model with the interaction given in a bilinear form of fields and currents of two objects. We derive the solution of the Schr$\ddot{\text{o}}$dinger equation. In Sec. IV, we investigate the separability of the two objects based on the solution. We find the no-go result of generation of spacelike entanglement, and discuss its implications. In Sec. V, the conclusion is devoted. We use the natural units $\hbar=c=1$ in this paper.

The experimental setting of two matter-wave interferometers to test quantum gravity was proposed, which is called the QGEM proposal Bose2017 ; Marletto2017 ; Marshman2020 . In each interferometer, a single object is in a superposition of two trajectories. Fig.1 presents a rough configuration of trajectories of each object.

We assume that the two objects interact with each other by the Newtonian potential. The Hamiltonian of the objects is

where $m_{\text{A}}$ and $m_{\text{B}}$ are the masses of the objects A and B, $\hat{\bm{x}}_{\text{A}}$ and $\hat{\bm{x}}_{\text{B}}$ are each position, and the Hamiltonian $\hat{H}_{\text{A}}$ and $\hat{H}_{\text{B}}$ determine each trajectory of the objects. Each of the two objects at $t=0$ is in the spatially superposed state,

where $|\psi_{\text{R}}\rangle_{\text{A}}$ and $|\psi_{\text{L}}\rangle_{\text{A}}$ are the states with wave packets localized around positions $\bm{x}=\bm{x}_{\text{A}_{\text{R}}}(0)$ and $\bm{x}=\bm{x}_{\text{A}_{\text{L}}}(0)$ at $t=0$, respectively. Also, $|\psi_{\text{R}}\rangle_{\text{B}}$ and $|\psi_{\text{L}}\rangle_{\text{B}}$ are defined in the same manner. Those states satisfy ${}_{\text{A}}\langle\psi_{\text{R}}|\psi_{\text{L}}\rangle_{\text{A}}\approx 0$ and ${}_{\text{B}}\langle\psi_{\text{R}}|\psi_{\text{L}}\rangle_{\text{B}}\approx 0$ when each wave packet is sufficiently separated. The evolved state $|\psi_{\text{f}}\rangle$ at $t=t_{\text{f}}$ is

where T is the time-ordered product, and $\hat{\bm{x}}^{\text{I}}_{\text{A}}(t)=e^{it(\hat{H}_{\text{A}}+\hat{H}_{\text{B}})}\hat{\bm{x}}_{\text{A}}e^{-it(\hat{H}_{\text{A}}+\hat{H}_{\text{B}})}$ and $\hat{\bm{x}}^{\text{I}}_{\text{B}}(t)=e^{it(\hat{H}_{\text{A}}+\hat{H}_{\text{B}})}\hat{\bm{x}}_{\text{B}}e^{-it(\hat{H}_{\text{A}}+\hat{H}_{\text{B}})}$ are the position operators in the interaction picture. The phase shift

is given by the Newtonian potential between the two objects on the trajectories $\bm{x}=\bm{x}_{\text{A}_{\text{P}}}(t)$ and $\bm{x}=\bm{x}_{\text{B}_{\text{Q}}}(t)$ $(\text{P},\text{Q}=\text{R},\text{L})$. In the expression (3), we omitted the symbol of the tensor product as $|\,\cdot\,\rangle_{\text{A}}\otimes|\,\cdot\,\rangle_{\text{B}}=|\,\cdot\,\rangle_{\text{A}}|\,\cdot\,\rangle_{\text{B}}$. The approximation of the third line of Eq. (3) is given as

These equations are valid when the size of each wave packet is larger than de Broglie wave length of each object Ford1993 ; Breuer2001 . Choosing the masses, the distance between a pair of trajectories and the traveling time properly, we find that the state (3) is entangled. Hence the gravitational interaction can generate quantum entanglement. The key point in the QGEM proposal is that the spatially superposed objects can probe quantum entanglement. In the following sections, we will discuss the detection of entanglement of dynamical fields by using such objects.

