Bell Inequality in the Holographic EPR Pair

Physics Letters B 791 (2019) 73–79

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Bell inequality in the holographic EPR pair ∗ ∗∗ Jiunn-Wei Chen a,b, Sichun Sun a,c, , Yun-Long Zhang d,e,f, a Department of Physics, Center for Theoretical Sciences, and Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan b Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA c Department of Physics, Sapienza University of Rome, Rome I-00185, Italy d Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan e Asia Paciﬁc Center for Theoretical Physics, Pohang 790-784, Republic of Korea f Center for Quantum Spacetime, Sogang University, Seoul 121-742, Republic of Korea a r t i c l e i n f o a b s t r a c t

Article history: We study the Bell inequality in a holographic model of the casually disconnected Einstein–Podolsky– Received 21 November 2018 Rosen (EPR) pair. The Clauser–Horne–Shimony–Holt (CHSH) form of Bell inequality is constructed using Received in revised form 30 January 2019 holographic Schwinger–Keldysh (SK) correlators. We show that the manifestation of quantum correlation Accepted 11 February 2019 in Bell inequality can be holographically reproduced from the classical ﬂuctuations of dual accelerating Available online 13 February 2019 string in the bulk gravity. The violation of this holographic Bell inequality supports the essential quantum Editor: N. Lambert property of this holographic model of an EPR pair. Keywords: © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license Holographic Bell inequality (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. Holographic EPR pair String ﬂuctuations Nambu–Goto action AdS/CFT correspondence

1. Introduction dacena and Susskind proposed the ER=EPR conjecture [10,11], which suggested that the ER bridge can be interpreted as maxi- Bell inequality plays an important role in quantum physics [1]. mally entangled states of two black holes that form a complex EPR Because correlations in local classical theories are bounded by the pair. This conjecture was proposed to resolve the Almheiri–Marolf– Polchinski–Sully (AMPS) paradox without resorting to a firewall Bell inequality, which can be violated by the presence of the non- [12]. Interestingly, the ER=EPR conjecture implies that entangle- local entanglement in quantum mechanics. The violation of Bell ment, which is thought to be a quantum mechanical effect of the inequality in the entangled Einstein–Podolsky–Rosen (EPR) pair EPR pair, can be captured by a complex ER bridge in gravitational [2] indicates that two particles have an “instant interaction”, in theory. contrast to theories of hidden variables that preserve strict local- Inspired by the original ER=EPR conjecture, a concrete holo- ity [1–5]. The Bell inequality is a very general test in quantum graphic model of the EPR pair was proposed in Ref. [13], based physics. In addition to the Bell’s tests in laboratories, there are also on the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspon- discussions of detections in cosmological scales, to confirm that dence [14] and one exact solution of the accelerating string in whether the inflated primordial fluctuations are quantum mechan- AdS5 spacetime [15]. In the holographic model, the EPR pair with ical in nature [6–8]. two accelerating quasiparticles on the boundary is proposed to In general relativity, two distant black holes can be connected dual to an uniformly-accelerating open string in AdS5 spacetime by the wormhole or Einstein–Rosen (ER) bridge [9]. Recently, Mal- and the endpoints of the string are attached to the boundary. Therefore, it is a holographic realization of the EPR pair, with an effective wormhole induced on the worldsheet in the AdS bulk. * Corresponding author at: Department of Physics, Sapienza University of Rome, One should notice that the holographic setting is different from Rome I-00185, Italy. the original ER=EPR conjecture in which both ER and EPR live in ** Corresponding author at: Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan. the same spacetime dimensions. E-mail addresses: [email protected] (J.-W. Chen), [email protected] In this paper, we focus on the Bell inequality in the holographic (S. Sun), [email protected] (Y.-L. Zhang). EPR model. We will show that the Bell inequality violated by the https://doi.org/10.1016/j.physletb.2019.02.012 0370-2693/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 74 J.