# Entanglement from dissipation and holographic interpretation

## Abstract

In this work we study a dissipative field theory where the dissipation process is manifestly related to dynamical entanglement and put it in the holographic context. Such endeavour is realized by further development of a canonical approach to study quantum dissipation, which consists of doubling the degrees of freedom of the original system by defining an auxiliary one. A time dependent entanglement entropy for the vacumm state is calculated and a geometrical interpretation of the auxiliary system and the entropy is given in the context of the AdS/CFT correspondence using the Ryu–Takayanagi formula. We show that the dissipative dynamics is controlled by the entanglement entropy and there are two distinct stages: in the early times the holographic interpretation requires some deviation from classical General Relativity; in the later times the quantum system is described as a wormhole, a solution of the Einstein’s equations near to a maximally extended black hole with two asymptotically AdS boundaries. We focus our holographic analysis in this regime, and suggest a mechanism similar to teleportation protocol to exchange (quantum) information between the two CFTs on the boundaries (see Maldacena et al. in Fortschr Phys 65(5):1700034, arXiv:1704.05333 [hep-th], 2017).

## 1 Introduction

The AdS/CFT correspondence is the most successful application of the holographic principle, and it plays a very important role in the study of the non perturbative sector of a class of Yang–Mills theories. It allows to calculate non-perturbative vacuum expectation values of Yang–Mills operators via tree level calculation in the perturbative sector of type IIB string theory on Anti-de Sitter background. In fact, the conjecture enables calculations of Yang–Mills expectation values in terms of classical propagators on asymptotically AdS geometries. On the other hand, since the AdS/CFT correspondence is a duality, it would be possible to read the holographic dictionary in another way and use the Yang–Mills computations to infer classical properties of the bulk geometry. A more challenging application is to use the holographic dictionary to analyze quantum aspects of gravity. However, our understanding on the basic principle of the holography, even in the context of string theory, is still limited. It is well-known that, to address fundamental issues in quantum gravity such as black hole singularity or quantum cosmology, it is necessary to develop a more general framework for the holography, which allows us to go beyond the Anti-de Sitter spacetime and explore non-trivial generalizations of the AdS/CFT correspondence.

*A*is the area of the minimal surface in the bulk whose boundary is given by \(\partial \Omega \). Their proposal has been checked in many ways and this relation is proved in [2] for a spherical symmetry case and for a more general case in [3]. Quantum corrections to this area law formula are considered in [4, 5] and the time evolution of the entropy was studied in [6, 7]. Also, a holographic entanglement entropy in a higher derivative gravity theory has been obtained in several works, in particular in [8, 9, 10, 11, 12, 13].

The main goal of the present work is to explore the RT formula in the context of dissipative quantum field theories. Dissipative quantum field theories have been studied in the last years for several reasons, and one of these reasons is the experimental results that came out of the Relativistic Heavy Ion Collider (RHIC). The experimental results suggest that the quark gluon plasma is “strongly coupled” with a very small value of \(\eta /s\), where \(\eta \) is the shear viscosity and *s* the thermodynamical entropy density. This result motivates the development of finite temperature holographic techniques to calculate transport coefficients, and a universal result for the shear viscosity, \(\eta /s= 1/(4\pi )\), was derived in [14]. On the other hand, using the Bañados–Teitelboim–Zanelli (BTZ) black hole [15] as its holographic dual, the Brownian motion of a heavy quark in a finite temperature Conformal Field Theory (CFT) was studied in [16] in the context of Langevin equation. The holographic Schwinger–Keldysh method was developed for this case in [17] and the Thermo Field Dynamics (TFD) in [18]. One of the interesting results is that there is a drag force on the fluctuating external quark even at zero temperature, owing to radiation. This was studied in detail in [19], where the same result for the zero temperature term is obtained for pure AdS in all dimensions. This suggests that, even at zero temperature, important dissipative effects can be read from holographic techniques.

A close relation between dissipation and entanglement was pointed out in [20], where it was reported an experiment where dissipation generates continuously entanglement between two macroscopic objects. Concerning the quantum treatment of dissipative systems, the quantum theory of damped linear harmonic oscillators can lead to a vacuum state that is in fact an entangled state [21]. By putting forward the Feshbach–Ticochinski approach in [22], it was realized a relationship between the canonical quantization of the damped harmonic oscillator and the TFD formalism, where the damped harmonic oscillator was interpreted as the thermal vacuum and the TFD entropy operator appears naturally as the entanglement entropy operator, in a suitable extension to quantum fields. Following these ideas, we are going to use in this work the RT formula to give a holographic interpretation of the relationship between entanglement and dissipation, concerning conformal field theories at zero and finite temperature.

