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We propose a novel method to probe gravity-induced entanglement. We consider the gravitational interaction between a particle trapped in a shallow potential and a harmonic oscillator. The harmonic oscillator is in a quantum superposition of two frequencies and only one of these states can excite the trapped particle via resonance. Once the excited particle is detected, the quantum state of the oscillator is collapsed, which can be observed as the sudden disappearance of the superposition of oscillator frequencies. Thus, the sudden decoherence, which is only triggered by particle detection, can be a smoking gun evidence of gravity-induced entanglement. Since the probability of particle excitation increases linearly with time, the total probability is multiplied by repeating experiments. We will also discuss experimental implementations using optomechanics., Comment: 16 pages, 5 figures

The open dynamics of quantum particles in relativistic scattering is investigated. In particular, we consider the scattering process of quantum particles coupled to an environment initially in a vacuum state. Tracing out the environment and using the unitarity of S-operator, we find the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator describing the evolution of the particles. The GKSL generator is exemplified by focusing on the concrete processes: one is the decay of scalar particle ($\phi \rightarrow \chi \chi$), and the others are the pair annihilation and the $2\rightarrow 2$ scattering of scalar particles ($\phi \phi \rightarrow \chi \chi$ and $\phi \phi \rightarrow \phi \phi$). The GKSL generator for $\phi \rightarrow \chi \chi$ has a parameter with the coupling between $\phi$ and $\chi$ and the mass of both fields. The GKSL generator associated with $\phi \phi \rightarrow \chi \chi$ is characterized by a Lorentz-invariant function of initial momenta. Especially, in the pair annihilation process, we show that the probability of pair annihilation varies depending on the superposition state of incident scalar $\phi$ particles. Furthermore, we observe that the GKSL generators derived in this paper have Poincar\'e symmetry. This means that the description by the GKSL generator with Poincar\'e symmetry is effective for the asymptotic behavior of open quantum dynamics in the long-term processes of interest., Comment: 18 pages + appendices + references, 10 figures, 2 tables

We analyze the framework recently proposed by Oppenheim et al. to model relativistic quantum fields coupled to relativistic, classical, stochastic fields (in particular, as a model of quantum matter coupled to ``classical gravity''). Perhaps surprisingly, we find that we can define and calculate scattering probabilities which are Lorentz-covariant and conserve total probability, at least at tree level. As a concrete example, we analyze $2 \to 2$ scattering of quantum matter mediated by a classical Yukawa field. Mapping this to a gravitational coupling in the non-relativistic limit, and assuming that we can treat large objects as point masses, we find that the simplest possible ``classical-quantum'' gravity theory constructed this way gives predictions for $2 \to 2$ gravitational scattering which are inconsistent with simple observations of, e.g., spacecraft undergoing slingshot maneuvers. We comment on lessons learned for attempts to couple quantum matter to ``non-quantum'' gravity, or more generally, for attempts to couple relativistic quantum and classical systems., Comment: 28 pages + appendices, 4 figures

In this study, we investigate the decoherence of a spatially superposed electrically neutral spin-$\frac12$ particle in the presence of a relativistic quantum electromagnetic field in Minkowski spacetime. We demonstrate that decoherence due to the spin-magnetic field coupling can be categorized into two distinct factors: local decoherence, originating from the two-point correlation functions along each branch of the superposed trajectories, and nonlocal decoherence, which arises from the correlation functions between the two superposed trajectories. These effects are linked to phase damping and amplitude damping. We also show that if the quantum field is prepared in a thermal state, decoherence monotonically increases with the field temperature., Comment: (v1)17 pages, 4 figures (v2) updated: published version

We report the feasibility of detecting the gravity-induced entanglement (GIE) with optomechanical systems, which is the first investigation that clarifies the feasible experimental parameters to achieve a signal-to-noise ratio of S/N=1. Our proposal focuses on GIE generation between optomechanical mirrors, coupled via gravitational interactions, under continuous measurement, feedback control, and Kalman filtering process, which matured in connection with the field of gravitational wave observations. We solved the Riccati equation to evaluate the time evolution of the conditional covariance matrix for optomechanical mirrors that estimated the minimum variance of the motions. The results demonstrate that GIE is generated faster than a well-known time scale without optomechanical coupling. The fast generation of entanglement is associated with quantum-state squeezing by the Kalman filtering process, which is an advantage of using optomechanical systems to experimentally detect GIE., Comment: 5 pages, 4 figures