-W. Chen et al. / Physics Letters B 791 (2019) 73–79 π π EPR pair living on the boundary can be captured by the perturba- θA = 0,θA = ,θB = θB − , (7) tions of the dual accelerating string with an effective wormhole on 2 2 the worldsheet living in the AdS bulk spacetime. To test the en- then we have the relation depended on the direction of B, B , tanglement of the EPR pair, Bell inequality is a natural choice as √ π it provides a sharp test of entanglement. Since in the holographic ψs|Cs|ψs=−2 2cos θB − . (8) model of an EPR model [13], it shows that the two quasi-particles 4 are indeed entangled with each other, which indicates the viola- For 0<θB <π/2, the Bell inequality |Cs| ≤ 2can be violated, and tion of the Bell inequality. It is reasonable that one quantity in the we√ reach the maximal violation at θB = π/4, with an extra factor holographic model can be identified as the correlator in the Bell of 2. inequality, and our proposal is that the holographic Schwinger– Keldysh (SK) correlator can just play the role. Thus, the violation Bell’s test in field theory Before we move to the holographic model, of holographic Bell inequality supports the quantum entanglement it is instructive to discuss the Bell’s test in the context of the quan- property of this holographic model of an EPR pair. tum field theory. The test can be described by a process with the In the following section 2, we firstly review the Bell inequality transition amplitude in quantum mechanics as well as in the field theory. In section 3, we summarize the holographic model of the EPR pair. Our main ↑ ↑ A =A B |T [P A P B ]|ψs, (9) result is in section 4, where we construct the Bell inequality based on this holographic model. We discuss and conclude the results in where a spin singlet state of Eq. (4)is measured by the projection section 5. In the appendix A, we give more details on the deriva- operators tion of the holographic Schwinger–Keldysh correlators. 1 + n A ·σ 1 + nB ·σ P A = , P B = , (10) 2. Bell inequality in quantum theories 2 2 ↑ ↑ to the final state |A B which is both spin up in the n and n In this section, we will review the essence of Bell inequality, A B directions. Note that the operators are time-ordered since one can which is captured in the Clauser–Horne–Shimony–Holt (CHSH) cor- either measure P or P first. Squaring the amplitude to get the relation parametrizations [3]. The entangled states made of a pair A B probability of spin 1/2particles are detected by two observers, Alice and Bob, respectively. While the generalization to particles of higher spin is 2 † P ∝|A| = ψs|T [P A P B ] |X X|T [P A P B ]|ψs straightforward. The operators correspond to measuring the spin X along various axes with outcomes of eigenvalues ±1. Performing =ψ |T [P P ]|ψ , (11) the operations A and A on the first particle at Alice’s location, s A B s ↑ ↑ and operations B and B on the second particle at Bob’s loca- where only |A B contributes in the complete set of state |X in tion. With the Pauli matrices σ = (σ , σ , σ ), and the unit vector x y z the first line and we have used the commutativity of P A and P B , n = (nx, ny, nz) to indicate the spatial direction of the measure- 2 = 2 = as well as P A P A and P B P B for projectors. ment, we have the following operators The above discussion shows that the Bell’s test is measuring a time-ordered Green’s function which does not have to vanish when A = n · A = n · (1) s A σ , s A σ , P A and P B are outside of each other’s light cone. A familiar exam- = · = · ple of this is Feynman propagator which is also a time-ordered Bs nB σ , Bs nB σ . (2) Green’s function. Typically, this dramatic property of the Green’s Then the CHSH correlation formulation is introduced as function does not show up in physical observable. It is very interesting that Bell’s test relates those Green’s functions to observables = + + − Cs As Bs As Bs As Bs As Bs , (3) through Eq. (11). In the following holographic EPR model, we will which is a linear combination of crossed expectation values of the study the contour time-ordered Schwinger–Keldysh correlators of measurements. the EPR pair holographically. In a local theory with hidden variables, the formula is bounded | | ≤ 3. Holographic model of an EPR pair by the Bell inequality Cs 2. While in quantum mechanics,√ this inequality can be violated, with a higher bound |Cs| ≤ 2 2[4] In this section, we summarize the holographic model of an (see also [6]for the cosmological case). For example, if we choose EPR pair in Ref. [13]. The holographic model proposed that an the entanglement state of a spin singlet entangled color singlet quark anti-quark (q–q¯) pair in N = 4su- persymmetric Yang–Mills theory (SYM) can be holographically de- | =√1 |↑ ⊗ |↓ − |↓ ⊗ |↑ ψs , (4) scribed by an open string with both of its endpoints attached to 2 the boundary of AdS . The string connecting the pair is dual to the = 5 and take the measurements along the (x, y) plane, i.e. nA color flux tube between