The Fall of Black Hole Firewall: Natural Nonmaximal Entanglement for Page Curve

The black hole firewall conjecture is based on Page curve hypothesis, which claims that entanglement between black hole and Hawking radiation is almost maximum. The hypothesis is inspired by Lubkin-Lloyd-Pagels-Page theorem for degenerate systems with zero Hamiltonian. Adopting canonical typicality for nondegenerate systems with nonvanishing Hamiltonians, the entanglement becomes nonmaximal,and energetic singularities (firewalls) do not emerge for general systems. For static thermal pure states of black hole and Hawking radiation, entanglement entropy equals thermal entropy of the smaller system.


Introduction
The interesting possibility of black hole firewalls was proposed from a viewpoint of quantum information [1] and has attracted much attention. In the firewall conjecture, black hole horizons are not a smooth region even for free fall observers who attempt to pass through it. On the horizon the observers see highly energetic quantum walls (firewalls) before they collide against it and burn up.
Essentially the firewall conjecture is based on the Page curve hypothesis of black hole evaporation [2], and the hypothesis comes from the Lubkin-Lloyd-Pagels-Page theorem (LLPP theorem) [3][4] [5]. According to the LLPP theorem, quantum entanglement between two macroscopic systems S I and S II is almost maximum in a typical pure state |Ψ I,II of the composite Hilbert space H I ⊗ H II , assuming that dimension N II of H II is much larger than dimension N I of H I . The reduced density operator (quantum state) ρ I = Tr II [|Ψ I,II Ψ| I,II ] of S I almost equalsÎ/N I , whereÎ is unit matrix acting on H I . Inspired by this theorem, Page came up with a fascinating scenario for information leakage of evaporating black holes. He thinks that evaporation of a macroscopic black hole in an initial pure state is modeled by two quantum systems B and R with finite, but time dependent dimensions N B and N R . B represents internal degrees of freedom of the black hole, and R represents the Hawking radiation out of the black hole. It may be possible that the finiteness of N B and N R is justified if quantum gravity is taken account of. Such a quantum effect may truncate the degrees of freedom in a higher energy scale than Planck energy, like string theory. In condensed matter physics, total energy E is quite low. Thus high-energy density of states around the cutoff scale of the system becomes irrelevant. So it is enough to treat a finite dimensional Hilbert space to describe the physics.
The essence of Page's hypothesis is summarized into the following propositions for entanglement between B and R: (I) When N R ≫ N B (or N R ≪ N B ), B and R in a typical pure state of quantum gravity share almost maximal entanglement. In other words, a typical quantum state of the smaller system among B and R is almost proportional to unit matrixÎ.
(II) Entanglement entropy S EE of the smaller system among B and R is equal to its thermal coarse-grained entropy. Proposition (I) is clearly motivated by the LLPP theorem. Combining (I) and (II), it is deduced that S EE between B and R takes almost the maximum value and equals the Bekenstein-Hawking entropy of black holes S B = A/(4G) after the Page time, at which decreasing S B equals increasing thermal entropy S R of the Hawking radiation. The Page time is estimated as about 53% of the lifetime of evaporating black holes, and the mass at the Page time is about 77% of initial mass [2]. Thus the black hole remains macroscopic at the Page time, and its semi-classical picture is valid. Black holes after the Page time are referred to as "old". Before the Page time, S EE is equal to S R , and the black holes are referred to as "young". Since time evolution of S B and S R is computed in an established semi-classical way, this argument provides a prediction for the time curve of S EE during the evaporation. This is the Page curve. Its schematic figure is given in figure 1.
