Bell Inequality in the Holographic EPR Pair

We study the Bell inequality in a holographic model of the casually disconnected Einstein-Podolsky-Rosen (EPR) pair. The Clauser-Horne-Shimony-Holt(CHSH) form of Bell inequality is constructed using holographic Schwinger-Keldysh (SK) correlators. We show that the manifestation of quantum correlation in Bell inequality can be holographically reproduced from the classical fluctuations of dual accelerating string in the bulk gravity. The violation of this holographic Bell inequality supports the essential quantum property of this holographic model of an EPR pair.

Interestingly, the ER=EPR conjecture implies that entanglement, which is thought to be a quantum mechanical effect, of the EPR pair, can be captured by a classical theory, at least when the EPR pair is very massive. In this case, it is classical gravity with an ER bridge, although more general EPR may require the ingredients of quantum gravity. To test this extraordinary claim, Bell inequality is a natural choice as it provides a sharp test of entanglement. In this work, instead of working with the original ER=EPR setup, we employ a concrete holographic model of the EPR pair proposed in Refs. [13,14] based on the Anti-de Sitter/Conformal Field Theory(AdS/CFT) correspondence [15]. The two particles of the boundary EPR pair are connected by a string in the AdS background of the bulk with an ER bridge on the string worldsheet. Therefore, this is a holographic realization of ER=EPR. We demonstrate that the Bell inequality violated by the EPR pair living at the boundary can also be violated by the gravitational theory with an ER bridge living in the bulk. Although the holographic setting is different from the original ER=EPR conjecture in which both ER and EPR live in the same spacetime dimensions, our study does shed light on how entanglement can be captured by a classical theory.
Bell Inequality. -The essence of Bell inequality is captured in the CHSH correlation parametrizations [3] which is reviewed here briefly. The entangled states made of a pair of spin 1/2 particles (the generalization to particles of higher spin is straight forward) are detected by two observers, Alice and Bob, respectively. The operators correspond to measuring the spin along various axes with outcomes of eigenvalues ±1. Performing the operations A and A on the first particle at Alice's location, and operations B and B on the second particle at Bob's location. With the Pauli matrices σ = (σ x , σ y , σ z ), and the unit vector n = (n x , n y , n z ) to indicate the spatial direction of the measurement, we have the following operators Then the CHSH correlation formulation is introduced as which is a linear combination of crossed expectation values of the measurements. In a local theory with hidden variables, the formula is bounded by the Bell inequality | C s | ≤ 2. While in quantum mechanics, this inequality can be violated, with arXiv:1612.09513v2 [hep-th] 21 Jun 2017 a higher bound | C s | ≤ 2 √ 2 [4] (see also [6]). For example, if we choose the entanglement state of a spin singlet and take the measurements along the (x, y) plane, i.e. n A = (cos θ A , sin θ A , 0) etc., it is straightforward to show Here cos θ AB = n A · n B depends on the relative angle of the measurements. And from (3) we have In particular, if we fix the direction of A and A , as well as the angle between B and B as π/2 then we have the relation depended on the direction of B, B , For 0<θ B <π/2, the Bell inequality | C s | ≤ 2 can be violated, and we reach the maximal violation at θ B = π/4, with an extra factor of √ 2. Bell's Test in Field Theory. -Before we move to holography, it is instructive to discuss the Bell's test in the context of the boundary quantum field theory. The test can be described by a process with the transition amplitude where a spin singlet state of Eq.(4) is measured by the projection operators to the final state |A ↑ B ↑ which is both spin up in the n A and n B directions. Note that the operators are time ordered since one can either measure P A or P B first. Squaring the amplitude to get the probability where only |A ↑ B ↑ contributes in the complete set of state |X in the first line and we have used the commutativity of P A and P B , as well as P 2 A = P A and P 2 B = P B for projectors.
The above discussion shows that the Bell's test is measuring a time ordered Green's function which does not have to vanish when P A and P B are outside of each other's light cone. A familiar example of this is Feynman propagator which is also a time ordered Green's function. Typically, this dramatic property of the Green's function does not show up in physical observable. It is very interesting that Bell's test relates those Green's functions to observables through Eq. (11). In the following holographic model, it is also a time order Green's function that we will compute.
Holographic EPR and Bell inequality. -In this section, we employ the holographic model of Ref. [13] to study the time ordered Green's function of Eq.(11) and the spin-spin correlation of the Bell's test.
The model proposed that an entangled color singlet quark anti-quark (q-q) pair in N =4 supersymmetric Young-Mills theory (SYM) can be described by an open string with both of its endpoints attached to the boundary of AdS 5 . The string connecting the pair is dual to the color flux tube between the two quarks, with a 1/r Coulomb potential as required by the scale invariance of boundary theory. Note that there is no confinement in this theory, therefore the pair can separate arbitrarily far away from each other. Various studies of related models can also be found in [16][17][18][19][20][21][22].