the two quarks, with a 1/r Coulomb po- (cos θA , sin θA , 0) etc., it is straightforward to show tential as required by the scale invariance of boundary theory. Note that there is no confinement in this theory, therefore the pair can s ≡ | | =− G AB ψs As Bs ψs cos θAB. (5) separate arbitrarily far away from each other. It was shown in [16] that the holographic EPR model corre- Here cos θAB = n A ·nB depends on the relative angle of the mea- sponds to the Lorentzian continuation of the holographic Schwing- surements. And from (3)we have er model. Since the particles are produced from the vacuum, they are necessarily entangled with one another, no matter what the ψs|Cs|ψs=−cos θAB− cos θAB − cos θA B + cos θA B . (6) actual nature of the particles, may they be quarks and antiquarks In particular, if we fix the direction of A and A , as well as the and even charged W± bosons. Various studies of related models angle between B and B as π/2, can also be found in [17–21]. J.-W. Chen et al. / Physics Letters B 791 (2019) 73–79 75

There are numerical solutions of the string shapes with different boundary behaviors [22–24]. But it is more convenient for us to work with the analytic solution for an accelerating string treated in the probe limit such that the back reaction to the AdS geometry is neglected [15]. In this analytic solution, the open string is also accelerated on the Poincáre patch of the AdS5 spacetime L2 ds2 = − dt2 + dw2 + (dx2 + dy2 + dz2) , (12) w2 with the AdS radius L and extra dimension with coordinate w. The string solution in the AdS5 bulk is governed by the semicircle 2 = 2 + 2 − 2 z t b w . (13) Fig. 1. Schematic diagram for the test quark (q)–antiquark (q¯) EPR pair in holographic model at a fixed time t > 0. From (14), the trajectory of left (right) world- For infinitely heavy test quarks, the quark and anti-quark live on sheet horizon is on w = b, which depicts the intersect of the string with the world = ± the AdS boundary w 0. They are accelerating√ along the z direc- volume horizon seen by q (q¯) in its co-moving frame. The Bell inequality test is per- tion, respectively, with the trajectories z =± t2 + b2. Therefore, formed by the measurements at Alice and Bob’s locations. The AdS black brane is the two entangled particles are out of causal contact with each present only in the finite temperature case, which we will briefly discuss in the last other the whole time. For the dynamical heavy quarks with the fi- section. nite mass m, the string needs to end on a flavor probe brane [25, 2 = T s L force that acts on the spins of the particles. We expect similar 26]at a small but finite zm m < b, where T s is the tension of the string. A constant electric field E = m/b on the flavor brane is measurement can be carried out in the holographic EPR model, responsible for accelerating the quasiparticles [13]. which introduces fluctuations to the world lines of quasiparticles which set the boundary conditions for the worldsheet of the string String fluctuations To consider the string fluctuations, we trans- fluctuations. Let xã = (τ˜, r˜) be the new worldsheet coordinates form the solution to the two co-moving frames of the accelerating in the comoving frame, then the string fluctuation is described ˜ ˜ μ ˜ ˜ ˜ ˜ ˜ ˜ ˜ particles with coordinates (τ˜, r, x, y, z) via X (τ˜, r) = τ˜, r, qi(τ˜, r) , with i = (x, y, z). When the fluctuations ˜ ˜ ˜ qi 1, the Nambu–Goto action of string with tension T s becomes |z|=b 1 − re˜ z cosh τ˜, t = b 1 − re˜ z sinh τ˜, √ ˜ d ˜dr˜ 1 = ˜ z = ˜ = ˜ 2 τ ˜ ˜ ˜ ˜ ˜˙ ˜˙ ij w b re , x b x, y b y. (14) S NG −T s L 1 + 2rf(r)q q − qiq j h , 2r˜3/2 i j 2 f (r˜) These two frames, which cover the regions z ≥ 0 and z ≤ 0sepa- (16) rately, are accelerating frames with a constant acceleration a = 1/b ∂q˜ ˙ ∂q˜ ˜ ˜ where q˜ ≡ i , q˜ ≡ i and hij = diag[1, e2z, e2z]. The classical along opposite directions of z. And (14)only maps the upper part i ∂r˜ i ∂τ˜ of the string (0 < w < b) into the proper frames of the acceler- equations of motion for the fluctuations on the string are ˜ ating particles with 0 < r < 1. Plugging this transformation (14) ˙ 2 f q˜ q˜ in the string solution (13), one finds the string configuration be- i − i = ∂r˜ ∂τ˜ 0. (17) comes z˜ = 0for both frames of the quark and anti-quark. Under r˜1/2 2 f r˜3/2 this transformation (14), the metric (12) becomes Focus on the transverse fluctuations along i = x˜, y˜ , 2 2 L 1 dr˜ − ˜ 2= − ˜ ˜ 2+ + 2z ˜2+ ˜ 2 + ˜2 d ds f (r)dτ e dx dy dz , (15) ˜ ˜ ω −iωτ˜ ˜ ˜ r˜ 4r˜ f (r˜) qi(τ˜, r) = e qi(ω)Yω(r), (18) 2π where f (r˜) = 1 − r˜. There are the bifurcate Killing horizons at ˜ r˜ = 1 associated with both frames of the accelerating quark and where qi(ω) is the Fourier