The firewall conjecture arises basically from (I). For an old black hole, Hawking radiation R is decomposed into A, which is emitted after the Page time, and C, which is emitted before the Page time. This is depicted in ŽůůĂƉƐŝŶŐ ^ŚĞůů where ǫ is ultraviolet cutoff (lattice spacing). Apparently, when ǫ → 0, this diverges like 1/ǫ 2 on the horizon for the typical stateρ AB ∝Î A ⊗Î B because of the correlation loss, and a firewall emerges. Besides the strong subadditivity paradox [8][1] is often worried about in the context of the firewall paradox. Let us suppose a strong subadditivity inequality of the von Neumann entropy for an old black hole A, late radation B and early radiation C: Assuming the no drama conditions S AB = 0 and S ABC = S C in the references [8] and [1] yields As long as the old black hole continuously emits the stored information after the Page time, the purity of AC system increases, and S AC decreases in time. Thus S A > S AC holds. This leads to an apparent contradiction that S C ≤ 0. Actually the early radiation has positive thermal entropy S C > 0. This seems to mean a breakdown of the no drama condition and suggests the existence of firewalls. However it is already known that the paradox is clearly avoided in moving mirror models [9] and long-lived remnant models [10]. In [9], it is pointed out that spacial locality among the subsystems A, B and C is ill-defined. Consequently the AB system inevitably has nonvanishing entanglement with zero-point fluctuation of the radiation field, and S AB = 0 does not hold even if we postulate the no drama condition for the horizon in a physical sense. This remains true in black hole evaporation. For the long-lived remnant models, S AC does not decrease even after the Page time, but rather increases until the last burst of the evaporating black hole. Thus S A > S AC is not satisfied and the paradox is evaded. Similarly, any evaporation scenario, in which no information is emitted out of the black hole until the last burst, is free of the strong subadditibity paradox at least. Though the strong subadditivity paradox can be avoided, the long-lived remnant models [11][12] [13] are supposed to have other flaws [14]. The energy of the remnant is of order of the planck mass, but in order to store the huge amount of information, the remnants seem to possess almost infinite degenaracy. The tremendous degeneracy may break the past great success of many experiments and observations via loop effects in particle scattering processes and species summation in partition functions for thermal equilibrium in early universe [13] [14]. In this paper, in order to avoid those flaws and firewalls simultaneously, we consider an alternative scenario. In the scenario, all the information come out at the last burst. Of course, the total energy of the last ray out of the black hole is merely of order of Planck mass and very tiny. However, as stressed first by Wilczek [15], an outgoing zero-point fluctuation flow of quantum fields, which is adjacent to the last ray, can share the huge amount of entanglement between the Hawking radiation emitted before. This fluctuation flow in a local vacuum region has zero energy, but transports the information to the future null infinity without any contradiction. At the last burst, quantum gravity critically affects the horizon. Hence the no drama condition is no longer required. Thus we do not need to care the strong subadditiviy paradox even if the entropy S AC suddenly decreases at the last burst. In this paper, first of all, it is pointed out that proposition (I) does not hold if nondegeneracy of energy eigenstates of the total system is taken account of. The typical states have to be exponentially close to Gibbs states with finite temperatures. The entanglement between AB and C becomes nonmaximal. Therefore, without breaking monogamy of entanglement, A is able to share entanglement with B, and simultaneously with C. The entanglement between A and B yields a correlation that makes the horizon smooth, and no firewall appears. Though it has been proven that such nonmaximal entanglement prevents the emergence of firewalls in moving mirror models [9], more stringent arguments are provided for general systems in this paper. Our result means that the ordinary used microcanonical states in the arguments of [1] and [2] are far from typical for quantum entanglement between a black hole and its Hawking radiation. In section 2, we briefly review a general formulation of canonical typicality with nonzero Hamiltonians [16,17,18,19,20]. Our discussion is based on [17]. From the rigorous results, it turns out that proposition (I) does not hold for general systems which satisfy natural conditions. Therefore it turns out that, after the Page time, the evapolating old black hole must evolve in Gibbs states with high precision as long as the pure state of the black hole and the Hawking radiation is typical. In general, heat capacity computed from a partition function Z(1/T ) of a Gibbs state must be nonnegative as whereĤ is the Hamiltonian of the system, T is temperature and · = Tr · exp −Ĥ/T /Z(1/T ). However the black hole heat capacity is acutally negative due to the gravitational instability. For instance, Schwarzschild blacl holes, the energy E is its mass M and equals (8πGT ) −1 . The heat capacity is computed as negative: dE dT = − (8πGT 2 ) −1 < 0. Thus the pure state of the system is never typical until the last burst. This leads to inevitable modification of the Page curve. In section 3, we discuss proposition (II). In black hole evaporation, the proposition is implausible. From a viewpoint of semi-classical general relativity, it looks more fascinating to take an alternative for the Page curve. The entanglment entropy continues to increase even after the Page time. At the last burst, it suddenly goes to zero and all the information is retrieved. Finally, it is commented that typical entanglement entropy of a large AdS black hole and its Hawking radiation equals thermal entropy of the smaller system. We adopt natural unit, c = = k B = 1.