We investigate violations of Leggett-Garg inequalities (LGIs) for a harmonic oscillator and a (1+1)-dimensional chiral scalar field with coherent-state projectors, which is equivalent to a heterodyne-type measurement scheme. For the harmonic oscillator, we found that the vacuum and thermal states violated the LGIs by evaluating the two-time quasi-probability distribution function. In particular, we demonstrate that the value of the two-time quasi-probability reaches -0.123 for a squeezed coherent-state projector, which is equivalent to 98% of the L\"uders bound corresponding to the maximal violation of the LGIs. We also find a violation of the LGIs for the local mode of a quantum chiral scalar field by constructing a coherent-state projector similar to the harmonic oscillator case. In contrast to the harmonic oscillator, the periodicity in the time direction of the quasi-probability disappears, which is related to the existence of quantum entanglement between the local mode and its complementary degrees of freedom., Comment: 23 pages, 14 figures

Understanding the phenomenon of quantum superposition of gravitational fields induced by massive quantum particles is an important starting point for quantum gravity. The purpose of this study is to deepen our understanding of the phenomenon of quantum superposition of gravitational fields. To this end, we consider a trade-off relation of entanglement (monogamy relation) in a tripartite system consisting of two massive particles and a gravitational field that may be entangled with each other. Consequently, if two particles cannot exchange information mutually, they are in a separable state, and the particle and gravitational field are always entangled. Furthermore, even when two particles can send information to each other, there is a trade-off between the two particles and the gravitational field. We also investigate the behavior of the quantum superposition of the gravitational field using quantum discord. We find that quantum discord increases depending on the length scale of the particle superposition. Our results may help understand the relationship between the quantization of the gravitational field and the meaning of the quantum superposition of the gravitational field., Comment: 19 pages, 6 figures

We investigate what kind of Markovian quantum master equation (QME) in the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form is realized under Poincar\'{e} symmetry. The solution of the Markovian QME is given by a quantum dynamical semigroup, for which we introduce invariance under Poincar\'{e} transformations. Using the invariance of the dynamical semigroup and applying the unitary representation of Poincar\'{e} group, we derive the Markovian QME for a relativistic massive spin-0 particle. Introducing the field operator of the massive particle and examining its evolution, we find that the field follows a dissipative Klein-Gordon equation. In addition, we show that any two local operators for spacelike separated regions commute with each other. This means that the microcausality condition is satisfied for the dissipative model of the massive particle., Comment: 29 pages, 1 table

We evaluate the gravity-induced negativity volume of the generalized Wigner function in a hybrid system consisting of a particle in a two-localized superposition state and an oscillator. The generalized Wigner function can capture the nonclassicality of the system. The increase in the negativity volume of the generalized Wigner function can be an indicator of the entanglement generation, which is demonstrated in the hybrid system generating the gravity-induced entanglement in various initial states. Moreover, by comparing the behaviors of the negativity volume with the entanglement fidelity, we show that the nonclassical feature of entanglement is properly identified by the criterion based on the negativity volume of the generalized Wigner function when the oscillator is initially in a thermal state., Comment: 12 pages, 5 figures

We investigate the quantum signature of gravity in optomechanical systems under quantum control. We analyze the gravity-induced entanglement and squeezing in mechanical mirrors in a steady state. The behaviors and the conditions for generating the gravity-induced entanglement and squeezing are identified in the Fourier modes of the mechanical mirrors. The condition of generating the entanglement between the mirrors found in the present paper is more severe than that of the gravity-induced entanglement between output lights. The gravity-induced entanglement in optomechanical systems is an important milestone towards verifying the quantum nature of gravity, which should be verified in the future., Comment: 13 pages, 5 figures