There are numerical solutions of the string shapes with different boundary behaviors [23][24][25]. 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 [14]. In the analytic solution the open string is also accelerated on the Poincáre patch of the AdS 5 with the AdS radius L and extra dimension w. The string solution in the AdS 5 bulk is given by The quark and anti-quark live on the AdS boundary w = 0. They are accelerating along the ±z direction, respectively, with the solution z = ± √ t 2 + b 2 . Therefore, the two entangled particles are out of causal contact with each other the whole time.
String fluctuations.-To consider the string fluctuations, we transform the solution to the co-moving spacetime (τ ,r, x, y,z) of the accelerating quarks via These two frames, which cover the regions z ≥ 0 and z ≤ 0 separately, are accelerating frames with a constant acceleration a = 1/b along opposite directions of z. And (14) only maps the upper part of the string (0 < w < b) into the proper frames of the accelerating quarks with 0 <r < 1. Plug this transformation (14) in the string solution (13), one finds the string configuration becomes z = 0 for both frames of the quark and anti-quark. Under this transformation (14), the metric (12) becomes where f (r) = 1 −r. Furthermore, with respect to the time τ =bτ , the Hawking temperature T H = 1 2πb matches with the Unruh temperature T a = a 2π [14,22] and we have set the reduced Planck constant and Boltzmann constant k B to be unit. Therefore, there is an event horizon atr = 1 associated with the quark and another event horizon also atr = 1 associated with the anti-quark. As shown in Fig. 1, the two horizons are connected by part of the string which can be seen as ER bridge.Hence it is suggested to be a holographic realization of the ER=EPR conjecture [9,10]. However, in the original conjecture, both ER and EPR live in the same spacetime dimensions, unlike the holographic model, EPR lives at the boundary while ER lives in the bulk.
The spin measurement of the quarks can be carried out by the Stern-Gerlach type experiment which applied a magnetic field gradience to generate a force that acts on the spins of the quarks. This introduces fluctuations to the world lines of quarks which set the boundary conditions for the worldsheet of the string fluctuations. Let (τ ,r) be the new worldsheet coordinates in the current frame, then the string fluctuation is X µ (τ ,r) = τ ,r,q i (τ ,r) , with i = (x,ỹ,z). Whenq i 1, the Nambu-Goto action of string with tension T s becomes whereq i ≡ ∂qi ∂r ,q i ≡ ∂qi ∂τ and h ij = diag[1, e 2z , e 2z ]. The equations of motion for the fluctuations on the string are Focusing on the transverse fluctuations i =x,ỹ,  (14), the trajectory of left(right) worldsheet horizon is on w = b, which depicts the intersect of the string with the world volume horizon seen by q(q) in its co-moving frame. The Bell inequality test is performed by spin measurements at Alice and Bob's locations. The AdS black brane is present only in the finite temperature case, which we briefly discuss in the last section.
bq i . In the low frequency limit ω → 0, it can be obtained analytically as and we have used the fact that T s L 2 = √ λ 2π . What we need for the Bell's test is the contour time ordered Schwinger-Keldysh (SK) Green's function, where F i A and F j B are separately defined on the causally disconnected left and right wedges of the Penrose diagram, corresponding to the boundaries of different patches of the AdS space. This off-diagonal SK propagator is examined in the Supplemental Material and found to be related to the holographic retarded Green's function similar to what was found in Refs. [26][27][28] but with different settings. For fluctuations coming from two causally separated quarks of an EPR pair along x, y directions, and in the low frequency limit ω → 0, which indicates that the spatial correlator G ij AB ∝ δ ij . The √ λ factor is consistent with the observation that the entanglement entropy of the entangled pair is of order √ λ [13]. It is also interesting that this SK correlator does not vanish when the quarks are separated at long distance. This is consistent with the non-local nature of entanglement. However, the SK correlator vanishes when the acceleration a becomes zero and the EPR pair is always infinitely far apart. Thus, we can only approach the zero acceleration limit after we identify the spin correlation with the normalized operators as in the following Eq.(23) in which a dependence cancels.
To study the correlators, we normalize the operators such that only the dependence on the spin wave function remains: The mixed measurements for correlators in CHSH correlation formulation become Together with the similar normalization of the operators A F and B F , the CHSH correlation formulations becomes For example, when θ AB = θ AB = θ A B = π/4, and θ A B = 3π/4, we can reach the maximum value 2 √ 2. In this derivation, we see the bulk string fluctuations, which come from classical gravity, reproduce the quantum entanglement of an EPR pair on the boundary. Technically, this result relies on only two ingredients. The first one is that the observable in Bell's test is a time ordered Greens function as shown in Eq. ( 11). And it is well known that the time ordered Green's function does not have to vanish when the measurements P A and P B are outside of each other's light cone. Mathematically, this is because the behaviors of the SK correlators outside the two horizons need to be correlated, otherwise the solution is not smooth insides the horizons. The second one is that the equation of motion of the classical string, Eq.(17), has no coupling betweenq x andq y such that Eq. (22) follows. This can be obtained as long as the string does not experience a force to propagate the fluctuation in the x-direction to the y-direction which breaks parity in general. It seems once these two conditions are satisfied, it does not matter whether there is an ER bridge in the bulk. Hence it is conceivable Bell inequality can still be violated in a holographic model where the EPR pair does not accelerate, similar to how holographic entanglement entropy is computed in a static system [29,30].