In order to handle dissipation using a canonical approach, one needs settings which are wider than those of usual applications of quantum field theory and the AdS/CFT correspondence. In particular, if one introduces dissipation from the outset in the usual settings, two immediate problems come to light. The first one is the Lorentz covariance breaking: dissipation is a typical process that breaks the invariance under boost transformations. Just because there is a deep relation between dissipation and the “arrow of time”, a dissipative process defines a natural preferred frame. Although it is possible to write a covariant action, the expectation values break the Lorentz covariance [23, 24, 25]. Also, as thermal effects are taken into account, the thermal bath’s frame works as the preferred frame. The second problem is the loss of unitarity. However, this problem can be circumvented and it is possible to preserve the unitarity and make usual quantum mechanics in a finite volume limit. In order to do such endeavour, we shall work therefore with Hamiltonian theories in which dissipative effects are caused by interactions with additional degrees of freedom [22, 23, 24]. In this case, the AdS/CFT correspondence allows an elegant interpretation of the auxiliary degrees of freedom. The explanation follows the same spirit of Israel’s work in black holes using TFD [26]. In TFD, in order to take care of thermal expectation values, the original system is duplicated and the thermal vacuum is defined via a Bogoliubov transformation, which actually entangles the system and its copy. In maximally extended black hole solutions, the auxiliary degrees of freedom are interpreted as fields living behind the horizon; for AdS black holes, this defines two asymptotic conformal theories. In references [27, 28], a mechanism to send a signal between the boundaries is presented. From a teleportation protocol, an effective potential reproduces an “attractive force” that brings the boundaries closer. The potencial is set up with the product of two hermitian operators, one on each side. In our work, we obtain a similar potential that couples the two boundaries.

We consider a particular coupling between two systems that will drive dissipation in one of the boundaries. Note that what defines the system and the auxiliary system is just the geometric region where the measures are taken. We show that, for one class of asymptotic observers, the vacuum state is an entangled state and we calculate the entanglement entropy in a “canonical way”. Actually, we show that the entanglement entropy can be calculated via expectation value of an operator, defined as the entanglement entropy operator, which is in fact a representation for the modular Hamiltonian [29], playing a dynamical role in this system. In fact, it will be seen that the time evolution of the vacuum entangled state is generated by the time derivative of that operator. So, after establishing a geometric description for the dissipative conformal field theory (DCFT), we can use the RT formula to infer that the time evolution of the minimal surface is controlled by the time dependence of the entangled state.

We observe at least two distinct holographic stages: for the first one, at very early times, some type of deviation from classical Einstein’s gravity should be considered in order to describe changes of the spacetime topology [30]; in the second one, for later times, we can study the leading large-N effects by considering a classical solution of the Einstein’s equations as the holographic dual. The final sections of the present paper will be devoted to study the second holographic stage.

In order to find out the geometric picture of the dissipative scenario presented here, we look for an asymptotically AdS gravitational system that reproduces the same dissipative dynamics on the boundary. It is known for a long time that scalar field coupled to Friedmann–Robertson–Walker (FRW) metric has a kind of dissipative behavior [31, 32, 33, 34, 35]. In particular, the equation of motion has the same damping term of the DCFT studied here. In [36] it was shown that, using a particular coordinate transformation, it is possible to have FRW metric on the boundary from a BTZ black hole on the bulk. So, this appears to be the geometric system that we are looking for. However, it will be shown that the time dependent entanglement entropy derived here demands that the original metric is not BTZ, but a sort of Vaidya solution in the adiabatic approximation; that is, a BTZ black hole with a slowly time dependent mass. Keeping this scenario in mind, the RT formula allows to find a natural relationship between dissipation, entanglement, thermodynamics and black hole physics. However, as it will be discussed in Sect. 7, for \(AdS_3\) the holographic picture based on the FRW boundary is just an approximation of the DCFT. In the asymptotic limit, the entangled state derived here can be seen as an approximation of the TFD vacuum.