Investigating the quantum nature of gravity is an important issue in modern physics. Recently, studies pertaining to the quantum superposition of gravitational potential have garnered significant interest. Inspired by Mari \textit{et al.} [Sci. Rep. {\bf 6} 22777 (2016)] and Baym and Ozawa [Proc. Natl. Acad. Sci. U.S.A. {\bf 106}, 3035 (2009)], Belenchia \textit{et al.} [Phys. Rev. D {\bf 98}, 126009 (2018)] considered a gedanken experiment involving such a quantum superposition and mentioned that the superposition renders causality and complementarity inconsistent. They resolved this inconsistency by considering the quantized dynamical degrees of freedom of gravity. This suggests a strong relationship between the quantum superposition of the gravitational potential and the quantization of the gravitational field. In our previous study [Phys. Rev. D {\bf 106}, 125002 (2022)], we have shown that the quantum uncertainty of a field guarantees the consistency between causality and complementarity. In this study, we focus on the entanglement between two particles' states due to the electromagnetic/gravitational potential and investigate its relationship with quantum uncertainty, causality, and complementarity. Our numerical analyses show that the quantum uncertainty of the electromagnetic/gravitational field results in vacuum fluctuations and prohibits the entanglement between two particles' states when causality is satisfied. We further demonstrate that complementarity holds when the particles do not get entangled. The uncertainty relation does not cause the entanglement between two particles' states, which guarantees complementarity., Comment: 15pages, 5 figures

No experimental evidence of the quantum nature of gravity has been observed yet and a realistic setup with improved sensitivity is eagerly awaited. We find two effects, which can substantially enhance the signal of gravity-induced quantum entanglement, by examining an optomechanical system in which two oscillators gravitationally couple and one composes an optical cavity. The first effect comes from a higher-order term of the optomechanical interaction and generates the signal at the first order of the gravitational coupling in contrast to the second order results in previous works. The second effect is the resonance between the two oscillators. If their frequencies are close enough, the weak gravitational coupling effectively strengthens. Combining these two effects, the signal in the interference visibility could be amplified by a factor of $10^{24}$ for our optimistic parameters. The two effects would be useful in seeking feasible experimental setups to probe quantum gravity signals., Comment: 18 pages, 7 figures, accepted version in PRD

We investigated the quantum state of an optomechanical suspended mirror under continuous measurement and feedback control using Wiener filtering. We focus on the impact of the two-mode theory of suspended mirror on the quantum state, which is described by the pendulum and rotational modes. It is derived from the beam model coupled to the cavity light in the low-frequency regime, including the internal friction of the beam and the finite size effect of the mirror. We constructed a Wiener filter for the two-mode theory and predicted the quantum state by evaluating the conditional covariance matrix using Wiener filter analysis. The results demonstrate that multimode analysis may play an important role in generating the quantum squeezed state. We also point out the possibility that one-mode analysis can be a good approximation by choosing the range of the Fourier space in the Wiener filter analysis., Comment: 34 pages, 14 figures

We consider how the reduced dynamics of an open quantum system coupled to an environment admits the Poincar\'e symmetry. The reduced dynamics is described by a dynamical map, which is given by tracing out the environment from the total unitary evolution without initial correlations. We investigate the dynamical map which is invariant under the Poincar\'e group. Based on the unitary representation theory of the Poincar\'e group, we develop a systematic way to give such a dynamical map. Using this way, we derive the dynamical map of a massive particle with a finite spin and a massless particle with a finite spin and a nonzero momentum. The dynamical map of a spinless massive particle is exemplified and the conservation of the Poincar\'e generators is discussed. We then find the map with the Poincar\'e invariance and the four-momentum conservation. Further, we show that the conservation of the angular momentum and the boost operator makes the map of a spinless massive particle unitary, Comment: 31 pages, 1 table

Pendulums have long been used as force sensors due to their ultimately low dissipation (high-quality factor) characteristic. They are widely used in the measurement of the gravitational constant, detection of gravitational waves, and determination of ultralight dark matter. Furthermore, it is expected that the quantum nature of gravity will be demonstrated by performing quantum control for macroscopic pendulums. Recently, we have demonstrated that quantum entanglement between two pendulums can be generated using an optical spring [D. Miki, N. Matsumoto, A. Matsumura, T. Shichijo, Y. Sugiyama, K. Yamamoto, and N. Yamamoto, arXiv:2210.13169 (2022)]; however, we have ignored that an optical spring can reduce the quality factor (Q-factor) by applying normal-mode splitting between the pendulum and rotational modes possessing relatively high dissipation. Herein, we analyze a system composed of a cylinder suspended using a beam (a suspended mirror, i.e., a pendulum) and an optical spring to consider normal-mode splitting. The reduction in Q-factor is determined only by the beam parameters: the ratio of the radius of the mirror to the length of the beam, and the ratio of the frequency of the rotational mode to the pendulum mode in the absence of cavity photons. In our analysis, we find that the reduction factor $4.38$ is reproduced, which is consistent with the experimental result in Matsumoto \textit{et al.} [N. Matsumoto, S. B. Catan$\tilde{\text{o}}$-Lopez, M. Sugawara, S. Suzuki, N. Abe, K. Komori, Y. Michimura, Y. Aso, and K. Edamatsu, Phys. Rev. Lett. 122, 071101 (2019)]. Our analysis shows that low dissipation (high quality) can be reached using an optical spring for the realistic pendulum system considering the rotational degree of freedom., Comment: 13pages, 5 figures

