FField-induced entanglement in spatially superposed objects
Akira Matsumura ∗ Department of Physics, Kyushu University, Fukuoka, 819-0395, Japan
Abstract
We discuss the generation of field-induced entanglement between two objects each in a super-position of two trajectories. The objects have currents coupled to local quantum fields, and thecurrents are evaluated by using classical values associated with each trajectory of the objects. Ifthe fields have only dynamical degrees of freedom and satisfy the microcausality condition, we showthat the objects initially in a product state cannot be entangled when they are spacelike separated.This means that such quantum fields do not work as mediators to generate spacelike entanglementbetween the two superposed objects. ∗ Electronic address: [email protected]
Typeset by REVTEX 1 a r X i v : . [ qu a n t - ph ] F e b ontents I. Introduction II. Bose-Marletto-Vedral proposal to test quantum gravity III. Model Hamiltonian for fields and objects IV. No generation of spacelike entanglement between two objects V. Conclusion Acknowledgments References I. INTRODUCTION
The full picture of quantum gravity [1–4], which unifies general relativity and quantummechanics, is still unclear. This is attributed to the lack of theoretical and experimentalapproaches to connect gravitaional and quantum phenomena. However, with the recentdevelopment of various quantum technologies [5–8], there have been attempts to clarifyquantum natures of gravity (for example, see [9] and the references therein, or the recentworks [12–30]). In such works, the Bose-Marletto-Vedral (BMV) proposal [10, 11], whichfocuses on gravity-induced entanglement, has been attracting attentions. In the proposal,the authors considered two objects each in a superposition of two trajectories and assumedthe Newtonian potential between them. The gravitational interactions generate the entan-glement between the two objects. They argued that the gravity-induced entanglement wouldverify quantum gravity. However, it is not clear about the quantum nature of dynamicaldegrees of freedom for the Einstein’s gravity (spin-2 gravitons). This is because the New-tonian potential is obtained only from the constraint conditions in the Einstein’s equation(the Poisson equation).The interesting point in the BMV proposal is that two spatially superposed objects canbe a probe to test quantum entanglement. This is analogous to entanglement harvestingprotocols [31–39] by the Unruh-DeWitt detector. The Unruh-DeWitt detector is constructed2y 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 knownthat the spacelike entanglement of a vacuum state induces the entanglement between thetwo spatially separated detectors (for example, see [31]).In this paper, we investigate how two superposed objects are capable to probe the en-tanglement of quantum fields. We assume that the fields have only dynamical degrees offreedom and any constraint equations are not imposed on the whole Hibert space of theobjects and the fields. We consider the superposed objects which do not interact each otherand whose currents locally couple with the fields. By assigning classical values to the cur-rents along the objects’ trajectory, we compute the time evolution of the reduced densityoperator of the objects. For the case where the objects are spatially separated, we find thatthey remain disentangled if the microcausality condition holds for the quantum fields. Inother words, such quantum fields cannot be mediators to generate spacelike entanglementfor the two superposed objects. Our analysis also presents possible extensions of the objects’model to