Nonmaximality of Entanglement in Canonical Typicality
In this section, we claim that proposition (I) is not satisfied for general systems with nondegenerate Hamiltonian. When pure states are randomly sampled in a sub-Hilbert space with fixed total energy, a typical state is not maximally entangled. Then the corresponding state of the smaller subsystem is not the completely mixed state which is proportional to the unit matrix, but a Gibbs state. Let us think two finite quantum systems S 1 and S 2 , whose dimensions of Hilbert spaces H 1 and H 2 are denoted by D 1 and D 2 , respectively. Let us think a pure state |Ψ 12 in H 1 ⊗ H 2 . Its density operatorρ 12 = |Ψ 12 Ψ| 12 is a D 1 D 2 × D 1 D 2 Hermitian matrix. Thus it can be expanded uniquely in terms of a basis of U(D 1 D 2 ) Hermitian generators Î ⊗Î,Ĝ nµ :ρ whereĜ nµ are traceless and satisfy Tr Ĝ nµĜn ′ µ ′ = D 1 D 2 δ nn ′ and Ĝ nµ = Tr ρ 12Ĝnµ = Ψ| 12Ĝnµ |Ψ 12 .
The set ofĜ nµ consist of basis generatorsT n andR µ for each sub-Hilbert space.T n andR µ are traceless and Hermitian, and obey the following normalization: The remaining generators are given bŷ The reduced quantum stateρ 1 for S 1 is computed aŝ It is worth noting thatρ 1 is uniquely fixed by measuring only D 2 1 − 1 expectation values ofĜ n0 with respect to |Ψ 12 . This is because Ĝ n0 = Ψ| 12Ĝn0 |Ψ 12 = Ψ| 12 T n ⊗Î |Ψ 12 = Tr 1 ρ 1Tn = T n holds. In fact,ρ 1 is written aŝ In order to analyze the smaller system S 1 , expectation values of other components ofĜ nµ are not required.
Next let us think a nondegenerate HamiltonianĤ for the composite system.Ĥ takes its general form of whereĤ 1 andĤ 2 are free Hamiltonians, andV 12 is an interaction term between the subsystems. For simplicity, we ignore ⊗ andÎ in later equations such thatĤ 1 ⊗Î is abbreviated asĤ 1 . The normalized eigenstates ofĤ with eigenvalue E j are denoted by |E j : We define a set ∆(E) of energy indices for a macroscopically large total energy E and a positive number δ as Let us introduce a sub-Hilbert space H ∆(E) , which is spanned by {|E j |j ∈ ∆(E)} and its dimension is denoted by D. Microcanonical energy shell is defined as the set of pure states in H ∆(E) . It should be stressed that H ∆(E) is not a tensor product H B ⊗ H R of any sub-Hilbert spaces H B of H 1 and H R of H 2 . In order to understand this, let us suppose a case in whichV 12 is negiligibly small. Then Hence, the tensor product structure assumption of the Page curve hypothesis is not appropriate for descriptions of black hole evaporation.
