Quantum entanglement and Hawking temperature

The thermodynamic entropy of an isolated system is given by its von Neumann entropy. Over the last few years, there is an intense activity to understand thermodynamic entropy from the principles of quantum mechanics. More specifically, is there a relation between the (von Neumann) entropy of entanglement between a system and some (separate) environment is related to the thermodynamic entropy? It is difficult to obtain the relation for many body systems, hence, most of the work in the literature has focused on small number systems. In this work, we consider black-holes --- that are simple yet macroscopic systems --- and show that a direct connection could not be made between the entropy of entanglement and the Hawking temperature. In this work, within the adiabatic approximation, we explicitly show that the Hawking temperature is indeed given by the rate of change of the entropy of entanglement across a black hole's horizon with regard to the system energy. This is yet another numerical evidence to understand the key features of black hole thermodynamics from the viewpoint of quantum information theory.


INTRODUCTION
Equilibrium statistical mechanics, as pioneered by Boltzmann and Gibbs, provides a relation between the density operator of a system under various external conditions and the emergent properties of the matter interms of the thermodynamical quantities. More specifically, equilibrium statistical mechanics is governed by the phase space distribution whose knowledge enables one to find various (close to) equilibrium properties of a system [1] In the same spirit, over the last three decades, the key question that has been asked in different areas of physics is: What is the relationship between the quantum entanglement and the emergent properties of the matter? [2][3][4]. For instance: (i) The fundamental law of quantum information processing states that entanglement cannot be increased by local operations [5,6]. This law is identical to the second law that says that thermodynamical entropy cannot decrease in an isolated system. One hopes that thermodynamics may help in understanding quantum theory [7]. (ii) In the case of quantum phase transitions, the interactions induce quantum entanglement. Also, the same interactions play a significant role in determining the emergent properties of the system. There are many attempts to find the relationship between quantum entanglement and quantum phase transition [8][9][10]. (iii) In the case of gravity, it is emerging that the physics across event horizons plays an important role and the first connections were seen by Bekenstein [11][12][13]. Jacobson [14] showed that the thermal properties across such horizons reproduce the entire structure of Einstein's equations. Since then a number of proposals have been made along the lines of emergent gravity [15,16]. (see also Ref. [17].) However, our understanding of quantum entanglement is still very limited. In fact, quantum entanglement can been unambiguously quantified only for bipartite sys-tems [18,19]. While the bipartite system is an approximation for applications to condensed matter systems, however, in the case of black-holes, the event horizon provides a natural boundary. Besides, quantum entanglement and black-hole entropy/temperature are both purely quantum effects. Over the last three decades, there has been a large body of literature to associate the microscopic origin of black-hole entropy due to quantum entanglement of modes across the horizon [20][21][22][23][24][25][26].
However, entanglement as a source of black-hole entropy has a couple of drawbacks: (i) The proportionality constant depends on the ultra-violet cut-off and the number of fields present. Recently, it was shown that due to new scaling symmetries of the entanglement, the ultraviolet divergence can be mapped to an IR divergence [27] and (ii) It is not clear whether the entanglement entropy lead to the laws of black-hole mechanics. More specifically, whether one can recover Hawking temperature from the quantum entanglement of modes across the horizon.
In this work, we explicitly show that the microcanonical temperature obtained from entanglement entropy is identical to the Hawking temperature and satisfies the first law of black-hole mechanics. To evaluate entanglement entropy of the massless scalar field propagating in (D + 2)−dimensional spherically symmetric space-time, we use the direct approach -discretize the Hamiltonian and evaluate the reduced density matrix in the real space -instead of the conformal field theory techniques [28]. The principal reason is that, as shown recently [27], entanglement entropy may have more symmetries than the classical Lagrangian/Hamiltonian of the system.
To remove the spurious effects due to the coordinate singularity at the horizon 1 , we consider Lemaître coordi-nate which is explicitly time-dependent [31]. One of the features that we exploit in our computation is the following: For a fixed Lemaître time coordinate, Hamiltonian of the scalar field in Schwarzschild space-time reduces to the scalar field Hamiltonian in flat space-time [32]. We perturbatively evolve the Hamiltonian about the fixed Lemaître time and obtain the entanglement entropy at different times. We show that all times, the entanglement entropy satisfies the area law, however, the value of the entropy is different at different times. We explicitly show that ratio of the change in the energy to the change in the entropy is identical to the Hawking temperature for 4-and 6-dimensional Schwarzschild, Reissner-Nordström and 6-dimensional Boulware-Deser [33].