This study is aimed at investigating the feasibility of generating quantum entanglement between macroscopic mechanical mirrors in optomechanical systems while under continuous measurement and feedback control. We carefully derive a covariance matrix for mechanical mirrors in a steady state, employing the Kalman filtering problem with an assumed dominant cavity photon dissipation, such that the common and differential modes of the mirrors are squeezed by the action of measuring the output light beams. We demonstrate that entanglement between the mechanical mirrors is generated when the states of the common and differential modes are squeezed with high purity in an asymmetric manner. Our results also show that quantum entanglement between $7$ mg mirrors is achievable in the short term., Comment: 20 pages, 10 figures

It has long been debated whether gravity should be quantized or not. Recently, the authors in [Sci. Rep. 6, 22777 (2016); Proc. Natl. Acad. Sci. U.S.A. 106, 3035 (2009)] discussed the inconsistency between causality and complementarity in a Gedankenexperiment involving the quantum superposition of massive/ charged bodies, and Belenchia et al. [Phys. Rev. D 98, 126009 (2018); Int. J. Mod. Phys. D 28, 1943001 (2019)] resolved the inconsistency by requiring the quantum radiation and vacuum fluctuations of gravitational/electromagnetic field. Stimulated by their works, we reanalyze the consistency between the two physical properties, causality and complementarity, according to the quantum field theory. In this analysis, we consider a Gedankenexperiment inspired by [Sci. Rep. 6, 22777 (2016); Proc. Natl. Acad. Sci. U.S.A. 106, 3035 (2009); Phys. Rev. D 98, 126009 (2018); Int. J. Mod. Phys. D 28, 1943001 (2019)], in which two charged particles coupled with a photon field are in a superposition of two trajectories. First, we observe that causality is satisfied by the retarded propagation of the photon field. Next, by introducing an inequality between visibility and which-path information, we show that the quantum radiation and vacuum fluctuations of the photon field ensure complementarity. We further find that the Robertson inequality associated with the photon field leads to the consistency between causality and complementarity in our Gedankenexperiment. Finally, we mention that a similar feature appears in the quantum field of gravity., Comment: 14 pages, 3 figures

We investigate the quantum nature of gravity in terms of the coherence of quantum objects. As a basic setting, we consider two gravitating objects each in a superposition state of two paths. The evolution of objects is described by the completely positive and trace-preserving (CPTP) map with a population-preserving property. This property reflects that the probability of objects being on each path is preserved. We use the $\ell_1$-norm of coherence to quantify the coherence of objects. In the present paper, the quantum nature of gravity is characterized by an entangling map, which is a CPTP map with the capacity to create entanglement. We introduce the entangling-map witness as an observable to test whether a given map is entangling. We show that, whenever the gravitating objects initially have a finite amount of the $\ell_1$-norm of coherence, the witness tests the entangling map due to gravity. Interestingly, we find that the witness can test such a quantum nature of gravity, even when the objects do not get entangled. This means that the coherence of gravitating objects always becomes the source of the entangling map due to gravity. We further discuss a decoherence effect and an experimental perspective in the present approach., Comment: 18 pages, 6 figures

The Bose-Marletto-Vedral (BMV) experiment [3, 4] is a proposal for testing the quantum nature of gravity with entanglement due to Newtonian gravity. This proposal has stimulated controversy on how the entanglement due to Newtonian gravity is related to the essence of quantum gravity and the existence of gravitons. Motivated by this, we analyze the entanglement generation between two charged particles coupled to a photon field. We assume that each particle is in a superposition of two trajectories and that the photon field is initially in a coherent state. Based on covariant quantum electrodynamics, the formula for the entanglement negativity of the charged particles is derived for the first time. Adopting simple analytic trajectories of the particles, we demonstrate the entanglement between them. It is observed that the entanglement is suppressed by the decoherence due to the vacuum fluctuations of the photon field. We also find that the effect of quantum superposition of bremsstrahlung appears in the entanglement negativity formula. The similar structures between the gravity theory and electromagnetic theory suggests that a similar feature may be observed in the entanglement generation by quantum gravitational radiation., Comment: 34 pages, 16 figures