verify the spacelike entanglement of fields; object with the fluctuation of internaldegrees of freedom, multi-objects or multi-trajectories model.This paper is organized as follows. In Sec. II, the BMV proposal to test quantumgravity and its thoretical approach are reviewed. In Sec. III, we introduce the model withthe interaction given in a bilinear form of fields and currents of two objects. We derivethe solution of the Schr¨odinger equation. In Sec. IV, we investigate the separability ofthe two objects by using the solution. We find the no-go result of generation of spacelikeentanglement, and discuss its implications. In Sec. V, the conclusion is devoted. We usethe natural units (cid:126) = c = 1 in this paper. II. BOSE-MARLETTO-VEDRAL PROPOSAL TO TEST QUANTUM GRAVITY
The experimental setting of two matter-wave interferometers to test quantum gravity wasproposed, which is called the BMV proposal [10, 11]. In each interferometer, a single objectis in a superposition of two trajectories. Fig.1 presents a rough configuration of trajectoriesof each object. We assume that the two objects interact with each other by the Newtonian3 bject A Object B
FIG. 1: A configuration of the trajectories of the objects A and B. For the BMV proposal, theentanglement is generated between the objects by the gravitational interaction. potential. The Hamiltonian of the objects isˆ H BMV = ˆ H A + ˆ H B + ˆ V AB , ˆ V AB = − Gm A m B | ˆ x A − ˆ x B | , (1)where m A and m B are the masses of the objects A and B, ˆ x A and ˆ x B are each position,and the Hamiltonian ˆ H A and ˆ H B determine each trajectory of the objects. Each of the twoobjects at t = 0 is in the spatially superposed state, | ψ in (cid:105) = 1 √ | ψ R (cid:105) A + | ψ L (cid:105) A ) ⊗ √ | ψ R (cid:105) B + | ψ L (cid:105) B ) , (2)where | ψ R (cid:105) A and | ψ L (cid:105) A are the states with wave packets localized around positions x = x A R (0) and x = x A L (0) at t = 0, respectively. Also, | ψ R (cid:105) B and | ψ L (cid:105) B are defined in the samemanner. When each wave packet is sufficiently separated, those states satisfy A (cid:104) ψ R | ψ L (cid:105) A ≈ B (cid:104) ψ R | ψ L (cid:105) B ≈
0. The evolved state | ψ ( t f ) (cid:105) at t = t f is | ψ ( t f ) (cid:105) = e − it f ˆ H BMV | ψ in (cid:105) = e − it f ( ˆ H A + ˆ H B ) T exp (cid:104) i (cid:90) t f dt Gm A m B | ˆ x IA ( t ) − ˆ x IB ( t ) | (cid:105) | ψ in (cid:105)≈ e − it f ( ˆ H A + ˆ H B ) (cid:88) P , Q=R , L e i Φ PQ | ψ P (cid:105) A | ψ Q (cid:105) B , (3)where T is the time-ordered product, and ˆ x IA ( t ) = e it ( ˆ H A + ˆ H B ) ˆ x A e − it ( ˆ H A + ˆ H B ) and ˆ x IB ( t ) = e it ( ˆ H A + ˆ H B ) ˆ x B e − it ( ˆ H A + ˆ H B ) are the position operators in the interaction picture. The phaseshift, Φ PQ = (cid:90) t f dt Gm A m B | x A P ( t ) − x B Q ( t ) | , (4)4s determined by the Newtonian potential between the two objects on the trajectories x = x A P ( t ) and x = x B Q ( t ) (P , Q = R , L). In the expression (3), we omitted the symbol of thetensor product as | · (cid:105) A ⊗ | · (cid:105) B = | · (cid:105) A | · (cid:105) B . The approximation of the third line of Eq. (3) isgiven as ˆ x IA ( t ) | ψ P (cid:105) A ≈ x A P ( t ) | ψ P (cid:105) A , ˆ x IB ( t ) | ψ Q (cid:105) B ≈ x B Q ( t ) | ψ Q (cid:105) B . (5)These equations are valid when each object is sufficiently localized, for example, when thesize of each wave packet is larger than de Broglie wave length of each object [40, 41].Choosing the masses, the distance between a pair of trajectories and the traveling timeproperly, we find that the state (3) is entangled. Hence the gravitational interaction cangenerate quantum entanglement. The key point in the BMV proposal is that the spatiallysuperposed objects can probe quantum entanglement. In the following sections, we willdiscuss the detection of entanglement of dynamical fields by such objects. III. MODEL HAMILTONIAN FOR FIELDS AND OBJECTS
