Investigation of black hole complementarity in AdS 2 black holes

Black hole complementarity plays a pivotal role in resolving the information loss paradox by treating Hawking radiation as carriers of information, apart from the complicated mechanisms involved in decoding information from this radiation. The thought experiment proposed by Susskind and Thorlacius, as well as the criteria set forth by Hayden and Preskill, provide deep insights into the intricate relationship of black hole complementarity between ﬁducial and infalling observers. We execute the Alice-Bob thought experiment in the context of two-dimensional anti-de Sitter black holes. It turns out that information cloning can be avoided in the case of a large black hole. According to the Hayden-Preskill criteria, the scale parameter associated with the explicit breaking of the one-dimensional group of reparametrizations must signiﬁcantly exceed the squared mass of the black hole to eﬀectively prevent information cloning.


I. INTRODUCTION
Hawking showed that semi-classical black holes emit thermal radiation and eventually evaporate [1,2].This fact raises the question of whether the quantum information contained in the matter that collapsed to form the black hole is lost forever [3].Subsequently, several alternative theories have been proposed to resolve the information loss paradox.
These theories encompass the idea that most of the information is emitted through Hawking radiation [4][5][6][7][8][9], the notion that black holes do not evaporate completely [10][11][12][13], the hypothesis that vast amounts of information are released during the final stages of evaporation [14], and the introduction of alternative horizon-free and non-singular structures like fuzzballs [15,16].A recent alternative viewpoint is also discussed in Ref. [17].
In particular, the concept of black hole complementarity (BHC) [6] has been discussed to reconcile the Hawking radiation carrying information with the equivalence principle in general relativity.It suggests that a freely falling observer referred to as Alice crossing the horizon would detect nothing out of the ordinary, while a fixed observer referred to as Bob would detect a significant feature known as the "stretched horizon."In fact, Susskind and Thorlacius [8] noticed that BHC does not contradict the principle of quantum mechanics.Consequently, they showed that an observer who falls into a black hole after extracting information from the Hawking radiation would not be able to detect the same information inside the black hole before reaching the spacelike singularity.They assumed that information absorbed by the black hole can be reconstructed after the Page time [9].Additionally, Hayden and Preskill [18] proposed an alternative viewpoint.They suggested that if information is thrown into a black hole after the Page time, it can still be recovered after the scrambling time, which is typically less than the Page time.Importantly, Alice and Bob in the black hole interior are also causally disconnected, and thus, the duplication of information can be evaded.In connection with BHC, there have also been some related studies [19][20][21].
On the other hand, according to AdS/CFT correspondence [22,23], the principle of unitarity within conformal field theory strongly implies that observers located in the bulk region should not detect any violations of unitarity.In relation to BHC, investigations concerning the three-dimensional Bañados-Teitelboim-Zanelli black hole [24] have affirmed the validity of BHC [25].In this respect, one may question whether BHC is still valid or not in two-dimensional AdS black holes obtained from the Jackiw-Teitelboim (JT) model [26,27] and the Almheiri-Polchinski (AP) model [28].
In this paper, we study the validity of BHC in two-dimensional AdS black holes after the Page time by executing the thought experiment presented by Susskind and Thorlacius, and then investigate it in the regime of the Hayden-Preskill's criteria using the scrambling time.The purpose of this paper is to find out some conditions to evade the duplication of information from two thought experiments distinguished by the Page time and the scrambling time.The organization of the paper is as follows.In Sec.II, we introduce a dilaton gravity model and explain the singularity structure of the AdS 2 black hole with the dilaton field.For the well-defined thought experiment, we focus on the black hole with the spacelike singularity.In Sec.III, we investigate BHC for the model based on the Susskind-Thorlacius thought experiment, and find that BHC is safe for a large black hole.In Sec.IV, we also examine BHC by employing the scrambling time in the Hayden-Preskill's criteria.We show that the scale parameter in the dilaton field should be larger than the squared mass of the black hole in order to evade the cloning of quantum information.We give our conclusions and discuss our results in Sec.V.In Appendix A, we discuss the scale parameter in the JT model with the timelike singularity.