In this section, we introduce a model of two obhects and fields to examine the detection of entanglement of the fields. In the Schr$\ddot{\text{o}}$dinger picture, we consider the Hamiltonian of two objects A and B and fields as

where the Hamiltonians $\hat{H}_{\text{A}}$, $\hat{H}_{\text{B}}$ and $\hat{H}_{\text{F}}$ determine each dynamics of the objects A, B and the fields. The vectors $\hat{\bm{J}}_{\text{A}}$ and $\hat{\bm{J}}_{\text{B}}$ are current operators with respect to the objects A and B, and $\hat{\bm{\phi}}$ is the field operator. The inner product $\bm{J}\cdot\bm{\phi}$ is defined by $\sum_{k}J^{k}\phi_{k}$ with labels $k$.

We assume that the fields have only dynamical degrees of freedom and there are no constraint equations on the whole Hilbert space. The field operators are represented on a physical Hilbert space $\mathcal{H}_{\text{F}}$ without negative norm states. In gauge field theories, there are formalisms using an unphysical Hilbert space of fields with gauge degrees of freedom Weinberg1996 . The fact that there are no negative norm states will be used to derive our result in the next section.

We note that the Hamiltonian (6) does not completely represent that in the linearized Einstein theory. At first glance, by choosing the component of currents $\hat{J}^{k}_{\text{A}}$, $\hat{J}^{k}_{\text{B}}$ and the fields $\hat{\phi}_{k}$ as the energy-momentum tensor $\hat{T}^{\mu\nu}$ and the metric perturbation $\hat{h}_{\mu\nu}$ properly, the local interaction $\hat{V}$ seems to be that in the linearized Einstein theory. This is not correct since the fields and those Hilbert space $\mathcal{H}_{\text{F}}$ are assumed not to have gauge degrees of freedom and negative norm states. Also, even for the transverse traceless gauge ($\hat{h}_{\mu\nu}$ have only physical modes), the Hamiltonian (6) is not admitted in the linearized Einstein theory. This is because, from the constraints of the Einstein equation, the non-dynamical parts of the metric perturbation give nonlocal interactions such as the Newtonian potential. However, there are no nonlocal interactions between the two objects in our model.

The almost same argument holds for the quantum electromagnetic dynamics, but we can admit an effective model described by the Hamiltonian (6). Let us consider that the objects A and B without total electric charges and with the electric dipole moments $\hat{\bm{d}}_{\text{A}}$ and $\hat{\bm{d}}_{\text{B}}$, respectively. For the distant objects, the Coulomb potential between them is neglected, and the local coupling to an electric field $\hat{\bm{E}}$ can be dominant. By assigning the field operator $\hat{\bm{\phi}}$ and the currents $\hat{\bm{J}}_{\text{A}}$ and $\hat{\bm{J}}_{\text{B}}$ to $\hat{\bm{E}}$, $\hat{\bm{d}}_{\text{A}}\delta^{3}(\bm{x}-\bm{x}_{\text{A}})$ and $\hat{\bm{d}}_{\text{B}}\delta^{3}(\bm{x}-\bm{x}_{\text{B}})$, our model describes the objects with the dipole coupling to the electric field at the positions $\bm{x}=\bm{x}_{\text{A}}$ and $\bm{x}=\bm{x}_{\text{B}}$. In Pozas-Kerstjens2016 , a similar model with time-dependent couplings and spatially smearing functions was considered as the Unruh-DeWitt detector model.

We consider that each object at $t=0$ is in a superposition of two local states $|\psi_{\text{R}}\rangle$ and $|\psi_{\text{L}}\rangle$ with $\langle\psi_{\text{P}}|\psi_{\text{P}^{\prime}}\rangle\approx\delta_{\text{PP}^{\prime}}$ ($\text{P},\text{P}^{\prime}=\text{R},\text{L}$). As the mentioned above, the interation of the model (6) can describe dipole coupling in the quantum electrodynamics. Each object may have some internal degrees of freedom such as electric dipole moments. We assume that the internal degrees of freedom of each objects at $t=0$ is in states $|a\rangle_{\text{Ai}}$ and $|b\rangle_{\text{Bi}}$, respectively. The objects move on the trajectories determined by the Hamiltonian $\hat{H}_{\text{A}}$ and $\hat{H}_{\text{B}}$ (see Fig. 1). The current operators $\hat{\bm{J}}^{\text{I}}_{\text{A}}(t,\bm{x})=e^{i\hat{H}_{0}t}\hat{\bm{J}}_{\text{A}}(\bm{x})e^{-i\hat{H}_{0}t}$ and $\hat{\bm{J}}^{\text{I}}_{\text{B}}(t,\bm{x})=e^{i\hat{H}_{0}t}\hat{\bm{J}}_{\text{B}}(\bm{x})e^{-i\hat{H}_{0}t}$ in the interaction picture defined with $\hat{H}_{0}=\hat{H}_{\text{A}}+\hat{H}_{\text{B}}+\hat{H}_{\text{F}}$ are approximated by the local values :