This work is organized as follows: in Sect. 2 we present the dissipative model; in Sects. 3 and 4 the time dependent entropy is canonically computed; in Sect. 5 we show how we deal with the time dependence of the entropy in holographic computations; in Sect. 6 the holographic model is constructed; and Sect. 7 is devoted to the conclusions.

## 2 The dissipative model

*A*inside another system \(\bar{A}\) (bath, medium, or environment), whose degrees of freedom are unknown. The system \(\bar{A}\) interacts with the system

*A*such that, from the \(\phi \)-field perspective, there is a dissipative process which can be described by the equation

^{1}The Eq. (2) is just the resulting dynamics as seen by the A system.

We want to draw attention for the similarity of this construction with the rules of TFD: the field \(\psi \) can be considered the TFD’s double of the system, and it can be interpreted as a derivation of the TFD picture. In this context, the field \(\psi \) evolves in the inverse time direction. In the usual interpretation, if the coupling is switched off (\(\gamma \rightarrow 0\)), we recover two non-interacting (free) CFT’s whose states describe spacetimes with two locally asymptotically AdS regions [30]. In particular, the ground state \(|0>_\phi \otimes |0>_\psi \) represents two disconnected copies of the exact AdS spacetime [38], and the thermal vacuum^{2} is the Kruskal extension of an eternal AdS black hole^{3} [40]. Furthermore, the AdS/CFT dictionary implies that, in this example, the gravity dual theory is very stringy, and therefore quantum gravity interprets it as quantizing strings on AdS such that these metrics shall be recovered in the proper large N-limit. In the following sections, the behavior of the entropy will refer us to two different scenarios. We will focus on the large \(\gamma t\) limit, where it will be possible to make an approximation compatible with the large N limit, and the holographic picture can be understood in terms of Vaidya black holes, written in a particular coordinate system. Also note that the Lagrangian (4) is different from the ones studied in [41, 42] just because we are interested in the dissipative process and its relation with the entanglement entropy. As the dissipative process imposes a privileged direction of time, Lorentz invariance is broken [23, 24, 25]. However, the entanglement entropy for models with broken Lorentz symmetry has the same behavior as the relativistic ones [43].

*k*is real,

^{4}we have that \( \Gamma = \frac{\gamma }{2}\) and \(\omega _{k} = \pm \,\sqrt{k^2 - \frac{\gamma ^2}{4}}\). In summary, the general solution is plane waves damped by a decaying factor \(e^{-\frac{\gamma }{2}t}\). In contrast, the solution for the field \(\psi \) has a growing factor \(e^{\frac{\gamma }{2}t}\); however, by taking the physical time parameter \(-t\), it has the correct damping behavior.

^{5}

## 3 Entanglement entropy and dissipative dynamics

*t*, in a finite volume, can also be obtained as

^{6}First of all, considering the result of their expectation value in the time evolved vacuum,

*t*can be written in terms of \(\mathcal{W}_{n}(t)\)

*B*-degrees of freedom, we obtain the time-dependent reduced matrix density [50]

*A*-operator \(\mathcal{O}_A\) in \(\left| 0(t) \right\rangle \) can be written as

*S*achieves its maximum value for some time

*t*. This might be recognized precisely as the second law of thermodynamics in a quantum-mechanical sense. For instance, in the example given previously, we can observe explicitly that

^{7}

### 3.1 DCFT renormalized entanglement entropy

## 4 The time dependent holographic computations

The renormalized entropy found in (44) would be 4*G* times the area of the minimal surface in the dual holographic space whose boundary is anchored by \(\partial \Omega \). However, one does not know a priori how much of the variation of this entropy is due to the dual metric change or to the extremal surface, or both. So, let us discuss this point in detail.

^{8}In particular, \(\left| 0(t=0)\right\rangle = \left. \left| 0 \right\rangle \!\right\rangle \) corresponds to the globally AdS spacetime; actually, \(\left. \left| 0\right\rangle \!\right\rangle = \left| 0\right\rangle _A\otimes \left| 0\right\rangle _B\) is dual to two disconnected copies of AdS as in Fig. 1 (see [30, 38] for interpretations of such geometries). On the other hand, the entanglement entropy is given by the RT formula,

*t*(or \(\gamma t\)) as a parameter for the states’ metrics and compute

*t*(or \(\gamma t\)).