For ordinary physical systems with large volume V , D becomes exponentially large like exp (γV ) for with a positive constant γ. Taking a small value of δ gives us a naive picture of for the energy shell, which often appears in standard textbooks of statistical mechanics. Note that δ-dependence for final results of statistical mechanics is irrelevant in general. In fact, δ does not necessarily have to be small in the later discussion, since the density of states e S(E) , where S is the entropy, is a very rapidly increasing function and the eigenstates close to the upper bound E give dominant contribution, as depicted in figure 3. Therefore, for simplicity, it is also possible to take an energy shell, say, Any pure state in H ∆(E) is written as where c j satisfy the normalization condition, j∈∆(E) |c j | 2 = 1. In order to analyze canonical typicality for H ∆(E) , let us introduce a uniform probability distribution for c j as is computed as The ensemble average of a quantum expectation value Ô of an observablê O in |Ψ 12 in eq. (2) is denoted by Ô . The statistical deviation from Ô is given by As proven in Appendix, the ensemble mean square error of Ô is upperbounded as where the operator norm Ô 2 represents the maximum absolute value of the eigenvalues ofÔ 2 . By takingÔ =Ĝ n0 , we have [17] δĜ n0 It should be stressed that Ĝ 2 n0 = T 2 n is independent of D 2 , though D grows exponentially as exp (γV 2 (D 2 )) with respect to volume V 2 (D 2 ) of S 2 . Hence the right hand side in eq. (4) becomes negligibly small as exp (−γV 2 (D 2 )) for large D 2 with D 1 fixed. Because the statistical fluctuation is so small, typical values of Ĝ n0 are very close to the central value Ĝ n0 . This implies thatρ 1 for typical |Ψ 12 in H ∆(E) coincides with its ensemble average stateρ 1 with almost certainty. In fact, we are able prove that the ensemble deviation ofρ 1 is estimated [17] as the right hand side of which decays rapidly due to D divergence as D 2 becomes large. Note thatρ 1 is given bŷ Due toV 12 , energy is exchanged between S 1 and S 2 as shown in figure 4. If the contribution ofV 12 is negligibly small compared toĤ 1 andĤ 2 , The sum, H 1 +Ĥ 2 , is approximately conserved. Then, as proven in many textbooks of statistical mechanics,ρ 1 becomes a Gibbs state with a fixed temperature with high precision for D 2 ≫ D 1 : for ordinary physical systems. The difference between the typical state and the Gibbs state must be exponentially small: The inverse temperature β is determined by the total energy E, which is much less than the cutoff energy scale for black hole evaporation. Contrary to proposition (I), eq. (6) shows thatρ 1 is not proportional toÎ 1 if the temperature is finite, unlessĤ 1 = 0. Note that a microcanonical stateρ m , which is proportional to the projection operatorÎ E onto the microcanonical energy shell, is far from typical, even though its von Neumann entropy I m is O(V ) and the difference from the Gibbs state entropy I c is merely O(ln V ). For the von Neumann entropy I t of a typical state, |I t − I c | is not O(ln V ), but exponentially small as O(exp (−γV )). 1 The entropy difference |I m − I c | = O(ln V ) is too large to regard the microcanonical state as a typical state. Nondegeneracy ofĤ provides nonmaximal entanglement between S 1 and S 2 to elude the strong subadditivity paradox. It has been pointed out [9] that such a nonmaximal entanglement appears in moving mirror models and avoids firewalls. The above argument extends the moving mirror result to general ones. This completely removes the reason for black hole firewall emergence in reference [1]. For evaporating old black holes, eq. (6) implies that state superposition of different energy black holes emerges in the total pure state of B and R, and generates much more entanglement, compared to a single black hole contribution with a fixed energy in the Page curve hypothesis. It should be noted that when the Gibbs state in eq. (6) is thermodynamically unstable due to the emergence of negative heat capacity, just like for asymptotically flat black hole spacetimes [22], typicality argument itself is unable to be applied to black hole evaporation, and never provides any correct insight. Quantum states in the evaporation are nontypical all the time, and proposition (I) loses its reasoning.
The result of canonical typicality with eq. (6) has been already commented by Harlow for weak interaction limit [7]. However it should be emphasized that Harlow does not give any proof of the typicality. Harlow also pointed out [7] that there remains a subtlety for the firewall removal even if we accept the canonical typicality. Let us recall the setup of gravitational collapse in figure 2. Naively, it may be expected that the quantum stateρ AB of late radiation A and black hole B for a typical state is approximated by The above tensor product structure of the state means no correlation between A and B even after taking account ofĤ nondegeneracy. Therefore, just as Figure 4: a canonical typicality setup for general two systems which preserves total energy and allows energy transportation between subsystems bi-directionally.