where $\hat{\bm{j}}^{\text{I}}_{\text{A}_{\text{P}}}(t,\bm{x})=\hat{\bm{s}}^{\text{I}}_{\text{A}}(t)\delta^{3}(\bm{x}-\bm{x}_{\text{A}_{\text{P}}}(t))$ and $\hat{\bm{j}}^{\text{I}}_{\text{B}_{\text{Q}}}(t,\bm{x})=\hat{\bm{s}}^{\text{I}}_{\text{B}}(t)\delta^{3}(\bm{x}-\bm{x}_{\text{B}_{\text{Q}}}(t))$ ($\text{P},\text{Q}=\text{R},\text{L}$) with the internal physical quantities $\hat{\bm{s}}^{\text{I}}_{\text{A}}(t)$ and $\hat{\bm{s}}^{\text{I}}_{\text{B}}(t)$ acting on the Hilbert spaces $\mathcal{H}_{\text{Ai}}$ and $\mathcal{H}_{\text{Bi}}$ of internal degrees of freedom, respectively. For example, if the objects have electric dipole moments and the fields are electric fields, the classical current $\hat{\bm{j}}^{\text{I}}_{\text{A}_{\text{P}}}(t,\bm{x})$ of the object A has the form $\hat{\bm{j}}^{\text{I}}_{\text{A}_{\text{P}}}(t,\bm{x})=\hat{\bm{d}}^{\text{I}}_{\text{A}}(t)\delta^{3}(\bm{x}-\bm{x}_{\text{A}_{\text{P}}}(t))$ with the electric dipole $\hat{\bm{d}}^{\text{I}}_{\text{A}}(t)(=\hat{\bm{s}}^{\text{I}}_{\text{A}}(t))$ in the interaction picture. The similar argument is made for the object B.

When the fields are in a state $|\chi\rangle_{\text{F}}$ at $t=0$, the state of the objects and the fields at $t=0$ is

where

where $|\alpha_{\text{R}}|^{2}+|\alpha_{\text{L}}|^{2}\approx 1$ and $|\beta_{\text{R}}|^{2}+|\beta_{\text{L}}|^{2}\approx 1$ holds since the state $|\psi_{\text{P}}\rangle$ satisfies $\langle\psi_{\text{P}}|\psi_{\text{P}^{\prime}}\rangle\approx\delta_{\text{PP}^{\prime}}$. Note that the initial product state may be not valid if there are constraint equations on the objects and fields. The solution of the Schr$\ddot{\text{o}}$dinger equation is

where $\hat{\bm{\phi}}^{\text{I}}(t,\bm{x})=e^{i\hat{H}_{0}t}\hat{\bm{\phi}}(\bm{x})e^{-i\hat{H}_{0}t}$. In the third line, we used the approximations (7) assigning the local currents and defined the unitary operator

The unitary operator $\hat{U}_{\text{PQ}}$ acts on not only fields’ state $|\chi\rangle_{\text{F}}$ but also the states of the internal degrees of freedom of the objects $|a\rangle_{\text{Ai}}$ and $|b\rangle_{\text{Bi}}$.

In the next section, we examine the entanglement between the two objects A and B using the solution Eq. (10). We will show no generation of entanglement for the trajectories of objects which are in spacelike regions. This argument follows by the microcausality of fields, which is independent of the dynamics of the fields.