II. TWO-DIMENSIONAL ADS BLACK HOLES
Let us consider a two-dimensional dilaton gravity model described by the action where Φ 2 = φ + φ 0 represents the dilaton field.Here, φ 0 is an arbitrary positive constant and ℓ is the radius of AdS spacetime.The dilaton gravity action (1) becomes combination of the topological term and the JT model [26,27].In particular, for φ 0 ∼ Q 2 and ℓ ∼ Q where Q is a magnetic charge, the model can describe the dimensionally reduced near extremal Reissner-Nordström (RN) black hole in the near horizon limit [29].For φ 0 = 1, it becomes the AP model [28].
In the conformal gauge of ds 2 = −e 2ρ(u,v) dudv, the equations of motion are obtained as follows: The Liouville equation ( 3) describes the spacetime with a negative constant curvature, and the length element can be expressed in terms of Poincarè coordinates as where the generalized coordinates x + (u) and x − (v) are monotonic functions.From Eq. ( 4) with the constraint equations ( 5), the dilaton field can be obtained as where M is the black hole mass and κ = π a .The scale parameter a responsible for explicitly breaking the one-dimensional reparametrization symmetry must remain positive to avoid a strong coupling singularity reaching the boundary within finite proper time [28].Upon using the coordinate transformations of one can rewrite the solutions ( 6) and (7) in the static form as Note that in Fig. 1, the metric in Poincaré coordinates (6) characterizes the entire black hole spacetime, whereas the metric in static coordinates ( 8) is confined to describing the exterior region of the black hole.
If the usual reflective boundary condition is chosen for a AdS black hole, the Hawking radiation from the black hole in the bulk can be reflected back into it, which means that the black hole does not evaporate and remains in thermal equilibrium with its Hawking radiation [30].To avoid this, one can choose an absorbing boundary condition by coupling a bulk scalar field representing the Hawking radiation to an auxiliary field at the boundary of AdS [31] since such a coupling permits energy to be transferred from the bulk field to the auxiliary field [32][33][34][35][36].In this respect, we will assume the absorbing boundary condition at infinity.To reveal the singularity structure in the global nature of the geometry, we consider the global coordinates y ± = y 0 ± y 1 .Then, the metric solution takes the form of and the dilaton field can be written as The dilaton singularity occurs at where Φ 2 becomes zero.In the Poincaré coordinates given by x ± = ℓ tan ( y ± ℓ ), the singularity curve (12) can be rewritten as where M c = φ 2 0 πa .When M < M c , the singularity becomes timelike and exists near the boundary.For M > M c , the singularity becomes spacelike, as illustrated in Fig. 1.Henceforth, we will consider scenarios where M > M c to preclude the presence of the timelike singularity.

III. THE THOUGHT EXPERIMENT BASED ON THE PAGE TIME
Let us start with the Alice and Bob thought experiment [6].Alice falls freely into the horizon of the collapsing black hole, carrying her information.On the other hand, Bob hovers outside the black hole and collects all the information emitted from the early Hawking radiation.According to Page's proposal [9], Bob can retrieve information about the collapsing matter, including Alice's information, after the Page time has elapsed.Once this information is extracted from the Hawking radiation, Bob proceeds to cross the event horizon, moving toward the singularity.Meanwhile, Alice sends a message encoded with her information to Bob before he reaches the spacelike singularity as shown in Fig. 1.If Bob receives her message before he touches the singularity, he observes the duplication of information and discovers a violation of unitarity in quantum mechanics.However, this issue may be resolved if the energy of the message Alice sends to Bob requires a scale far beyond the Planckian order.In other words, the backreaction caused by the super-Planckian energy significantly interrupts the background black hole geometry.Consequently, we can infer that Alice would be unable to send her message, thereby preventing any duplication of information on Bob's part.