transform of fluctuation on the bound- ˜ ˜ anti-quark. Furthermore, with respect to the time τ =bτ˜, the Un- ary directions and Yω(r) is the function of the radio direction r = 1 = a and frequency ω. ruh temperature Ta 2πb 2π [15,21]. We have set the reduced Planck constant h¯ and Boltzmann constant kB to be unit. The reg- The retarded Green’s function of the particle under effective i ij i ular thermal state with this temperature is similar to the Hartle– random force F (τ ) can be defined as iGR(τ ) = θ(τ )[F (τ ), Hawking state in this spacetime with the Unruh vacuum, and more F j(0)] [15]. In the AdS/CFT correspondence, F i (τ ) is the oper- details are addressed in the appendix A. ator conjugate to the fluctuations qi (τ ) of the fundamental string, ˜ As shown in Fig. 1, the two horizons are connected by part of where the dimensionful quantities are τ = bτ˜, qi = bqi . In the low the string which can be seen as an effective ER bridge on the string frequency limit ω → 0, it can be obtained analytically as worldsheet. It is suggested to be a holographic realization of the EPR pair in [13], although the dimensions are different from the 2T L2 ij s ˜ ˜ ˜ ij = GR(ω) =− f (r)Y−ω(r)∂˜Yω(r)δ ˜ original ER EPR conjecture [10,11]. In this holographic model, EPR b2r˜1/2 r r→0 + √ lives at the boundary with 3 1dimension while the “effective 2 + a λ ER” lives on the worldsheet in bulk with 4 1dimension. =− iωδij + O (ω2), (19) 2π 4. Constructing holographic Bell inequality √ 2 = λ and we have used the fact that T s L 2π . In this section, we construct Bell Inequality in the holographic What we need for the Bell’s test is the contour time-ordered model of an EPR pair in [13]. The spin measurement of the par- Schwinger–Keldysh (SK) Green’s function, ticles can usually be carried out by the Stern–Gerlach type ex- ij =F i F j periment which applied a magnetic field gradience to generate a iG AB(τ , x) A (τ , x) B (0) , (20) 76 J.-W. Chen et al. / Physics Letters B 791 (2019) 73–79

F i F j where A and B are separately defined on the causally discon- To study the holographic correlators, we normalize the force F i nected left and right wedges of the Penrose diagram, correspond- operators I such that only the dependence on directions remains ing to the boundaries of different patches of the AdS space. With = F x + F y F x F x 1/2 these conjectures, in [27], it was shown that in the semi-classical AF (cos θA A sin θA A )/ A B , limit of the AdS/CFT correspondence, the two-point correlators can = F x + F y F x F x 1/2 BF (cos θB B sin θB B )/ A B . (25) be calculated from the variations of the generating function of the boundary quantum field theory, which is identified with the to- Similar to (5), the mixed measurements for correlators in CHSH tal on-shell action of the dual field in the bulk gravity. In our correlation formulation become case, the bulk fields are the string fluctuations, which are gov- = − ≡ erned by the equations of motion from the Nambu–Goto action AF BF cos(θA θB ) cos θAB. (26) in the bulk gravity, with the corresponding boundary conditions Together with the similar normalization of the operators AF and I q˜ (τ˜, r˜)|˜→ =q˜ (τ˜). Thus, through the AdS/CFT correspondence in i r 0 i BF , the CHSH correlation formulation in (3) becomes semi-classical limit (or large-N, large ’t Hooft coupling limit of the dual field theory), the generating function of the holographic EPR CF =AF BF +AF BF +AF BF −AF BF pair could be assumed to be = cos θAB + cos θAB + cos θA B − cos θA B . (27) i AdS/CFT i [˜ I ˜ J ] ¯ S EPR h¯ S NG qi ,q j Z EPR ≡e h e . (21) For example, when θAB = θAB = θA B√= π/4, and θA B = 3π/4, we can reach the maximum value 2 2. It suggests that in this After considering the holographic advanced and retarded holographic EPR system, the CHSH formulation of Bell inequality Green’s functions in (A21)–(A22)in Appendix A, taking func- ≤ J CF 2can be violated, which supports the essential quantum tional derivatives of Z with respect to q˜ I and q˜ will yield the EPR i j property of the holographic EPR pair in Ref. [13]. Schwinger–Keldysh (SK) correlators in the holographic EPR pair