In this section, we investigate the generation of entanglement between the two objects. Before mentioning our result, we focus on two origins of the generation of entanglement.

First, it is important to consider whether the unitary evolution gives correlations between the objects or not. The Hamiltonian $\hat{H}_{0}=\hat{H}_{\text{A}}+\hat{H}_{\text{B}}+\hat{H}_{\text{F}}$ yields independent dynamics of each system, which give no correlations. On the other hand, the unitary evolution $\hat{U}_{\text{PQ}}$ Eq.(11) given by the local interaction $\hat{V}$ leads to the following process: the object A locally excites the fields, and then the excitaions propagate to the object B and alter the potential around it. This process gives effective interactions and induces correlations between the objects A and B. In fact, there are no such effects when the two objects are in spacelike separated regions (see Fig. 2).

If the fields in spacelike regions commute each other (the microcausality condition, for example, see Weinberg1995 ), we have

where note that $\hat{\bm{j}}^{\text{I}}_{\text{A}_{\text{P}}}(t,\bm{x})=\hat{\bm{s}}^{\text{I}}_{\text{A}}(t)\delta^{3}(\bm{x}-\bm{x}_{\text{A}_{\text{P}}}(t))$ and $\hat{\bm{j}}^{\text{I}}_{\text{B}_{\text{Q}}}(t^{\prime},\bm{y})=\hat{\bm{s}}^{\text{I}}_{\text{B}}(t^{\prime})\delta^{3}(\bm{y}-\bm{x}_{\text{B}_{\text{Q}}}(t^{\prime}))$ with the internal quantities $\hat{\bm{s}}^{\text{I}}_{\text{A}}(t)$ and $\hat{\bm{s}}^{\text{I}}_{\text{B}}(t)$ of each object. Then the unitary operator $\hat{U}_{\text{PQ}}$ is factorized into the local unitaries,

where $\hat{U}_{\text{A}_{\text{P}}}$ and $\hat{U}_{\text{B}_{\text{Q}}}$ are

The local unitaries $\hat{U}_{\text{A}_{\text{P}}}$ and $\hat{U}_{\text{B}_{\text{Q}}}$ act on the Hilbert spaces $\mathcal{H}_{\text{Ai}}\otimes\mathcal{H}_{\text{F}_{\text{A}}}$ and $\mathcal{H}_{\text{Bi}}\otimes\mathcal{H}_{\text{F}_{\text{B}}}$, where the total Hilbert space $\mathcal{H}_{\text{F}}$ of the fields is described by $\mathcal{H}_{\text{F}}=\mathcal{H}_{\text{F}_{\text{A}}}\otimes\mathcal{H}_{\text{F}_{\text{B}}}$. There are no interactions induced by the fields for the factorized evolution in Eq. (13), which does not generate entanglement between the two objects.

Another important point is quantum entanglement of fields’ state. The previous work Reznik2003 showed that a pair of Unruh-DeWitt detectors, even if they are spacelike separated, becomes entangled due to the entanglement of the vacuum of a relativistic field. Also, there are many works about the generation of entanglement for spacelike separated detectors in the context of entanglement harvesting protocol Lin2010 ; Salton2015 ; Pozas-Kerstjens2015 ; Pozas-Kerstjens2016 . These works mean that the entanglement of the state $|\chi\rangle_{\text{F}}$ of the fields can be a source of the entanglement of the objects.

However, in the following we find that the spacelike entanglement of fields cannot be generated in the state of the trajectories. The definition of entanglement as follows: a given state is not entangled if the density operator $\rho$ of a system has a separable form Werner1989 ; Nielsen2002 ; Horodecki2009 ,

where $p_{i}$ is a probability, $\rho_{i}$ and $\sigma_{i}$ are density operators of the subsystems. A state which cannot be written in such form is called entangled. We show that the state of the objects’ trajectories is written in a separable form. Tracing out the fields and the internal degrees of freedoms from the evolved state (10), for the case where the objects are in spacelike regions, the reduced density operator for the trajectories is