In this section, we introduce a model with a field-current bilinear interaction. In theSchr¨odinger picture, we consider the Hamiltonian of fields and two objects A and B asˆ H = ˆ H A + ˆ H B + ˆ H F + ˆ V , ˆ V = (cid:90) d x (cid:88) k (ˆ j k A ( x ) + ˆ j k B ( x )) ˆ φ k ( x ) , (6)where the Hamiltonians ˆ H A , ˆ H B and ˆ H F determine each dynamics of the objects A, B andthe fields. ˆ j k A and ˆ j k B are current operators with respect to the objects A and B, and ˆ φ k is thefield operator. We assume that the fields have only dynamical degrees of freedom and thereare no constraint equations on the whole Hilbert space. The field operators are representedon a physical Hilbert space H F without negative norm states. In gauge field theories, thereare formalisms using an unphysical Hilbert space of fields with gauge degrees of freedom[46]. The fact that there are no negative norm states will be used to derive our result in thenext section.We note that the Hamiltonian (6) does not completely represent that in the linearizedEinstein theory. At first glance, by choosing the currents ˆ j k A , ˆ j k B and the fields ˆ φ k as theenergy-momentum tensor ˆ T µν and the metirc perturbation ˆ h µν properly, the local interactionˆ V seems to be that in the linearized Einstein theory. This is not correct since the fields andthose Hilbert space H F are assumed not to have no gauge degrees of freedom and negative5orm states. Also, even for the transverse traceless gauge (ˆ h µν have only physical modes),the Hamiltonian (6) is not admitted in the linearized Einstein theory. This is because, fromthe constraints of the Einstein equation, the other components of the metric perturbationgive nonlocal interactions such as the Newtonian potential. However, there are no nonlocalinteractions between the two objects in our model.The almost same argument holds for the quantum electromagnetic dyanmics, but wecan admit an effective model described by the Hamiltonian (6). Let us consider that theobjects A and B without total electric charges have the electric dipole moments ˆ d A and ˆ d B ,respectively. For the distant objects, the Coulomb potential between them is neglected, andthe local coupling to an electric field ˆ E can be dominant. By assigning the field operator ˆ φ k and the currents ˆ j k A and ˆ j k B to ˆ E k , ˆ d k A δ ( x − x A ) and ˆ d k B δ ( x − x B ), our model describes theobjects with the dipole coupling to the electric field at the positions x = x A and x = x B .In [35], a similar model with time-dependent couplings and spatially smearing functions wasconsidered as the Unruh-DeWitt detector model.We consider that each object at t = 0 is in a superposition of two localilzed states | j R (cid:105) and | j L (cid:105) with (cid:104) j P | j P (cid:48) (cid:105) ≈ δ PP (cid:48) (P , P (cid:48) = R , L). The objects move on the trajectoriesgiven by the Hamiltonian ˆ H A and ˆ H B (see Fig. 1). We assume that the current operatorsˆ j k IA ( t, x ) = e i ˆ H t ˆ j k A ( x ) e − i ˆ H t and ˆ j k IB ( t, x ) = e i ˆ H t ˆ j k B ( x ) e − i ˆ H t in the interaction picture definedwith ˆ H = ˆ H A + ˆ H B + ˆ H F are approximated by the classical currents [40, 41]:ˆ j k IA ( t, x ) | j P (cid:105) A ≈ j k A P ( t, x ) | j P (cid:105) A , ˆ j k IB ( t, x ) | j Q (cid:105) B ≈ j k