*t*, or \( \gamma t\).

*t*shall be interpreted as a parameter rather than a spacetime coordinate. However, if the whole spacetime is built up as a foliation of spaces \(\{(\mathcal{N} , g_{jk})_t, \}_t \equiv M\), there is a frame where the parameter

*t*is the time coordinate and it is on the same foot as the other coordinates of the spacetime,

*t*is the time coordinate,

*r*is the radial coordinate, \(\Omega _{d-1}\) stands for the polar coordinates, and

*R*is interpreted as the AdS curvature. The exact AdS spacetime is given by \(h= 1 + R^{-2} r^2\) and the induced metric on the spheres \(r=\text{ constant }\) is \(g_{\Omega \Omega }(r,t) d\Omega _{d-1}^2 = r^2 d\Omega ^2_{d-1} \). Now assume that \(h(r) \ne 0 \;\; \forall \;\; r \ge 0\).

^{9}Therefore, the extremal surface is a sphere \(r_0(t)\) whose area is

*h*(

*r*) could be determined by solving the Einstein equations (EE) with this input, and the method would allow to determine the space metric from the behavior of the extremal surface.

As it will be clear in the next section, the later times behavior of the entropy indicates that the system thermalizes (see Eq. (50)) [46, 57]. So, the later times regime might be holographically modeled by a two sided geometry consisting of slight (time dependent) deformations from the maximally extended AdS-Schwarschild solution, whose boundaries (where \(\Phi \) and \(\Psi \) live) are causally disconnected by an event horizon (see reference [40]), at least classically. The following model is going to be built up more precisely along these lines of thought.

The example above is a one-sided geometry that can be interpreted as the dual to the state described by the reduced density matrix \(\rho _\Phi \equiv Tr_\Psi |0(t)\rangle \langle 0(t)|\) in the field theory. This can also be expressed as a two-sided geometry dual to the full state \(|0(t)\rangle \in \mathcal{H}_\Phi \otimes \mathcal{H}_\Psi \). For example, for the AdS-Schwarzschild solution with \(h(r) = 1- 2M/r^{d-2} + R^{-2} r^2\), this state is the thermal matrix density \( \rho (\beta )\), while its TFD representation \(|0(\beta )\rangle \) corresponds to the Kruskal extension of the solution (see the construction of [40]). This type of spacetimes describes a sort of wormhole such that the minimal area surface is clearly placed at the throat of the geometry (Fig. 2).

Moreover, if the metric is globally a General Relativity (GR) solution (for some physical energy-momentum tensor), the topological censorship theorems [58] state that there is an event horizon separating causally these two conformal boundaries placed at \(r \sim \pm \infty \). Nevertheless, if the boundary field theories are assumed to interact as in the example above (Eq. (72)), then the boundaries should be causally connected (see [59]). Classically, this conflict can be bypassed only if we assume that the solution above, with *h*(*r*) maximally extended to all the real values of *r* with a throat of radius \(\sqrt{\alpha }\) at \(r=0\), does not contain event horizons anywhere; that is, it can be only a solution of some deformation from GR dynamics (e.g., Lovelock theory [60], Lorentz violating gravities [61], etc.). However, a mechanism that makes the black hole traversable, based on a teleportation protocol, was presented in [28]. This mechanism is suitable to understand the holographic picture for the entropy in the asymptotic limit (\(\gamma t \gg 1\)).

## 5 Constructing a holographic dual model

Different types of dual geometries corresponding to the behavior described in the previous section could be built up. As emphasized in [52], it is not clear whether a system of two interacting CFTs can be realized holographically. In the case studied here, we have two peculiarities: a time-dependent entropy and the breaking of Lorentz symmetry by dissipative processes. In fact, we will present a different scenario of those studied in [41, 42] and [52]. Let us shed some light on the kind of model we intend to present. From Eq. (49), it can be seen that the early times behavior of the entropy is similar to the results of [41, 42], where it was calculated the entropy for two interacting theories. On the other hand, the asymptotic behavior of the entropy, defined in Eq. (50), suggests that there is thermalization. So, in the later times, there is a formation of a horizon.