Explicitly, the Page time for an evaporating black hole can be obtained using the Stefan- where σ is the Stefan-Boltzmann constant, and the temperature of the black hole is identified with T = √ κM π .Note that the thermal entropy is given by and this entropy is halved when the mass becomes Hence, the Page time can be calculated as After the Page time t P , Bob enters the black hole so that where Alice's initial location v A is set to zero for convenience.From Eq. ( 13), one can determine the coordinate when Bob reaches the singularity as Alice falls freely, the proper time she experiences between the horizon at x + = 1 √ κM and Using the uncertainty principle ∆τ ∆ω > 1 2 , we have where ∆x − A is a numerical constant that depends on Alice's initial data [18].Now, if the mass M is sufficiently greater than M c , the energy uncertainty becomes This implies that as long as M is substantially larger than M c , the scale of energy required to convey the message exceeds that of the black hole mass.Therefore, Alice cannot transmit her information to Bob, ensuring that information cloning is avoided.where M is the mass of the Schwarzschild black hole [8].
Similarly, if Bob jumps into the black hole after the scrambling time has elapsed, Alice needs to encode her message into the radiation with the order of the energy √ M [18].However, in our current model, the large mass limit is not enough to guarantee the no-cloning theorem.
The additional condition is that the scale parameter a must be significantly larger than the squared mass of the black hole.One might naturally inquire about what is the corresponding parameter to a in the RN side since the starting action (1) can be obtained from the nearhorizon limit of the near-extremal RN black hole which reduces to AdS 2 × S 2 [29].The constants M, φ 0 , and ℓ in our model are already expressed by the mass M (4) , the magnetic charge Q, and the Plank length ℓ P in the RN black hole: M = M (4) − Q ℓ P , φ 0 = πQ 2 2 , and ℓ = Qℓ P [29].In Appendix A, the identification of the scale parameter is elaborated as a = 2πQ 3 ℓ P .All these identifications are valid only under the near-extremal condition, which states M ≪ Q ℓ P , or equivalently M ≪ M c , under which a timelike singularity occurs.In the present thought experiments, however, we have assumed the condition of M > M c , under which a spacelike singularity occurs.Therefore, our conclusions on the prevention of information cloning in the two-dimensional AdS black hole are pertinent only when M > M c and do not extend to the geometry of the RN black hole.
FIG.1.A Penrose diagram illustrates the AdS 2 black hole along with dilaton singularities.In this diagram, both the light and dark gray regions represent the region described by the Poincaré patch(6).The dark gray area, in particular, indicates the exterior region of the black hole, which is described by the metric(8).The dilaton singularities are depicted by the zigzag curves.
IV. THE THOUGHT EXPERIMENT BASED ON THE SCRAMBLING TIMELet us now assume that the black hole is already maximally entangled with a quantum memory that Bob possesses.In other words, Bob can decode the initial quantum state encoded within the black hole.Now, Alice falls freely into the the event horizon of the black hole, her information remaining unknown to Bob.Meanwhile, Bob remains outside of the black hole and collects all the information emitted from the black hole.According to the proposal by Hayden and Preskill[18], Bob can retrieve Alice's information after the scrambling time has elapsed.Once Bob retrieves Alice's information from the Hawking radiation, he crosses the event horizon and proceeds toward the singularity.During this time, Alice attempts to transmit a message encoded with her information to Bob before he reaches the singularity.the scrambling time.For the black hole possessing a sufficiently large mass, duplication of quantum states is prevented after the Page time, as the energy required for transmitting the message would exceed the mass of the black hole.At the same time, the scale parameter a also should be sufficiently larger than the squared mass of the black hole to obey the Hayden-Preskill criteria.Consequently, we conclude that the quantum cloning in the twodimensional AdS black hole can be evaded for the large black hole with the large scale parameter a.It is worth nothing that in the case of the four-dimensional Schwarzschild black hole, the only condition required to avoid the duplication of information in the Alice-Bob experiment is to take the large mass limit.This holds true irrespective of the values chosen for the Page time and the scrambling time.Alice needs to encode her message into the radiation with the order of the energy1