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We review recent progress in understanding the entanglement entropy of gravitational configurations for anti-de Sitter gravity in two and three spacetime dimensions using the AdS/CFT correspondence. We derive simple expressions for the entanglement entropy of two- and three-dimensional black holes. In both cases, the leading term of the entanglement entropy in the large black hole mass expansion reproduces exactly the Bekenstein-Hawking entropy, whereas the subleading term behaves logarithmically. In particular, for the BTZ black hole the leading term of the entanglement entropy can be obtained from the large temperature expansion of the partition function of a broad class of 2D CFTs on the torus.

Entanglement is one of the most basic features of quantum mechanics. Historically, it has generated a long debate about the nondeterministic character of quantum mechanics. More recently, it has played a crucial role for the development of new areas of research, such as quantum information and quantum computing. Entanglement is a characterization of the spatial correlations between parts of a quantum system. It is measured by the entanglement entropy (EE), which is the von Neumann entropy arising when the degrees of freedom in an unobservable part of the system are traced over.

In recent years the notion of EE has been widely used for investigating general features of quantum field theory (QFT) and quantum phases of matter (e.g., spin chains and quantum liquids, [

As first recognized by ’t Hooft [

Unfortunately, any attempt to explain the BH entropy as due to quantum entanglement is plagued by both conceptual and technical difficulties.

First, the usual statistical interpretation of the BH entropy—aiming to explain the black hole entropy in terms of microstates—is conceptually very different from the EE, which measures the observer’s lack of information about the quantum state of the system in an inaccessible region of spacetime. The problem becomes even more involved going beyond the semiclassical approximation. In fact, in the usual Euclidean quantum gravity formulation the metric, except its boundary value, cannot be fixed a priori (see e.g., [

A possible shortcut for circumventing these difficulties is to consider gravity theories with CFT duals and to make use of the AdS/CFT correspondence to identify the black hole EE with the EE of the dual CFT. This approach allows to reduce the computation of the black hole EE to calculations in a field theory where spacetime geometry is not dynamical.

In this paper we use this approach to compute the EE of two-dimensional (2D) and three-dimensional (3D) AdS black holes. Two- and three-dimensional AdS gravity allows for a dual description in terms of 2D CFT. The calculation of the black hole EE is reduced to the calculation of the EE of a 2D CFT, for which very simple expressions are known.

We derive explicit formulas for the entanglement entropy of 2D and 3D AdS black holes. In both cases, the leading term of the EE in the large black hole mass expansion reproduces exactly the BH entropy, whereas the subleading term behaves logarithmically.

This work is mainly based on [

Quantum entanglement gives a measure of spatial correlations between parts of a system and it is measured by the entanglement entropy. In this paper we are mainly interested in the EE of a quantum field theory (QFT). In particular, we consider, following the discussion in [

The density matrix describing the subsystem

As discussed also in [

Using the general formulation described in

For

The three different forms of the 2D spacetime.

We can also consider a 2D CFT at finite temperature

It is important to stress that the cylinder

In

According to the holographic principle, suggested by ’t Hooft [

One of the most fruitful realizations of the holographic principle is the AdS/CFT correspondence, which was conjectured by Maldacena [

Essentially, the AdS/CFT correspondence can be interpreted as a relation between partition functions in the bulk and correlation functions on the boundary. The gravity partition function in the bulk turns out to be equal to the correlation functions of the operators

The AdS/CFT correspondence implies that we can describe a boundary quantum field theory in terms of a bulk gravity theory. In particular infrared (IR) effects in the bulk theory describing a

Following Susskind and Witten in [

This UV/IR connection is at the heart of the holographic requirement that the number of degrees of freedom should be of order the area of the boundary measured in Planck units [

As already mentioned in the Introduction, in the last years the notion of EE has been used with success as a tool for understanding quantum phases of matter, e.g., for spin chains and quantum liquids [

Because of its geometric nature, EE is also a natural candidate for trying to tackle one of the most difficult problems of black hole physics: the microscopic origin of the Bekenstein-Hawking entropy. The notion of quantum entanglement comes naturally in the play, since the horizon of a black hole divides spacetime into two subsystems, such that observers inside the horizon cannot communicate the results of their measurements to observers outside. This led ’t Hooft to conjecture in [

Another basic, conceptual, problem is the difficulty to relate the usual statistical interpretation of the BH entropy—in terms of black hole microstates—with the meaning of EE, which measures the observer’s lack of information about the quantum state of the system in inaccessible regions of spacetime. This problem becomes even more severe when one goes beyond the semiclassical approximation. As mentioned in the Introduction, the very notion of EE for pure quantum gravity is not easy to define: the main obstruction comes from the fact that in the Euclidean quantum gravity formulation [

A possible way out of these difficulties is to consider gravity theories with CFT duals (see e.g., [

There are several advantages in pursuing this approach. As explained in

There are two main drawbacks of this approach. The first is related to the fact that the AdS/CFT correspondence is holographic. Spatial correlations in the bulk gravity theory are codified in a highly nonlocal way in the correlations of the boundary CFT. This is particularly evident in the UV/IR relation, which relates large distances on AdS space with the short distances behavior of the boundary CFT [

The second strong limitation of the holographic approach is that it works only for gravity models with CFT duals. The most important class of such models is AdS gravity. In

Two-dimensional models of gravity have been used as toy models for dealing in a simplified context with complicated problems of black hole physics and quantum gravity. In particular, they can be used to describe the radial modes (the

2D AdS black holes are classical solutions of a 2D dilaton-gravity theory, which in the simplest case is described by the action