B Q ( t, x ) | j Q (cid:105) B , (7)where j k A P ( t, x ) and j k B Q ( t, x ) (P , Q = R , L) have nontrivial values only on the classicaltrajectories x = x A P ( t ) and x = x B Q ( t ), respectively. For example, if the objects haveelectric dipole moments and the fields are electric fields, the classical current j k A P ( t, x ) of theobject A has the form j k A P ( t, x ) = d k A ( t ) δ ( x − x A P ( t )) with the electric dipole d k A ( t ). Thelocalized state | j P (cid:105) A is | j P (cid:105) A = | ψ P (cid:105) A | d (cid:105) A , where | ψ P (cid:105) A is the state with a local wave packetand | d (cid:105) A is the state of the electric dipole of the object A. For these states, the position andthe electric dipole of the object A are assumed not to fluctuate; ˆ x IA ( t ) | ψ P (cid:105) A ≈ x A P ( t ) | ψ P (cid:105) A and ˆ d I k A ( t ) | d (cid:105) A ≈ d k A ( t ) | d (cid:105) A , where ˆ x IA ( t ) = e i ˆ H A t ˆ x A e − i ˆ H A t and ˆ d I k A ( t ) = e i ˆ H A t ˆ d k A e − i ˆ H A t . Thesimilar argument is made for the object B.When the fields are in an state | χ (cid:105) F at t = 0, the state of the objects and the fields at6 = 0 is | Ψ in (cid:105) = | α (cid:105) A | β (cid:105) B | χ (cid:105) F , | α (cid:105) A = α R | j R (cid:105) A + α L | j L (cid:105) A , | β (cid:105) B = β R | j R (cid:105) B + β L | j L (cid:105) B , (8)where | α R | + | α L | ≈ | β R | + | β L | ≈ | j P (cid:105) satisfies (cid:104) j P | j P (cid:48) (cid:105) ≈ δ PP (cid:48) .Note that the initial product state may be not valid if there are constraint equations on theobjects and fields. The solution of the Schr¨odinger equation is | Ψ( t f ) (cid:105) = e − i ˆ Ht f | Ψ in (cid:105) = e − i ˆ H t f T exp (cid:104) − i (cid:90) t f dt (cid:90) d x (cid:88) k (ˆ j k IA ( t, x ) + ˆ j k IB ( t, x )) ˆ φ I k ( t, x ) (cid:105) | Ψ in (cid:105)≈ e − i ˆ H t f (cid:88) P , Q=R , L α P β Q | j P (cid:105) A | j Q (cid:105) B ˆ U PQ | χ (cid:105) F , (9)where ˆ φ I k ( t, x ) = e i ˆ H t ˆ φ k ( x ) e − i ˆ H t . In the third line, we used the approximations (7) assign-ing the classical currents and defind the unitary operator ˆ U PQ ,ˆ U PQ = T exp (cid:104) − i (cid:90) t f dt (cid:90) d x (cid:88) k ( j k A P ( t, x ) + j k B Q ( t, x )) ˆ φ I k ( t, x ) (cid:105) . (10)In the next section, we examine the entanglement between the two objects A and B usingthe solution Eq. (9). We will show no generation of entanglement between the objects inspacelike regions. This argument follows by the microcausality of fields, which is independentof the dynamics of the fields. IV. NO GENERATION OF SPACELIKE ENTANGLEMENT BETWEEN TWOOBJECTS
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 betweenthe objects or not. The Hamiltonian ˆ H = ˆ H A + ˆ H B + ˆ H F yields independent dynamicsof each system, which give no correlations. On the other hand, the unitary evolution ˆ U PQ Eq.(10) given by the local interaction ˆ V leads to the following process: the object A locallyexcites the fields, and then the excitaions propagate to the object B and alter the potentialaround it. This process gives effective interactions and induces correlations between the7 ull conesObject A Object B FIG. 2: A configuration of trajectories of each object, which is in spatially separated regions. objects A and B. In fact, there are no such effects when the two objects are in spacelikeseparated regions (see Fig. 2). If the fields in spacelike regions commute each other (themicrocausality condition, for example, see [45]), we have (cid:104)(cid:90) d x (cid:88) k j k A P ( t, x ) ˆ φ I k ( t, x ) , (cid:90) d y (cid:88) (cid:96) j (cid:96) B Q ( t (cid:48) , y ) ˆ φ I (cid:96) ( t (cid:48) , y ) (cid:105) = 0 , (11)where note that j k A P ( t, x ) ∝ δ ( x − x A P ( t )) and j (cid:96) B Q ( t (cid:48) , y ) ∝ δ ( y − x B Q ( t (cid:48) )). Then theunitary operator ˆ U PQ is factorized into the local unitaries,ˆ U PQ = ˆ U A P ⊗ ˆ U B Q , (12)where ˆ U A P and ˆ U B Q areˆ U A P = T exp (cid:104) − i (cid:90) t f dt (cid:90) d x (cid:88) k j k A P ( t, x ) ˆ φ I k ( t, x ) (cid:105) , (13)ˆ U B Q = T exp (cid:104) − i (cid:90) t f dt (cid:48) (cid:90) d y (cid:88) (cid:96) j (cid:96) B Q ( t (cid:48) , y ) ˆ φ I (cid:96) ( t (cid:48) , y ) (cid:105) . (14)The local unitaries ˆ U A P and ˆ U B Q act on the Hilbert spaces H F A and H F B of the fields fortwo spacelike regions, respectively (the total Hilbert space H F of the fields is describedby H F = H F A ⊗ H F B ). There are no interactions induced by the fields for the factorizedevolution in Eq. (12), which does not generate entanglement between the two objects.Another important point is quantum entanglement of fields’ state. The previous work [31]showed that a pair of Unruh-DeWitt detectors, even if they are spacelike separated, becomesentangled due to the entanglement of the vacuum of a relativistic field. Also, there are manyworks about the generation of entanglement for spacelike separated detectors in the context8f entanglement harvesting protocol [32–35]. These works mean that the entanglement ofthe state | χ (cid:105) 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 bedetected by the two objects of interest. The definition of entanglement as follows: a givenstate is not entangled if the density operator ρ of a system has a separable form [42–44], ρ = (cid:88) i p i ρ i ⊗ σ i , (15)where p i is a probability, ρ i and σ i are density operators of the subsystems. A state whichcannot be written in such form is called entangled. We show that the reduced densityoperator of the two objects is written in a separable form. Tracing out the fields from theevolved state (9), the reduced density operator of the objects in spacelike regions is ρ = (cid:88) P , P (cid:48) =R , L (cid:88) Q , Q (cid:48) =R , L α P α ∗ P (cid:48) β Q β ∗ Q (cid:48) F (cid:104) χ | ˆ U † A P (cid:48) ˆ U A P ⊗ ˆ U † B Q (cid:48) ˆ U B Q | χ (cid:105) F | j P (cid:105) A (cid:104) j P (cid:48) | ⊗ | j Q (cid:105) B (cid:104) j Q (cid:48) | , (16)where we used Eq. (12), and the evolution operator e − i ˆ H t f was ignored because the lo-cal unitary transformation does not change the entanglement of the objects. The unitaryoperator ˆ V AP (cid:48) P = ˆ U † A P (cid:48) ˆ U A P appearing in (16) satisfiesˆ V ARR = ˆ V ALL = ˆ I A , ˆ V ALR = ˆ V A † RL = ( ˆ V ARL ) − , (17)and hence all of the unitaries ˆ V ARR , ˆ V ARL , ˆ V ALR and ˆ V ALL commute each other. This means thatˆ V AP (cid:48) P has the following spectral decomposition,ˆ V AP (cid:48) P = (cid:90) e iθ P (cid:48) P ( λ ) d ˆ µ F A ( λ ) , (18)where ˆ µ F A is an operater-valued measure on the Hilbert space H F A , which is a part of thefields’ Hilbert space H F . The real phase θ P (cid:48) P ( λ ) has the antisymmetric property θ P (cid:48) P ( λ ) = − θ PP (cid:48) ( λ ), which reflects Eq. (17). As the number of trajectories for each object is two, thenumber of independent components of θ P (cid:48) P ( λ ) is one. Hence the phase is always written as θ P (cid:48) P ( λ ) = θ RL ( λ )( n P − n P (cid:48) ) , (19)where n R = 0 and n L = 1. From the above facts, we find that the reduced density operator9 is separable, ρ = (cid:88) P , P (cid:48) =R , L (cid:88) Q , Q (cid:48) =R , L α P α ∗ P (cid:48) β