SCALAR FIELD HAMILTONIAN IN LEMAÎTRE COORDINATES
Th action for the massless, real scalar field Φ(x µ ) propagating in (D + 2)−dimensional space-time is where g µν is the Lemaître metric and is given by where τ, ξ are the time and radial components in Lemaître coordinates, respectively, r is the radial distance in Schwarzschild coordinate and dΩ D is the D−dimensional angular line-element. τ and ξ are related by [31] ξ − τ = dr In order for the line-element (2) to describe a black hole, the space-time must contain a singularity (say at r = 0) and have horizons. In this work, we assume that the space-time contains one non-degenerate event-horizon at r h and that it is asymptotically flat. The specific form of f (r) corresponds to different space-time.
The symmetry of the Lemaître metric (2) allows us to decompose the normal modes of the scalar field as: and Z lm i are the real hyper-spherical harmonics. We define the following dimensionless pa- Substituting Eq. (2) in the action (1) and using the orthogonal properties of Z lm i , we get, Noting that the line-element (2) is explicitly timedependent and that the radial coordinate ξ is related to τ via relation (3), we expand the action in Eq:(4) via infinitesimal transformation of ξ and τ [34]: Defining canonical conjugate momenta, we obtain Hamiltonian upto second order as where H 0 , V 1 and V 2 are defined in Appendix (A)(Appendix A contains detailed calculations). The above Hamiltonian is key equation regarding which we would like to stress the following points: First, in the limit of → 0, the Hamiltonian reduces to that of a free scalar field propagating in flat space-time [32]. In other words, the zeroth order Hamiltonian is identical for all the space-times. Higher order terms contain information about the global space-time structure and, more importantly, the horizon properties. Second, the Lemaître coordinate is intrinsically time-dependent; the expansion of the Hamiltonian corresponds to the perturbation about the Lemaître time. Evaluation of the entanglement entropy (EE) for different values of corresponds to different values of Lemaître time. Third, it is not possible to obtain a closed form analytic expression for the density matrix (tracing out the quantum degrees of freedom associated with the scalar field inside a spherical region of radius r h ) and hence, we need to resort to numerical methods. In order to do that we take a spatially uniform radial grid, {r j }, with a =r j+1 −r j . We discretize the Hamiltonian (23) for different space-times.
The procedure to obtain the entanglement entropy for different is similar to the one discussed in Refs. [21,24].
In this work we assume that the quantum state corresponding to the descritized Hamiltonian (23) is the ground state with wave-function Ψ GS (x 1 , . . . , x n ; y 1 , . . . , y N −n ). The reduced density matrix ρ( y, y ) is obtained by tracing over the first n of the N oscillators (Π n i=1 dx i ) Ψ * GS (x 1 , . . . , x n ; y )Ψ GS (x 1 , . . . , x n ; y ). In this work, we use Rényi entropy to be the measure of entanglement. In the limit of α → 1, S α reduces to von Neumann entropy. Also, Rényi entropy provides a convergent alternative to the measure of entanglement for all space-time dimensions [26]. For D = 2, we obtain entanglement entropy for α = 1, while for higher dimensions we obtain for different values of α. Fourth, in analogy with microcanonical ensemble picture of equilibrium statistical mechanics, evaluation of the Hamiltonian (23) at different , corresponds to setting the system at different internal energies. In analogy we define entanglement temperature [35]: Slope of EE(∆S α /∆ ) Slope of energy(∆E/∆ ) (8) In other words, we calculate the change in the ground state energy (entanglement entropy) for different values of and find the ratio of the change in the ground state energy and change in the entanglement entropy. While the entanglement entropy and the energy diverge, their ratio is a non-divergent quantity. To understand this, let us do a dimensional analysis where na is the horizon radius. In the thermodynamic limit, by setting L finite with N → ∞ and a → 0, T EE in Eq. (10) is finite.
For large N , we show that, in the natural units, the above calculated temperature is identical to Hawking temperature for the corresponding black-hole [36]: In the rest of this letter, for different black-hole spacetimes we obtain T EE numerically 2 and show that they match with Hawking temperature.