where we used Eq. (13) and introduced $|\chi^{\prime}\rangle=|\chi\rangle_{\text{F}}|a\rangle_{\text{Ai}}|b\rangle_{\text{Bi}}$ as a short notation. The evolution operator $e^{-i\hat{H}_{0}t_{\text{f}}}$ was ignored because each degree of freedom just evolves independently by the free Hamiltonian $\hat{H}_{0}$. The unitary operator $\hat{V}^{\text{A}}_{\text{P}^{\prime}\text{P}}=\hat{U}^{\dagger}_{\text{A}_{\text{P}^{\prime}}}\hat{U}_{\text{A}_{\text{P}}}$ appearing in (17) satisfies

and hence all of the unitaries $\hat{V}^{\text{A}}_{\text{RR}},\hat{V}^{\text{A}}_{\text{RL}},\hat{V}^{\text{A}}_{\text{LR}}$ and $\hat{V}^{\text{A}}_{\text{LL}}$ commute each other. This means that $\hat{V}^{\text{A}}_{\text{P}^{\prime}\text{P}}$ has the following spectral decomposition,

where $\hat{\mu}_{\text{Ai}\otimes\text{F}_{\text{A}}}$ is an operater-valued measure on the Hilbert space $\mathcal{H}_{\text{Ai}}\otimes\mathcal{H}_{\text{F}_{\text{A}}}$. The real phase $\theta_{\text{P}^{\prime}\text{P}}(\lambda)$ has the antisymmetric property $\theta_{\text{P}^{\prime}\text{P}}(\lambda)=-\theta_{\text{P}\text{P}^{\prime}}(\lambda)$, which reflects Eq. (18). As the number of trajectories for each object is two, the number of independent components of $\theta_{\text{P}^{\prime}\text{P}}(\lambda)$ is one. Hence the phase is always written as

where $n_{\text{R}}=0$ and $n_{\text{L}}=1$. From the above facts, we find that the reduced density operator $\rho$ is separable,

where we used Eqs. (19) and (20) and defined the probability measure $\mu$ with $d\mu(\lambda)=\langle\chi^{\prime}|d\hat{\mu}_{\text{Ai}\otimes\text{F}_{\text{A}}}(\lambda)|\chi^{\prime}\rangle$, the state $|\psi(\lambda)\rangle_{\text{A}}$ and the density operator $\sigma_{\text{B}}(\lambda)$ as

Here, we emphasize that the Hilbert space $\mathcal{H}_{\text{F}}$ of the fields has no negative norm states, which was mentioned below Eq. (6). The fact leads to the inequalities $\mu(\lambda)\geq 0$ and $\sigma_{\text{B}}(\lambda)\geq 0$ and gurantees that $\mu(\lambda)$ and $\sigma_{\text{B}}(\lambda)$ are a probability measure and a density operator, respectively. Hence the separablity of the state of the objects’ trajectories holds. If gauge degrees of freedom are included in the fields, the Hilbert space $\mathcal{H}_{\text{F}}$ may have a negative norm state and the separability is not always guaranteed.

The separability of the objects does not depend on the dynamics of fields and the details of classical trajectories. Also, the seprability holds even for the case where the objects’ state of the internal degrees of freedom and the fields are initially in a mixed state. Our result means that the fields do not play a role of quantum mediators to generate the spacelike entanglement among the trajectories of such objects.

We compare our result with the no-go theorems in Marletto2017 ; Simidzija2018 on generation of entanglement. The theorem in Marletto2017 argued that two systems mediated by classical systems with only a single observable (this is the meaning of “classical” for that claim) have no entanglement. For our model, the mediators are the fields, which may have noncommutative observables, for example, the field operator and its conjugate. In this sense, the fields can be quantum systems in general. However, there are no generations of spacelike entanglement.