The black hole mass, temperature and Bekenstein-Hawking (BH) entropy are given by [

The 2D gravity theory has a dual description in terms of a chiral CFT with with central charge [

This AdS/CFT correspondence has been used to give a microscopical meaning to the thermodynamic entropy of the 2D AdS black hole. The BH entropy (

It has been observed that in two dimensions black hole entropy can be ascribed to quantum entanglement if 2D Newton constant is wholly induced by quantum fluctuations of matter fields [

However, two obstacles prevent a direct application of Equation (

Equation (

The calculations leading to Equation (

Owing to these geometric features, in the black hole case we cannot give a direct meaning to both the measures

The second difficulty can be circumvented using an appropriate coordinate system and a regularization procedure; the first difficulty can be solved using, instead of Equation (

The coordinate system

The regularized boundary is at finite proper distance from the horizon. Since

Putting all together we obtain from Equation (

The black hole EE (

In terms of the 2D CFT we have to trace over the degrees of freedom outside the spacelike slice

Thus, the EE of the 2D CFT in the curved background given by the AdS solution with negative mass has exactly the form given by Equation (

Let us now consider the large mass behavior

For

There are several arguments indicating that our derivation of the EE of 2D AdS black holes could be extended to black holes in 3D AdS spacetime,

In the case of 2D AdS gravity, the dual theory has the form of a chiral CFT, whereas in the 3D case, owing to different boundary conditions, the dual theory is a 2D CFT with both left and right movers.

In this section of the paper we investigate quantum entanglement in the context of 3D AdS gravity, in particular the Bañados-Teitelboim-Zanelli (BTZ) black hole, using the AdS

We will show that the AdS/CFT correspondence, and in particular the UV/IR relation, allows us to identify in a natural way

Classical, pure AdS

The classical solutions of 3D gravity corresponding to hyperbolic orbits of SL

The separating element between the two classes of solutions discussed above corresponds to parabolic orbits of SL

For

If we rescale the coordinates in Equation (

In order to find the holographic EE of the solution (

It is well known that the partition function of a 2D CFT on the complex has to be invariant for transformation of the modular group PSL

Let us briefly review the well-known duality between the BTZ black hole and AdS

In the asymptotic

Passing to consider the Euclidean solution with the conical singularity (

Because

As a consequence of the AdS/CFT correspondence, the EE expressed in Equations (

Owing to the holographic nature of the correspondence, the bulk interpretation of these parameters requires careful investigation. As discussed in

The UV/IR connection allows to identify the UV cut-off

The bulk interpretation of the parameter

The AdS/CFT correspondence and the IR/UV connection allow us to give to the EE (

It is interesting to notice that the identification

From Equations (

Let us now consider the classical solution of 3D AdS gravity given by Equation (

An important point, which we have only partially addressed, concerns the role played by the classical solutions of 3D AdS gravity describing conical singularities of the spacetime. Since they represent singular geometries, they cannot be part of the physical spectrum of pure 3D AdS gravity (although they may play a role for gravity interacting with pointlike matter). On the other hand, they are related with the BTZ black hole solutions by modular transformations, and one can associate an EE to them. All this could be very useful for shedding light on the phase transition (analogue to the Hawking-Page transition of four-dimensional gravity [

The spinless BTZ black hole (

The EE (

The holographic EE (

Macroscopic,

The logarithmic correction in Equation (

In principle, one could also consider the regime

The derivation of the EE for the spinless BTZ black hole can be easily extended to the rotating BTZ solution, obtaining (see [

Let us now briefly comment on the relationship between our approach and that of [

In this section we show that the leading term of the holographic EE for the BTZ black hole can be obtained from the large temperature expansion of the partition function of a broad class of 2D CFTs on the torus [

In the previous section we have discussed the holographic EE of gravitational configurations in 3D AdS spacetime. In our approach the EE of the boundary CFT,

First of all, we work in the region of validity of the gravity description of the AdS/CFT correspondence, when the AdS length

Moreover, considering curvature effects much smaller than the curvature

The other regime we have investigated so far is

One may now wonder about the regime

The most direct way to learn something about the relationship between the two regimes

Here we will use a simple, albeit not completely general, approach to this problem. We will show that, for the most common 2D CFTs (free bosons, free fermions, minimal models and Wess-Zumino-Witten models), the asymptotic, large temperature

The partition function of the CFT on the torus,

We are interested in the asymptotic expansion of

The leading term in the large temperature expansion of the thermal entropy for the four CFT classes on the torus considered in this section is

We have derived, using the AdS/CFT correspondence, simple and general formulas for the entanglement entropy of 2D and 3D AdS black holes. The picture that emerges is intriguing but also not completely surprising. The EE of 2D and 3D AdS black holes reduces to the Bekenstein-Hawking entropy—and the subleading terms have the universally predicted logarithmic behavior—only for macroscopic black holes, when thermal correlations dominate. Our result indicates that the EE for a black hole is a semiclassical concept with physical meaning only in the region

It is important to point out that we have not performed a direct calculation of the EE of matter fields in the BTZ background or other bulk gravitational configurations. In particular, in the 2D and 3D bulk we consider pure gravity with no extra matter fields (pure gravity in 3D has no physical propagating degrees of freedom, so we also have no gravitons).

A highly nontrivial check that our procedure gives the correct answer is represented by the exact matching of our results for the logarithmic subleading terms in Equation (

The main drawback of our approach is its limited range of validity. It works only for black holes in AdS spacetime. Moreover, we have only considered the 2D and 3D case; the extension to four-dimensional AdS black holes is in principle possible but computationally more involved.