We explore a general feature of the interaction mediated by the gravitational fields of spatially superposed masses. For this purpose, based on quantum information theory, we characterize the evolution of two particles each in a superposition state of paths. The evolution is assumed to be given by a completely positive trace-preserving (CPTP) map. We further assume that the probability of particle being on each path is unchanged during the evolution. This property is called population-preserving. We examine when a population-preserving CPTP map can create entanglement in terms of separable operations, which form a large class of local operations and classical communication (LOCC). In general, entanglement is not always generated by inseparable or non-LOCC operations, and one can consider a model of gravity described by an inseparable operation which does not create entanglement. However, we find that a population-preserving CPTP map is inseparable if and only if it can create entanglement. This means that the above model of gravity is incompatible with the possible evolution of spatially superposed masses., Comment: 21 pages, 4 figures

In this study, we determine a violation of the Leggett-Garg inequalities due to gravitational interaction in a hybrid system consisting of a harmonic oscillator and a spatially localized superposed particle. The violation of the Leggett-Garg inequalities is discussed using the two-time quasiprobability in connection with the entanglement negativity generated by gravitational interaction. It is demonstrated that the entanglement suppresses the violation of the Leggett-Garg inequalities when one of the two times of the quasiprobability t_1 is chosen as the initial time. Further, it is shown that the Leggett-Garg inequalities are generally violated due to gravitational interaction by properly choosing the configuration of the parameters, including t_1 and t_2, which are the times of the two-time quasiprobability. The feasibility of detecting violations of the Leggett-Garg inequalities in hybrid systems is also discussed., Comment: 14 pages, accepted for publication in the Physical Review A

We develop an entanglement criterion with third- and fourth-order cumulants to detect the entanglement of non-Gaussian states. The efficiency of the entanglement criterion is investigated for gravitating mirrors in optomechanical systems. We show that the entangled regime of the mirrors is enlarged by the third- and fourth-order cumulants. We also discuss the limitations of the entanglement criterion for mirrors in a highly non-Gaussian state., Comment: 14 pages, 7 figures

Quantum coherence is one of the most striking features of quantum mechanics rooted in the superposition principle. Recently it has been demonstrated that it is possible to harvest the quantum coherence from a coherent scalar field. In order to explore a new method of detecting axion dark matter, we consider a point-like Unruh-DeWitt detector coupled to the axion field and quantify a coherent measure of the detector. We show that the detector can harvest the quantum coherence from the axion dark matter. To be more precise, we consider a two-level of electron system in an atom as the detector. In this case, we obtain the coherence measure $C=2.2\times 10^{-6}\gamma\, \left(T/1{\rm s}\right)$ where $T$ and $\gamma$ are an observation time and the Lorentz factor. At the same time, the axion mass $m_a$ we can probe is determined by the energy gap of the detector., Comment: 16pages

We discuss the generation of field-induced entanglement between two objects each in a superposition of two trajectories.The objects have currents coupled to local quantum fields, and the currents are evaluated around each trajectory of the objects.The fields have only dynamical degrees of freedom and satisfy the microcausality condition. We find that the superposed state of trajectories cannot be entangled when the objects are spacelike separated.This means that the quantum fields do not generate spacelike entanglement in the superposition of two trajectories of each object., Comment: 16 pages, 2 figures

We analyze the dynamics of a gravity-induced entanglement for N massive particles. Considering the linear configuration of these particles, we investigate the entanglement between a specific pair of particles under the influence of the gravitational interaction between the massive particles. As the particle number increases, the specific particle pair decoheres more easily due to the gravitational interaction with other particles. The time scale of the gravity-induced decoherence is analytically determined. We also discuss the entanglement dynamics of initially entangled particles, which exemplify the monogamy of the gravity-induced entanglement., Comment: 24 pages, 7 figures, accepted for publication in PRD