J 5. Conclusion and discussions ¯ 2 2 δ2 S [q˜ I , q˜ ] ij ≡ h δ ln Z EPR NG i j iG IJ , (22) i2 ˜ I ˜ J ˜ I ˜ J In this paper, using Bell inequality as a sharp test of entan- δ(qi )δ(q j ) δ(qi )δ(q j ) glement, we study a holographic model with an EPR pair at the where the semiclassical limit in (21) has been used. boundary theory and an effective wormhole on the string world- This off-diagonal SK correlator for the EPR pair is found to be sheet in the bulk. By revealing how Bell inequality is violated in related to the holographic retarded Green’s function the EPR pair through the holographic duality from the on-shell string fluctuations in higher dimension bulk gravity, our study −ω/(2Ta) ij −2e ij iG (ω) = ImG (ω) , (23) supports the essential property of this holographic model of EPR AB −ω/T R 1 − e a pair in Ref. [13]. We also explained that why a higher dimen- which has been examined in the equation (A23). It is similar to sional classical theory can reproduce the quantum property of the what was found for the boundary field theory in Refs. [27–29] boundary theory. Because from the boundary theory’s viewpoint, but that is from the on-shell action of the bulk gravity. From the the operators conjugated to the fluctuations of the EPR pair are F i F i quantized through path integral quantization. While in the semi- boundary theory’s viewpoint, the operators A , B in (20) con- ˜ A ˜ B classical limit of AdS/CFT correspondence, the final result of the jugated to the sources qi , qi are quantized through path integral quantization. However, in the semi-classical limit of the AdS/CFT path integral of this quantum EPR theory is assumed to be the correspondence for the string worldsheet in (21), the final result of exponentiated classical on-shell Nambu–Goto action. Thus, from the path integral of this quantum EPR theory is assumed to be the the bulk theory’s viewpoint, the string in the AdS bulk is in the I J probe limit and the string fluctuations are governed by the classi- exponentiated classical on-shell Nambu–Goto action S NG[q˜ , q˜ ] in i j cal equations of motion, which are obtained from the variation of (A20). From the bulk theory’s viewpoint, the string in the AdS bulk Nambu–Goto action. is in the probe limit and the string fluctuations are still governed In our derivation, we see that the bulk string fluctuations in the by the classical equations of motion in (17), which are obtained AdS gravity produce the holographic correlators. We assume that from the variation of the Nambu–Goto action in (16). the holographic SK correlators play the role as that in the CHSH For fluctuations coming from two causally separated quarks of formulation of the Bell inequality in a dual EPR pair on the bound- an EPR pair along x and y directions, and in the low frequency ary, which can be violated. Technically, this result relies on only limit ω → 0, √ two ingredients. The first one is that the observable in Bell’s test 3 is a time-ordered Greens function as shown in Eq. (11). And it is xx yy λa xy yx iG = iG = , iG = iG = 0, (24) well known that the time-ordered Green’s function does not have AB AB 2π 2 AB AB √ to vanish when the measurements P A and P B are outside of each ij ∝ ij which indicates that the spatial correlator G AB δ . The λ factor other’s light cone. Mathematically, this is because the behaviors of is consistent with the observation√ that the entanglement entropy the SK correlators outside the two horizons need to be correlated, of the entangled pair is of order λ [13]. It is also interesting that otherwise the solution is not smooth inside the horizons. The sec- this SK correlator does not vanish when the particles are sepa- ond one is that the equation of motion of the classical string, rated at long distance. This is consistent with the non-local nature Eq. (17), has no coupling between q˜x and q˜ y such that Eq. (24) of entanglement. However, the SK correlator vanishes when the follows. This can be obtained as long as the string does not expe- acceleration a becomes zero, when the ER bridge on the world- rience a force to propagate the fluctuation in the x-direction to the sheet disappears. Although it is unclear that how the holographic y-direction which breaks parity in general. It seems once the two Bell inequality is evaluated in this case, we may only approach the conditions above are satisfied, it does not matter whether there is zero acceleration limit after we identify the correlation with the an effective wormhole on the worldsheet in the bulk. Hence it is normalized operators as in the following Eq. (25)in which a de- conceivable Bell inequality can still be violated in a holographic pendence cancels. model where the EPR pair does not accelerate, similar to how J.