At this point, it is also curious whether experimental observables associated with time ordered Green's functions in our spacetime dimensions, not just in the bulk of holography, can be found. If it is found, then entanglement can be described in a classical theory without holography. In view of the original ER=EPR conjecture, this seems not completely impossible.
Conclusion and Discussions. -The ER=EPR conjecture is proposed to resolve the black hole information paradox without introducing a firewall in the black hole. The conjecture implies that entanglement of the EPR pair, which is thought to be a quantum mechanical effect, can be captured by classical gravity through the ER bridge. Using Bell inequality as a sharp test of entanglement, we study a holographic model with an EPR pair at the boundary and an ER bridge in the bulk. By revealing how Bell inequality is violated by classical gravity in the bulk, our study sheds light on the possible conditions needed for the entanglement of the original ER=EPR. Since the original ER=EPR has both ER and EPR living in the same spacetime dimensions, it is curious whether experimental observables associated with classical time ordered Green's functions in our spacetime dimensions, not just in the bulk of holography, can be found For future work, it is interesting to consider the back reaction by the measurements and see whether the ER bridge is broken due to the energy injected by measurements. This might provide an opportunity to study the "wave function collapse" typically used to describe how measurements change the states.
Another interesting direction is the decoherence of the EPR pair in the environment. If the environmental effect can be described by thermal fluctuations, then we can add a black hole to the bulk of our model. When the distance of the EPR pair increases with time, the ER bridge also approaches the black brane horizon and then enters the horizon [31]. We expect the ER bridge breaks after it enters the horizon which might shed light on the decoherence process in the boundary field theory.
Acknowledgements. -We are grateful to A. Karch for many valuable comments and discussions. We thank helpful comments from D. Berenstein, F. L. Lin, and R. X. Miao. J. W. Chen and S. Sun are supported by the MOST and NTU-CTS at Taiwan. J.W. Chen is also partially supported by MIT MISTI program and the Kenda Foundation. Y. L. Zhang is supported by APCTP and CQUeST. * jwc@phys.ntu.edu.tw; † sichunssun@gmail.com; ‡ yunlong.zhang@apctp.org; †, ‡ corresponding authors.
Andx a = (τ ,r) are coordinates on the static string worldsheet. Without loss of generality, we can consider the string fluctuation as X µ (τ ,r) = τ ,r,q i (τ ,r) , which lead to an induced metric on the string worldsheet g ab =(∂ a X µ )(∂ b X ν )g µν . The Nambu-Goto action of string with tension T s is S= −T s dτ dr √ − det g ab . When the fluctuationq i 1, the action becomes whereq i ≡ ∂qi ∂r ,q i ≡ ∂qi ∂τ and h ij = diag[1, e 2z , e 2z ]. The equations of motion for the fluctuationsq i on the string are Performing a Fourier transform, whereq i (ω) is defined as the Fourier transform of fluctuation on the boundary, after choosing the normalization limr →0 Y ω (r) = 1. Then (28) becomes Requiring the in-falling boundary condition at the horizon, this equation is solved by The complex conjugate is the other solution with the outgoing boundary condition at the horizon. We need to extend these solutions into the Kruskal plane of the metric (26), with new coordinates U and V , which are initially defined in the right-quadrant {U < 0, V > 0}, with Andr * is placed outside the worldsheet horizon 0 <r < 1, with f (w 2 ) = 1 −w 2 . The full extension of the metric (26) and string worldsheet in the Kruskal plane is shown in Fig. 2. In the right-quadrant the two solutions near the horizon areq And in the left quadrant {U > 0, V < 0}, Similar to the Herzog-Son's prescription [27,33], two linear combinations are analytic over the full Kruskal plane, q + (ω) =q B + + e +πω/2qA + , which can be used as two bases for the string fluctuations, The coefficients a i (ω), b i (ω) can be determined by the two boundary valuesq A i (ω) andq B i (ω) of the solutions, a i (ω) = n ω −q A i (ω) + e πω/2qB i (ω) , with n ω = 1/(e πω − 1). The total boundary term of the Nambu-Goto action in terms of the string solution turns out to be q i (−ω,r)∂rq j (ω,r)δ ij , (45) After considering (42), it becomes where the holographic retarded and advanced Greens functions are defined as Taking functional derivatives of S ∂ with respect tõ q B i (ω) andq A j (ω) yields precisely the Schwinger-Keldysh correlators. This off-diagonal one is related to the retarded Green's function as (21) in the main text, with T a = 1 2πb = a 2π after restoring the physical units.