### 5.1 Early times

^{10}We can encompass these two disconnected copies in the same expression for the metric by the introduction of a “radial” global coordinate \(\varrho \), defined in the intervals \(\varrho < 0\) and \(\varrho > 0\), respectively.

^{11}Both manifolds are analytically completed by adding the limit points \(\varrho \rightarrow 0^\pm \). Expressed in terms of these coordinates, the metric in \(d+1\) dimensions reads

### 5.2 Later times

Based on these discussions, we will present in this section a gravitational system that fits to our dissipative theory. In particular, it has similar field’s equation of motion and it is described in a coordinate system covering a region inside the event horizon.

### 5.3 The BTZ black hole with a time-dependent boundary

*a*(

*t*) constrained by Einstein equations (see [36] for details). Comparing (84) with (86), we have

Note that, in general, the term \(H\dot{\phi }\) is absorbed in the frequency term by using the conformal time variable and rescaling the fields. However, if we keep the structure of Eq. (89), the approach of doubling the degrees of freedom to make canonical quantization used in [22] allows to explore the subtleties involving the definition of the vacuum of the dissipative theory. In our case, the auxiliary system is in fact a physical system, defined as the degrees of freedom behind the horizon.

*r*is less than \(r_e\) and the boundary system “sees” inside the horizon. According to RT prescription, in this coordinate system we have the following equation for the entanglement entropy [36]:

## 6 Gravity computations

In this part, we explain an approximated holographic method to analyze the DCFT. It assumes a holographic model for our dissipative system by a little time interval around (before) certain specific instant of its evolution \(t_0\), belonging to certain (stable) regime of interest.

*a*-coordinates. For this calculation, let us assume that there exists a \(t_0 \in \mathfrak {R}\) such that the holographic model simply stops to be a suitable description of our system, so the physical time parameter is taken to be \(t \le t_0\). We are going to assume that \(\mu \) in (80) varies slowly over time, such that the BTZ is in fact a sort of Vaidya solution in the adiabatic approximation, where \(\displaystyle \frac{\dot{\mu }}{\mu } \approx 0\) [6, 67] (see Appendix). The Vaidya solution is the simplest example of a time dependent gravitational background which is explored in the context of the RT conjecture [6, 68, 69]. It represents the black hole collapse and has a time dependent horizon defining a time dependent temperature [70, 71, 72, 73]. We assume also that for the \(AdS_3\), the holographic model describes only the system approximately, i.e. in a neighborhood (immediately before) of \(t_0\), so to first order

### 6.1 On the holographic energy density

### 6.2 Relating \(\gamma \) with temperature

The last term denotes the subleading contribution, which is proportional to \(\frac{\rho _0}{\gamma }\). This calculation can be viewed as a holographic check on the consistency that the entanglement in DCFT behaves as the entanglement of an ordinary CFT at finite temperature [57].

### 6.3 The approximated state, its time evolution and the breakdown of our \(AdS_3/DCFT_2\) description

Our approach discussed in the last section can be described as an approximation on the entangled state.

*t*approaches \(t_0\). This (approximated) equation allows us to interpret correctly our previous calculations, and to understand correctly the ranks of validity. Notice that the state on the right-hand side \(\left| 0_{a(t)}\right\rangle \) is the conformal ground state dual to the BTZ, in

*a*(

*t*)-coordinates.

Nevertheless, the form of the states (19) and (117), and the configuration of the systems are suggestively similar in many aspects. So a natural question that arises here is if there exists some way of slightly modifying the action/states of the double CFT proposed, in order to capture more of the dual gravitational description. The simplest possibility is to introduce a parameter, interpreted as a temperature from the A-subsystem, such that: (a) in the limit \(\beta \rightarrow 0\), the state agrees with 117, and (b) for low (temperature) scales \(\beta ^{-1} \rightarrow \,\sim \gamma \), one would recover the (main) dissipative/damping behavior (19). In the limit \(\gamma t \rightarrow 0\), the (sub)systems *A* and *B* decouple, and they can be seen as the standard TFD duplication in the state (117). This will be studied in a forthcoming work.