-W. Chen et al. / Physics Letters B 791 (2019) 73–79 77 holographic entanglement entropy is computed in a static system and ERC Ideas Advanced Grant (No. 267985) “DaMeSyFla”; Y. L. [30,31]. Zhang was supported by CQUeST, APCTP (funded by the Ministry In this paper we are only work with the holographic model of Science, ICT and Future Planning (MSIP), Gyeongsangbuk-do and of an EPR pair. For future work, it is interesting to consider the Pohang City) and Grant-in-Aid for JSPS international research fel- back reaction by the measurements and see whether the effec- low (18F18315). tive wormhole on the worldsheet is broken due to the energy injected by measurements. This might provide an opportunity to Appendix A. Holographic SK correlator study the “wave function collapse” typically used to describe how measurements change the states. Another interesting topic is the In this appendix, we give a detailed derivation of the relation decoherence of the EPR pair in the environment. If the environ- between holographic Schwinger–Keldysh correlator and retarded mental effect can be described by thermal fluctuations, then we Green function in equation (23). The derivation follows Ref. [27, can add a black hole to the bulk of our model. When the distance 38]. Let x˜μ = (τ˜, r˜, x˜, y˜ , z˜) be the dimensionless coordinates in the between the particles in the EPR pair increases with time, the co-moving frame of the accelerating particles via (14). The square effective wormhole on the worldsheet also approaches the black 2 μ ν of line element of the AdS spacetime becomes ds = g˜μνdx˜ dx˜ , brane horizon and then enters the horizon [32]. We expect the where string breaks after it enters the horizon which might shed light on the decoherence process in the boundary field theory. For the 2 L 1 −2z˜ −2z˜ g˜μν = diag − f (r˜), , e , e , 1 . (A1) current holographic model of EPR pair we studied, the SK correla- r˜ 4rf˜ (r˜) tors will vanish in the limit of zero acceleration a → 0 and there is no “effective ER bridge” on the worldsheet anymore. Although this And xã = (τ˜, r˜) are coordinates on the static string worldsheet. does not necessarily contradict with the ER=EPR conjecture, since Without loss of generality, we can consider the string fluctuation μ ˜ ˜ ˜ ˜ there may exists an ER bridge of the other sort for this static EPR as X (τ˜, r) = τ˜, r, qi(τ˜, r) , which lead to an induced metric on pair. = μ ν ˜ the string worldsheet gab (∂a X )(∂b X )gμν . The Nambu–Goto ac- Our work is complementary to Refs. [33–35]in which proper- ˜ ˜ tion of string with tension T s is S NG=−T s dτ dr − det gab. When ties of quantum mechanical entanglement are given geometrical ˜ the fluctuation qi 1, the action becomes interpretations using holography. For example, the no-cloning the- √ orem in the quantum mechanics is related to the no-go theorem d ˜dr˜ 1 λ τ ˜ ˜ ˜ ˜ ˜˙ ˜˙ ij S NG− 1+ 2rf(r)q q − qiq j h , (A2) for topology change in general relativity [33]. The un-observability 2π 2r˜3/2 i j 2 f (r˜) of entanglement for most of the observables is related to the un- √ ∂q˜ ˙ ∂q˜ ˜ ˜ detectability of the presence or absence of a wormhole for most where λ ≡ 2 T L2, q˜ ≡ i , q˜ ≡ i and hij = diag[1, e2z, e2z] π s i ∂r˜ i ∂τ˜ of the observables [34]. The entanglement conservation is related have been used. to the invariant of a certain area in general relativity [35]. It will ˜ The classical equations of motion for the fluctuations qi on the be interesting to further explore the connection of these results to string are our model on holographic Bell inequality. ˙ It will also be interesting to see whether our work can be gen- 2 f q˜ q˜ i − i = eralized to distinguish the quantum mechanical “squashed” [36] ∂r˜ ∂τ˜ 0. (A3) r˜1/2 2 f r˜3/2 and the classical Shannon [37] entanglement. While the violation of Bell inequality for an EPR pair implies the pair is entangled Performing a Fourier transform, quantum mechanically, it is non-trivial to generalize such a test d ˜ ˜ ω −iωτ˜ ˜ ˜ from an EPR pair to a many-body system described by the bound- qi(τ˜, r) = e qi(ω)Yω(r), (A4) ary CFT. We envision this kind of theory would be able to subtract 2π the classical (e.g. thermal) contribution to the Ryu–Takayanagi en- ˜ where qi(ω) is defined as the Fourier transform of fluctuation on tanglement entropy formula [30,31]. ˜ = the boundary, after choosing the normalization limr˜→0 Yω(r) 1. Finally, the strongest version of ER=EPR suggests that each EPR Then (A3) becomes pair is literally connected by an ER bridge [10], not just dual to an ER bridge. So both ER and EPR coexist in the same unified the- ˜ − ˜ ˜ 2 ˜ ˜ f (r) 2rf (r) ˜ ω Yω(r) ory of the same dimensions, in contrast to our holographic model Y (r) − Y (r) + = 0. (A5) ω 2rf˜ (r˜) ω 2 f (r˜)r˜3/2 where ER and EPR exist in two different theories of different dimensions. Moving the EPR pair of our model to the bulk (which Requiring the in-falling boundary condition at the horizon, this might involve quantizing the string fluctuations) might provide a equation is solved by handle on studying this problem. −iω/2 Yω(r˜) = (1 − r˜) Fω(r˜). (A6) Acknowledgements The complex conjugate