The no-go theorem in Ref. Simidzija2018 elucidates our result. We can rewrite Eq. (10) for the spacelike separated two objects as

where we used Eq. (13), and $|\Psi_{\text{in}}\rangle$ is the initial state given in Eq. (8). In this formula, we find the controlled unitary $\hat{U}_{\text{AF}}$,

Exactly speaking, $\hat{U}_{\text{AF}}$ has inverse only when it acts on the subspace spanned by $|\psi_{\text{R}}\rangle_{\text{A}}$ and $|\psi_{\text{L}}\rangle_{\text{A}}$ of the Hilbert space $\mathcal{H}_{\text{A}}$. In Ref. Simidzija2018 , the authors showed that the unitary evolution $\hat{U}=(\hat{U}_{\text{AS}}\otimes\hat{\mathbb{I}}_{\text{B}})(\hat{\mathbb{I}}_{\text{A}}\otimes\hat{U}_{\text{BS}})$ with the exponential of a Schmidt rank-1 operator $\hat{U}_{\text{AS}}=e^{-i\hat{m}_{\text{A}}\otimes\hat{X}_{\text{S}}}$ does not generate entanglement between the systems A and B. The systems A, B and S correspond to the objects A and B, and the fields F for our model. The controlled unitary $\hat{U}_{\text{AF}}$ is rewritten as the form

where $\hat{V}^{\text{A}}_{\text{RL}}=\hat{U}^{\dagger}_{\text{A}_{\text{R}}}\hat{U}_{\text{A}_{\text{L}}}$, and the self-adjoint operator $\hat{X}_{\text{F}}$ satisfies $e^{-i\hat{X}_{\text{F}}}=\hat{V}^{\text{A}}_{\text{RL}}\otimes\hat{\mathbb{I}}_{\text{F}_{\text{B}}}$, and $\hat{m}_{\text{A}}=0\times|\psi_{\text{R}}\rangle_{\text{A}}\langle\psi_{\text{R}}|+1\times|\psi_{\text{L}}\rangle_{\text{A}}\langle\psi_{\text{L}}|$. Since the entanglement between the two objects is invariant under the local unitary transformation $\hat{U}_{\text{A}_{\text{R}}}$, the controlled unitary $\hat{U}_{\text{AF}}$ plays the same role as the exponetial of a Schmidt rank-1 operator. Thus, our no-go result on generation of spacelike entanglement is a consequence of the no-go theorem in Simidzija2018 . Note that the no-go theorem can be applied under the approximation assigning local currents (7) and for the states of trajectories satisfying $\langle\psi_{\text{P}}|\psi_{\text{P}^{\prime}}\rangle\approx\delta_{\text{PP}^{\prime}}$. If these conditions do not hold, we need a further study on entanglement generation.

We comment on the extension of our model. It is well known that the spacelike entanglement of a field is extracted by the Unruh-DeWitt detectors Reznik2003 . Further, in Refs. Foo2021 ; Henderson2020 , the authors discussed an entanglement harvesting protocol by using the Unruh-DeWitt detectors with quantum superpositions of trajectories. The critical difference is that the states of trajectories are only focused on to show the separability. This means that the information of internal degrees of freedom are neccessary for an extraction of spacelike entanglement from fields. Further, it is worth considering a multi-partite Miki2020 or multi-trajectory Tilly2021 extended model of the QGEM proposal, since our result is based on the fact that each of two objects is superposed in two classical trajectories. It is interesting to characterize the advantage of many objects or trajectories for the generation of spacelike entanglement of fields.

In the QGEM proposal, it was demonstrated that two spatially superposed objects can be a probe of gravity-induced entanglement. We discussed how such objects probe state entanglement of quantum fields. We considered a pair of objects in a superposition of local states which couple with quantum fields. In this system, there are no constraints for the whole system and the fields have only dynamical degrees of freedom. From the entanglement analysis for the objects with the approximated currents evaluated on each trajectory, we found that the state of the trajectories cannot be entangled if the objects are in spacelike regions. This result is independent of the dynamics of fields and the detail of objects’ trajectories, which holds if the commutator of fields vanishes for spacelike separated regions (microcausality). The limitation for entanglement generation characterizes the behavior of the fields as quantum mediators between the two superposed objects. In other words, the position space of such objects cannot store the spacelike entanglement of fields. We can imagine several strategies ; use of the information of trajectories and internal degrees of freedom, and extensions with multiple objects and object superposed in multiple trajectories. It is important to discuss how the extensions are effective for the detection of spacelike entanglement. We need further research on quantum objects which play a crucial role in probing quantum nature of fields.