## 7 Conclusion

In this work, we have studied a close relation between dissipation and entanglement in the context of AdS/CFT correspondence. This kind of relation was experimentally verified in [20], where it was shown that dissipation generates continuously entanglement between two macroscopic objects. In order to study quantum dissipation in conformal field theories, we have followed the strategy of [22, 23, 24], which consists of doubling the degrees of freedom of the original system by defining an auxiliary one. In particular, using the canonical formalism presented in [22], we have defined an entropy operator, responsible of controlling the dissipative dynamics. The entropy operator, which naturally appears in the scenario studied here, turns up to be in fact the modular Hamiltonian. A geometrical and physical interpretation of the auxiliary system and entropy operator is given in the context of the AdS/CFT correspondence using the RT formula. One has two asymptotically AdS regions such that the two theories live on the respective asymptotic boundaries. The dissipation at one boundary is interpreted as an exchange of energy with the other boundary, which is controlled by a kind of constant coupling. The scenario is similar to the one presented in [30], where a teleportation protocol is explored in order to make the wormholes traversable, introducing interaction between the two asymptotic boundaries.

We have showed that the vacuum state evolves in time as an entangled state and the entanglement entropy was calculated. The entropy depends on time and has two distinct behaviors: in the early times, it is the typical entanglement entropy of two interacting theories; in the later times, the entropy’s behavior suggests thermalization. In order to give a geometric interpretation, we have looked for a gravitational system that reproduces a similar equation of motion in the boundary. Since this kind of dissipative behavior was studied long time ago in the context of scalar fields coupled to Friedmann–Robertson–Walker (FRW) metric with a damping term coming from the Hubble constant [31, 32, 33, 34, 35], then the BTZ black hole in the coordinate system developed in [36] is the natural system to study here, as it has a FRW metric in the boundary. However, in two dimensions a linear approximation must be done. In addiction, owing to the linear time dependence of the entropy, we have showed that the dual geometry must be not the BTZ black hole in the coordinates defined in [36], but an adiabatic approximation for the Vaidya black hole defined in the same coordinate system.

We have used the RT formula to relate the BTZ/Vaidya geometry’s throat area to the entropy. As the time evolution of the entangled state is controlled by the entropy operator, the RT conjecture allows us to relate the time evolution of the throat with the time evolution of the vacuum state. It should be emphasized that this kind of relation involving state/area is possible because, in the canonical formalism used here, there is a well defined entanglement entropy operator. A possible close relation of this operator with a kind of area operator, defined in the context of quantum gravity, deserves further development (for some suggestions in this sense, see for instance [50]).

In addition to the previously discussed deformation of the DCFT, there are some applications of this work that we intend to do. Another important sequence to this work is to use the approach developed here in the context of Janus solutions of supergravity (studied in [74]), where two CFTs defined on \(1+1\) dimensional half spaces are glued together over a \(0 + 1\) dimensional interface. If we can generalize the Janus deformation in order to include the dissipative deformation studied here, it will be possible to have a holographic interpretation of the dissipative system in terms of the time dependent Janus’ black hole. Also, it will be important to compute the retarded Green’s function in order to compare with the results of reference [75]. Finally, in the spirit of reference [28], it will be very interesting to study the specific teleportation protocol that reproduces the scenario presented here, as well as to apply the technology developed here in the \(AdS_2\) case, in order to find a possible connection between the DCFT and the Sachdev–Ye–Kitaev quantum mechanical model.

## Footnotes

- 1.
This may be viewed as a sort of quantum quenching [37], where an interaction is suddenly turned on.

- 2.
In a TFD description.

- 3.
- 4.
Otherwise we would have that \(\Gamma \) depends on

*k*. - 5.
- 6.
- 7.
In [51] it is argued that the entanglement entropy for a very small subsystem obeys a property which is analogous to the first law of thermodynamics; here this kind of relation naturally arises.

- 8.
We are assuming implicitly that the fields are in the adjoint representation of a SU(N) group in the large N limit.

- 9.
Here we are using radial coordinates such that \(g_{rr}^{-1} = g_{tt} \equiv h(r)\) for simplicity, but this is not crucial and depends on changes of the radial coordinates \(r \rightarrow r'(r)\).

- 10.
- 11.
The coordinate \(\varrho \) is defined in terms of the usual global radial coordinate \(\chi \) as \(\cosh ^2 \chi \equiv \displaystyle \frac{1}{1-\varrho ^2}\).

## Notes

### Acknowledgements

M. Botta Cantcheff would like to thank CONICET for financial support. The authors would like to thank Pedro Martinez for the help with the figures.

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