We investigate the phenomenon of gravity-induced entanglement in optomechanical systems. Assuming photon number conservation and the Newtonian potential expanded up to the quadratic order of the oscillator positions, we exactly solve the dynamics of the optomehcanical systems. Then, we find that the phase difference due to the Newtonian gravity leads to the large entanglement of photons in separated cavities. We clarify the generating mechanism of large gravity-induced entanglements in optomechanical systems in an exact manner. We also determine the characteristic time to generate the maximal entanglement of photons. Finally, by comparing the characteristic time with the decoherence time due to photon leakage, we evaluate the range of the dissipation rate required for testing the gravity-induced entanglement., Comment: 21 pages, 3 figures

We investigate the squeezing of primordial gravitational waves (PGWs) in terms of quantum discord. We construct a classical state of PGWs without quantum discord and compare it with the Bunch-Davies vacuum. Then it is shown that the oscillatory behavior of the angular-power spectrum of the cosmic microwave background (CMB) fluctuations induced by PGWs can be the signature of the quantum discord of PGWs. In addition, we discuss the effect of quantum decoherence on the entanglement and the quantum discord of PGWs for super-horizon modes. For the state of PGWs with decoherence effect, we examine the decoherence condition and the correlation condition introduced by C. Kiefer et al. (Class. Quantum Grav. 24 (2007) 1699). We show that the decoherence condition is not sufficient for the separability of PGWs and the correlation condition implies that the PGWs in the matter-dominated era have quantum discord., Comment: 33 pages, 10 figures

We investigate quantum correlations appearing for two qubit detectors which are initially uncorrelated and locally coupled to a massless scalar field in a vacuum state. Under the perturbation up to the second order in the coupling, the state of the detectors can be entangled through the interaction with the scalar field but satisfies the Bell-CHSH inequality. The violation of the Bell-CHSH inequality for such an entangled state is revealed by local filtering operations. In this paper, we construct the optimal filtering operations for the qubit detectors and derive the success probability of the filtering. The success probability characterizes the reliability of revealing the violation of the Bell-CHSH inequality by the filtering operations. Through these analyses, we demonstrate a trade-off relation between the success probability and the size of parameter region showing the violation of the Bell-CHSH inequality., Comment: 26 pages, 10 figures

We consider a quantum circuit model describing the evaporation process of black holes. We specifically examine the behavior of the multipartite entanglement represented by this model, and find that the entanglement structure depends on the black hole mass $M$ and the frequency of the Hawking radiation $\omega$. For sufficiently small values of $M\omega$, the black hole and the radiation system becomes a separable state after the Page time and a firewall-like structure appears. On the contrary, for larger values of $M\omega$, the entanglement between the black hole and the radiation is not lost. These behaviors imply that owing to the monogamy property of the multipartite entanglement, low frequency modes of the Hawking radiation destroys the quantum correlation between the black hole and the emitted Hawking radiation., Comment: 26 pages, accepted version in CQG

We discuss a parameter estimation problem using quantum decoherece in the double-slit interferometer. We consider a particle coupled to a massive scalar field after the particle passing through the double slit and solve the dynamics non-perturbatively for the coupling by the WKB approximation. This allows us to analyze the estimation problem which cannot be treated by master equation used in the research of quantum probe. In this model, the scalar field reduces the interference fringes of the particle and the fringe pattern depends on the field mass and coupling. To evaluate the contrast and the estimation precision obtained from the pattern, we introduce the interferometric visibility and the Fisher information matrix of the field mass and coupling. For the fringe pattern observed on the distant screen, we derive a simple relation between the visibility and the Fisher matrix. Also, focusing on the estimation precision of the mass, we find that the Fisher information characterizes the wave-particle duality in the double-slit interferometer., Comment: 25 pages, 8 figures

We investigate quantum entanglement between two symmetric spatialregions in de Sitter space with the Bunch-Davies vacuum. As a discretized model of the scalar field for numerical simulation, we consider a harmonic chain model. Using the coarse-grained variables for the scalar field, it is shown that the multipartite entanglement on the superhorizon scale exists by checking the monogamy relation for the negativity which quantifies the entanglement between the two regions. Further, we consider the continuous limit of this model without coarse-graining and find that non-zero values of the logarithmic negativity exist even if the distance between two spatial regions is larger than the Hubble horizon scale., Comment: 20 pages, 11 figures