Q β ∗ Q (cid:48) F (cid:104) χ | ˆ V AP (cid:48) P ⊗ ˆ U † B Q (cid:48) ˆ U B Q | χ (cid:105) F | j P (cid:105) A (cid:104) j P (cid:48) | ⊗ | j Q (cid:105) B (cid:104) j Q (cid:48) | = (cid:88) P , P (cid:48) =R , L (cid:88) Q , Q (cid:48) =R , L α P α ∗ P (cid:48) β Q β ∗ Q (cid:48) × (cid:90) e iθ RL ( λ )( n P − n P (cid:48) )F (cid:104) χ | d ˆ µ F A ( λ ) ⊗ ˆ U † B Q (cid:48) ˆ U B Q | χ (cid:105) F | j P (cid:105) A (cid:104) j P (cid:48) | ⊗ | j Q (cid:105) B (cid:104) j Q (cid:48) | = (cid:90) dµ ( λ ) | j ( λ ) (cid:105) A (cid:104) j ( λ ) | ⊗ σ B ( λ ) , (20)where we used Eqs. (18) and (19) and introduced the probability measure µ with dµ ( λ ) = F (cid:104) χ | d ˆ µ F A ( λ ) | χ (cid:105) F . The states | j ( λ ) (cid:105) A and σ B ( λ ) are | j ( λ ) (cid:105) A = (cid:88) P=R , L α P e iθ RL ( λ ) n P | j P (cid:105) A , (21) σ B ( α ) = 1 dµ ( λ ) (cid:88) Q , Q (cid:48) =R , L β Q β ∗ Q (cid:48) F (cid:104) χ | d ˆ µ F A ( λ ) ⊗ ˆ U † B Q (cid:48) ˆ U B Q | χ (cid:105) F | j Q (cid:105) B (cid:104) j Q (cid:48) | . (22)Here, we emphasize that the Hilbert space H F of the fields has no negative norm states, whichwas mentioned below Eq. (6). The fact leads to the inequalities µ ( λ ) ≥ σ B ( λ ) ≥ µ and σ B ( λ ) are a probability measure and a density operator, respectively, and thenthe separablity of the state holds. If gauge degrees of freedom are included in the fields, theHilbert space H F may have a negative norm state and the separability is not guaranteed.The separability of the objects does not depend on the dynamics of fields and the detailsof classical trajectories. Also, the seprability holds even for the case where the objects andthe fields are initially in a product mixed state. Our result means that the fields do notplay a role of quantum mediators to generate the spacelike entanglement between the twosuperposed objects.We compare our result with the no-go theorems in [11, 47] on generation of entanglement.The theorem in [11] argued that two systems mediated by classical systems with only asingle 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 quantumsystems in general. However, there are no generations of spacelike entanglement.The no-go theorem in Ref. [47] elucidates our result. We can rewrite Eq. (9) for the10pacelike separated two objects as | Ψ( t f ) (cid:105) = e − i ˆ H t f (cid:88) P , Q=R , L α P β Q | j P (cid:105) A | j Q (cid:105) B ˆ U PQ | χ (cid:105) F = e − i ˆ H t f (cid:88) P , Q=R , L α P β Q | j P (cid:105) A | j Q (cid:105) B ˆ U A P ⊗ ˆ U B Q | χ (cid:105) F = e − i ˆ H t f (cid:16) (cid:88) P=R , L | j P (cid:105) A (cid:104) j P | ⊗ ˆ U A P ⊗ ˆ I F B ⊗ ˆ I B (cid:17) ⊗ (cid:16) ˆ I A ⊗ (cid:88) Q=R , L | j Q (cid:105) B (cid:104) j Q | ⊗ ˆ I F A ⊗ ˆ U B Q (cid:17) | Ψ in (cid:105) , (23)where we used Eq. (12), and | Ψ in (cid:105) is the initial state given in Eq. (8). In this formula, wefind the controlled unitary ˆ U AF ,ˆ U AF = (cid:88) P=R , L | j P (cid:105) A (cid:104) j P | ⊗ ˆ U A P ⊗ ˆ I F B . (24)Exactly speaking, ˆ U AF has inverse only when it acts on the subspace spanned by | j R (cid:105) A and | j L (cid:105) A of the Hilbert space H A . In Ref. [47], the authors showed that the unitary evolutionˆ U = ( ˆ U AS ⊗ ˆ I B )(ˆ I A ⊗ ˆ U BS ) with the exponential of a Schmidt rank-1 operator ˆ U AS = e − i ˆ m A ⊗ ˆ X S does not generate entanglement between the systems A and B. The systems A, B and Scorrespond to the objects A and B, and the fields F for our model. Note that the controlledunitary ˆ U AF is rewritten as the formˆ U AF = ˆ U A R ( | j R (cid:105) A (cid:104) j R | ⊗ ˆ I F A ⊗ ˆ I F B + | j L (cid:105) A (cid:104) j L | ⊗ ˆ V ARL ⊗ ˆ I F B ) = ˆ U A R e − i ˆ m A ⊗ ˆ X F , (25)where ˆ V ARL = ˆ U † A R ˆ U A L , and the self-adjoint operator ˆ X F satisfies e − i ˆ X F = ˆ V ARL ⊗ ˆ I F B , andˆ m A = 0 × | j R (cid:105) A (cid:104) j R | + 1 × | j L (cid:105) A (cid:104) j L | . Since the entanglement between the two objects isinvariant under the local unitary transformation ˆ U A R for the fields, the controlled unitaryˆ U AF plays the same role as the exponetial of a Schmidt rank-1 operator. Thus, our no-go result for generation of spacelike entanglement is a consequence of the no-go theorem in[47]. Note that the no-go theorem can be applied under the approximation assigning classicalcurrents (7) and for the localized states satisfying (cid:104) j P | j P (cid:48) (cid:105) ≈ δ PP (cid:48) . If these conditions do nothold, we need a further study on entanglement generation.We comment on the extension of our model. It is well known that the spacelike entangle-ment of a field is extracted by the Unruh-DeWitt detectors [31]. Further, in Refs. [38, 39],the authors discussed an entanglement harvesting protocol by using the Unruh-DeWitt de-tectors with quantum superpositions of trajectories. The critical difference is that there11re no internal degrees of freedom (such as a spin or an energy level) which transit by theinteraction with quantum fields. This means that the approximation (7) is violated andthe objects’ state fluctuates. We expect that such degrees of freedom are neccessary foran extraction of spacelike entanglement from fields. Also, it is worth considering a multi-partite [29] or multi-trajectory [30] extended model of the BMV proposal, since our resultis based on the fact that each of two objects is superposed in two classical trajectories. It isinteresting to characterize the advantage of many objects or trajectories for the generationof spacelike entanglement of fields. V. CONCLUSION
In the BMV proposal, it was demonstrated that the Newtonian gravity can generate en-tanglement in two spatially superposed objects. Since the Newtonian gravity corresponds tonon-dynamical parts of the Einstein’s theory (scalar part), quantum natures of the other dy-namical components (spin-2 gravitons) have not been investigated in detail. As an attemptto clarify the role of dynamical fields, we discussed how such objects probe the entanglementof quantum fields with only physical degrees of freedom. We considered a pair of objectsin a spatially superposed state which locally couples with the fields by field-current bilinearinteractions. From the entanglement analysis for the objects under the approximation usingclassical currents, we found that the objects in spacelike regions cannot be entangled. Thisresult dose not depend on the dynamics of fields and the detail of the trajectories of two ob-jects, which holds if the commutator of spacelike separated fields vanishes (microcausality).The limitation for entanglement generation characterizes the behavior of the fields as quan-tum mediators between the two superposed objects. In other words, such objects have nocapacity to detect the spacelike entanglement of fields. We can imagine several extensions ofthe model of objects; objects with internal degrees of freedom fluctuated by quantum fields,multiple objects and object superposed in multiple trajectories. It is important to discusshow the extensions are effective for the detection of spacelike entanglement. We need furtherresearch on quantum objects which play a crucial role in probing quantum nature of fields.12 cknowledgments
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