We are grateful to A. Karch for many valuable comments and ∗ +iω/2 Y (r˜) = Y− (r˜) = (1 − r˜) F− (r˜) (A7) discussions. We thank helpful comments from D. Berenstein, F. L. ω ω ω Lin, and R. X. Miao, as well as many insightful comments from the is the other solution with the outgoing boundary condition at the referees, which all help us to ﬁnalize this version. J. W. Chen and horizon. S. Sun were supported by the Ministry of Science and Technology We need to extend these solutions into the Kruskal plane of (MOST) of Taiwan (No. 105-2811-M-002-130), Center for Theoret- the metric (A1), with new coordinates U and V , which are initially ical Sciences at the National Taiwan University (NTU-CTS). J. W. deﬁned in the right-quadrant {U < 0, V > 0}, with Chen was also partially supported by Massachusetts Institute of − ˜ − ˜∗ + ˜ − ˜∗ Technology-International Science and Technology Initiatives (MIT- U =−e 2τ e 2r , V =+e 2τ e 2r . (A8) MISTI) program and the Kenda Foundation. S. Sun was partially ∗ supported by MIUR in Italy under Contract (No. PRIN 2015P5SBHT) And r˜ is placed outside the worldsheet horizon 0 < r˜ < 1, 78 J.-W. Chen et al. / Physics Letters B 791 (2019) 73–79

where I, J=A, B and √ ˜ ij λ f (r) ij G (ω) =− Y− (r˜)∂˜Y (r˜)δ , (A21) R 2˜1/2 ω r ω r˜→0 π√b r f r˜ ij λ ( ) ˜ ˜ ij GA(ω) =− Yω(r)∂˜Y−ω(r)δ ˜ . (A22) πb2r˜1/2 r r→0 ij ij This can be compared with [28], where GR and GA are conjec- tured to be the holographic retarded and advanced Green’s func- Fig. 2. The Kruskal plane of the metric (15)in terms of coordinates U and V . The tions. From (A20)and (22), the off-diagonal one is related to the string worldsheet covers four quadrants, with the time-like boundary A (red) at the retarded Green’s function as (23)in the main text, left quadrant, and B (green) at the right quadrant. J [˜ I ˜ ] − /(2T ) √ S NG q , q −2e ω a r˜ √ ij ≡ i j = ij iG AB(ω) − ImGR (ω) . (A23) ∗ dw˜ 1 1 − r˜ δ(q˜ A )δ(q˜ B ) 1 − e ω/Ta r˜ ≡ = ln √ , (A9) i j f (w˜ 2) 2 1 + r˜ 0 If restoring the physical units, the exponential index becomes h¯ ω ha¯ exp[− ], and Ta = . Thus, the Planck constant does not with f (w˜ 2) = 1 − w˜ 2. The full extension of the metric (A1) and kB Ta 2πkB c appear in the correlators. string worldsheet in the Kruskal plane is shown in Fig. 2. In the right-quadrant the two solutions near the horizon are References − ˜ − q˜ B = e iωτ Y (r˜) ∼ e i(ω/2) ln(V ), (A10) − ω [1] J.S. Bell, On the Einstein–Podolsky–Rosen paradox, Physics 1 (1964) 195. B −iωτ˜ ∗ i(ω/2) ln(−U ) [2] A. Einstein, B. Podolsky, N. Rosen, Can quantum mechanical description of q˜+ = e Y (r˜) ∼ e . (A11) ω physical reality be considered complete?, Phys. Rev. 47 (1935) 777. And in the left quadrant {U > 0, V < 0}, [3]. J.F Clauser, M.A. Horne, A. Shimony, R.A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23 (1969) 880. − ˜ − − [4] B.S. Cirelson, Quantum generalizations of Bell’s inequality, Lett. Math. Phys. 4 q˜ A = e iωτ Y (r˜) ∼ e i(ω/2) ln( V ), (A12) − ω (1980) 93. A −iωτ˜ ∗ i(ω/2) ln(U ) [5] J.B. Hartle, Space–time quantum mechanics and the quantum mechanics of q˜+ = e Y (r˜) ∼ e . (A13) ω space–time, arXiv:gr-qc /9304006. Similar to the Herzog–Son’s prescription [27,39], two linear com- [6] J. Maldacena, A model with cosmological Bell inequalities, Fortschr. Phys. 64 (2016) 10, arXiv:1508 .01082 [hep -th]. binations are analytic over the full Kruskal plane, [7] S. Choudhury, S. Panda, R. Singh, Bell violation in the sky, Eur. Phys. J. C 77 (2) (2017) 60, arXiv:1607.00237 [hep -th]. ˜ ˜ ˜ B +πω/2 ˜ A q+(ω, r) = q+ + e q+, (A14) [8]. J.W Chen, S.H. Dai, D. Maity, S. Sun, Y.L. Zhang, Towards searching for entan- ˜ ˜ = ˜ B + −πω/2 ˜ A gled photons in the CMB sky, arXiv:1701.03437 [quant -ph]. q−(ω, r) q− e q−, (A15) [9] A. Einstein, N. Rosen, The particle problem in the general theory of relativity, Phys. Rev. 48 (1935) 73. which can be used as two bases for the string fluctuations, [10] J. Maldacena, L. Susskind, Cool horizons for entangled black holes, Fortschr. d Phys. 61 (2013) 781, arXiv:1306 .0533 [hep -th]. ˜ ˜ ω ˜ ˜ ˜ ˜ = qi(τ˜, r) = ai(ω)q+(ω, r) + bi(ω)q−(ω, r) . (A16) [11] L. Susskind, Copenhagen vs Everett, teleportation, and ER EPR, Fortschr. Phys. 2π 64 (6–7) (2016) 551, arXiv:1604 .02589 [hep -th]. [12] A. Almheiri, D. Marolf, J. Polchinski, J. Sully, Black holes: complementarity or The coefficients a (ω), b (ω) can be determined by the two bound- i i firewalls?, J. High Energy Phys. 1302 (2013) 062, arXiv:1207.3123 [hep -th]. ˜ A ˜ B ary values qi (ω) and qi (ω) of the solutions, [13] K. Jensen, A. Karch, Holographic dual of an Einstein–Podolsky–Rosen pair has a wormhole, Phys. Rev. Lett. 111 (21) (2013) 211602, arXiv:1307.1132 [hep -th]. = − ˜ A + πω/2 ˜ B [14] J.M. Maldacena, The large N limit of superconformal field theories and super- ai(ω) nω qi (ω) e qi (ω) , (A17) gravity, Adv. Theor. Math. Phys. 2 (1998) 231, arXiv:hep -th /9711200. = πω˜ A − πω/2 ˜ B bi(ω) nω e qi (ω) e qi (ω) , (A18) [15]. B.W Xiao, On the exact solution of the accelerating string in AdS(5) space, = πω − Phys. Lett. B 665 (2008) 173, arXiv:0804 .1343 [hep -th]. with nω 1/(e 1). [16] J. Sonner, Holographic Schwinger effect and the geometry of entanglement, In the following, we calculate the boundary formula of the on- Phys. Rev. Lett. 111 (21) (2013) 211603, arXiv:1307.6850 [hep -th]. shell Nambu–Goto action in (A2). After considering the classical [17] K. Jensen, A. Karch, B. Robinson, Holographic dual of a Hawking pair has a ˜ wormhole, Phys. Rev. D 90 (6) (2014) 064019, arXiv:1405 .2065 [hep -th]. equations of motions (A3) and integrating out the r direction, we [18] K. Jensen, J. Sonner, Wormholes and entanglement in holography, Int. J. Mod. obtain Phys. D 23 (12) (2014) 1442003, arXiv:1405 .4817 [hep -th]. √ [19] M. Chernicoff, A. Güijosa, J.F. Pedraza, Holographic EPR pairs, wormholes and f r˜ d λ ( ) ω ˜ ˜ ˜ ˜ ij radiation, J. High Energy Phys. 1310 (2013) 211, arXiv:1308 .3695 [hep -th]. S NG = − qi(−ω, r)∂˜q j(ω, r)δ ˜ . πb2r˜1/2 2π r r→0 [20] T. Hirayama, P.W. Kao, S. Kawamoto, F.L. Lin, Unruh effect and holography, Nucl. A B Phys. B 844 (2011) 1, arXiv:1001.1289 [hep -th]. (A19) [21] E. Caceres, M. Chernicoff, A. Guijosa, J.F. Pedraza, Quantum fluctuations and the Unruh effect in strongly-coupled conformal field theories, J. High Energy Phys. In (A16), we have obtained the analytic and regular solutions for 1006 (2010) 078, arXiv:1003 .5332 [hep -th]. the string fluctuations. Putting them back into (A19), the on-shell .[22] C.P Herzog, A. Karch, P. Kovtun, C. Kozcaz, L.G. Yaffe, Energy loss of a heavy quark moving through N=4 supersymmetric Yang–Mills plasma, J. High Energy Nambu–Goto action becomes Phys. 0607 (2006) 013, arXiv:hep -th /0605158. 1 d [23] M. Chernicoff, A. Guijosa, Acceleration, energy loss and screening in strongly- [˜ I ˜ J ]=− ω ˜ A − ˜ B + ˜ B − ˜ A S NG q , q q ( ω)q (ω) q ( ω)q (ω) coupled gauge theories, J. High Energy Phys. 0806 (2008) 005, arXiv:0803 .3070 i j 2 2π i j i j [hep -th]. ij ij M.[24] P. Chesler, K. Jensen, A. Karch, Jets in strongly-coupled N = 4 super Yang– × nω(1 + nω) GA(ω) − GR(ω) Mills theory, Phys. Rev. D 79 (2009) 025021, arXiv:0804 .3110 [hep -th]. + ˜ A − ˜ A + ij − ij [25] A. Karch, E. Katz, Adding flavor to AdS/CFT, J. High Energy Phys. 0206 (2002) qi ( ω)q j (ω) (1 n)GR(ω) nGA(ω) 043, arXiv:hep -th /0205236. ij ij + q˜ B (−ω)q˜ B (ω) nG (ω) − (1 + n)G (ω) , (A20) [26] A. Karch, S. Sun, Matrix flavor brane and dual Wilson line, Phys. Rev. D 89 (6) i j R A (2014) 066008, arXiv:1312 .2694 [hep -th]. J.-W. Chen et al. / Physics Letters B 791 (2019) 73–79 79

[27] C.P. Herzog, D.T. Son, Schwinger–Keldysh propagators from AdS/CFT correspon- [34] N. Bao, J. Pollack, G.N. Remmen, Wormhole and entanglement (non-)detection dence, J. High Energy Phys. 0303 (2003) 046, arXiv:hep -th /0212072. in the ER=EPR correspondence, J. High Energy Phys. 1511 (2015) 126, arXiv: [28]. D.T Son, A.O. Starinets, Minkowski-space correlators in AdS/CFT correspon- 1509 .05426 [hep -th]. dence: recipe and applications, J. High Energy Phys. 0209 (2002) 042, arXiv: [35] G. Remmen, N. Bao, J. Pollack, Entanglement conservation, ER=EPR, and a new hep -th /0205051. classical area theorem for wormholes, J. High Energy Phys. 1607 (2016) 048, [29]. D.T Son, A.O. Starinets, Viscosity, black holes, and quantum ﬁeld theory, Annu. arXiv:1604 .08217 [hep -th]. Rev. Nucl. Part. Sci. 57 (2007) 95, arXiv:0704 .0240 [hep -th]. [36] M. Christandl, A. Winter, Squashed entanglement: an additive entanglement [30] S. Ryu, T. Takayanagi, Holographic derivation of entanglement entropy from measure, J. Math. Phys. 45 (2004) 829, arXiv:quant -ph /0308088. AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602, arXiv:hep -th /0603001. [37] C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 [31] T. Numasawa, N. Shiba, T. Takayanagi, K. Watanabe, EPR pairs, local projections (1948) 623–656. and quantum teleportation in holography, J. High Energy Phys. 1608 (2016) [38] J. Casalderrey-Solana, D. Teaney, Transverse momentum broadening of a fast 077, arXiv:1604 .01772 [hep -th]. quark in a N=4 Yang–Mills plasma, J. High Energy Phys. 0704 (2007) 039, [32] F. Dominguez, C. Marquet, A.H. Mueller, B. Wu, B.W. Xiao, Nucl. Phys. A 811 arXiv:hep -th /0701123. (2008) 197, arXiv:0803 .3234 [nucl -th]. [39]G. W. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870. [33] N. Bao, J. Pollack, G.N. Remmen, Splitting spacetime and cloning qubits: link- ing no-go theorems across the ER=EPR duality, Fortschr. Phys. 63 (2015) 705, arXiv:1506 .08203 [hep -th].