Schwarzschild (SBH) black holes
The D +2-dimensional Schwarzschild black hole spacetime in dimensionless unitsr is given by the line element in Eq.(2) with f (r) is given by: In Fig.(1), we have plotted total energy (in dimensionless units) and EE versus for 4 and 6-dimensional Schwarzschild space-time, respectively. Note that for 4dimensions we have computed von Neumann entropy(  Rényi entropy for α = 1), while for 6-dimensions we have computed for α > 1.
We would like to point few things regarding the numerical results: First for every , Rényi entropy scales approximately as S α (r h /a) D . Second, EE and the total energy increases proportional to . Third, we have plotted the entropy corresponding to the black hole and that scales with . The entropy corresponding to the scalar field decreases with time ( ) as the total system is isolated.
Using the relation (8), we evaluate the temperature numerically. For 4-dimensions, in dimensionless units, we get T EE = 0.076 which is close to the value of the Hawking temperature 0.079. In the case of six dimensions, we have evaluated the temperature for different values of α. However, it is important to note that for different values of N , we obtain approximately the same value of entropy. The results are tabulated, see Table(1). For all N, temperature from entanglement entropy α = 10 matches with the Hawking temperature.

Reissner-Nordström (R-N) black holes
The four dimensional R-N black hole is given by the line element in Eq.(2), where f (r) is Q is the charge of the black hole. Note that we have rescaled the radius w.r.t the outer horizon (r h = M + M 2 − Q 2 ). Choosing q = Q/r h , we get and the black hole temperature in the unit of r h is T BH = (1−q 2 )/4π. The energy and EE for different q values have the same profile, which looks exactly like in the previous case and is shown in figure (2 ). As given in the table (1), T EE matches with the Hawking temperature.

Six dimensional Boulware-Deser(B-D) black holes
The six dimensional B-D black hole is given by the line element with f (r) [33] is given by where a = 12α GB ω −2/3 ,r = (rω 1/3 )/R h , and α GB is the Gauss-Bonnet coupling term andR h is a dimensionless horizon radius and ω, is related to the ADM mass. The black hole temperature in the unit ofR h is [37] T BH = 1 4π For the numerical studies, we set the value of α as 10 and 40 and run the code for the N = 100 lattice sites for different α values. The profile of energy and Rényi entropy are same as like in case of lower dimensional black holes. That is,entropy for the scalar field is a decreasing function of as shown in figure (3).

CONCLUSIONS
In this work, we have shown -for a fixed horizon radius -the ratio of the change in the entanglement entropy at different times and the change the ground state energy at different times is related to the Hawking temperature. It is important to note that entanglement and energy diverges in the limit of a → 0, however, the entanglement temperature is a finite quantity.
Our analysis also shows that the entanglement entropy satisfies all the properties of the black-hole entropy. First, like the black hole entropy, the entanglement entropy increases and never decreases. Second, the entanglement entropy and the temperature satisfies the first law of black-hole mechanics dE = T EE dS EE . It is important to  note that in higher-dimensions, the temperature is linked to the arbitrary Rényi parameter (α). Recently, Baez has given a physical understanding of the parameter in the canonical ensemble picture [38]. It may be interesting to find such an understanding in the microcanonical ensemble picture.
While the unitary quantum time-evolution is reversible and retains all information about the initial state, we have shown that the restriction of the degrees of freedom outside the event-horizon at all times leads to temperature analogous to Hawking temperature. Our analysis may have relevance to the eigen-state thermalization hypothesis [39][40][41], which we plan to explore in the future work. and the redefined field operators areΠ such that they satisfy the following canonical commutation relation Appendix-B : Plots of internal energy as well as EE as a function of for different black hole space-time In this Appendix, we give plots of EE for different space-times;    The blue dots are the numerical data and the red line is the best linear fit to the data.