the original story of firewalls, the expectation value of the kinetic energy density of radiation fields might be divergent. However, we claim that the worry is useless. In order to see no divergence, we go back to the above general formulation. The system Hamiltonian is given bŷ whereĤ A ,Ĥ B ,Ĥ C are free Hamiltonians for the late radiation, the black hole and the early radiation, andV AB ,V AC ,V BC are two-body interactions among A, B, C, andV ABC is a three-body interaction (if we have). The setup is depicted in figure 5. Let us think thatV AC +V BC +V ABC are negligibly small as usual in statistical mechanics setups. However, we do not neccesalily assume thatV AB is small. Thenρ AB does not take the form of eq. (7), but insteadρ The expression of eq. It can be easily understood if we notice that the decomposition of AB system into two parts (A and B) is arbitrary. If we choose the boundary of two systems differently, we have different subsystems A ′ and B ′ and different free HamiltoniansĤ ′ A ′ ,Ĥ ′ B ′ and interactionV ′ A ′ B ′ between them. But physics of the composite system does not change at all becausê holds in eq. (8). From the viewpoint of A ′ and B ′ ,V AB is an ordinary local operator of A ′ or B ′ . There is no cause to make Tr AB ρ ABVAB = Tr diverge. This remains true even if we take the limit ofV AB → 0 and recover the expression in eq. (7). After all, we have no reasoning for firewall emergence on the horizon from a viewpoint of quantum information.
Here it is worth emphasing that the LLPP theorem can be regarded as a special case of the canonical typicality. When we consider a nondegenerate Hamiltonian with a cut off, the density of states e S(E) has the maximum value at an energy close to the cut off (See figure 3), and the temperature is infinite at this point since T −1 = β = ∂S ∂E = 0. If we choose a state randomly from the whole Hilbert space without any energy condition, we almost always get a state with energy corresponding to the maximum density of states. Then the subsystem has the Gibbs state with β = 0, which is proportional to the identity operator. This is exactly what the LLPP theorem claims. Consequently, we obtain a very high energy state with infinite temperature. Therefore the firewall argument based on the LLPP theorem actually says that not only the horizon, but also the whole space is on fire. (Note that the argument in [1] does not use any special property of the horizon. Therefore it can be applied to an arbitrary partition of the space.) This wrong conclusion teaches us the importance of introducing a physical Hamiltonian and the energy conservation law to consider this problem.

Possible Modification of Page Curve
In this section, we revisit Page's proposition (II). In section 2, we explained that quantum stateρ 1 of the smaller system S 1 in a typical state |Ψ 12 of H ∆(E) equals a Gibbs thermal state. Because entanglement entropy is defined as S EE = − Tr 1 [ρ 1 lnρ 1 ], S EE is actually the same as thermal entropy when S 1 and S 2 exchanges energy via the boundary, and interactionV 12 is negligibly small compared to the free Hamiltonians. This condition of smallV 12 really holds in nearest-neighbor interaction cases, because the free Hamiltonians are proportional to volumes of S 1 and S 2 , andV 12 is merely proportional to the boundary area. Thus proposition (II) might sound convincing. However, we should not forget a crucial condition which makesρ 1 a typical state. Energy has to be transported not only from S 1 to S 2 , but also from S 2 to S 1 . In such a situation, all of energy eigenstates in H ∆(E) are able to contribute on the same footing with each other, as depicted in figure 6. U 12 is expected to generate very complicated time evolution, and may yield fast scrambling of the system in H ∆(E) . Here 'scrambling' means relaxation of nontypical initial states with zero entanglement into typical states with high entanglement. After the relaxation, it is very unlikely to find the system in a nontypical state again for ordinary systems. This makes the canonical typicality method in section 2 promising for late time.  makes sense after the relaxation. How about cases in which energy is transferred only from S 1 to S 2 ? In these cases, the energy transportation does not happen from S 2 to S 1 , as depicted in figure 7. This actually arises in black hole evaporation, because the outgoing Hawking radiation emitted by black holes does not come back. The radiation is not able to give any amount of energy to the black holes without putting a mirror outside the horizon, or assuming thermal equilibrium. As well as the negative heat capacity of evaporating black holes, the one way energy transportation is caused by the gravitational instability. Such a one-way dynamics of energy transportation makes the system remain in nontypical states before the last burst of the black hole. Generation of the Hawking radiation takes place in an outside region (a few times the black hole radius far from the horizon) with very small spacetime curvature. In an ordinary sense, the semi-classical treatment of the generation is justified, and the process is not random at all. Time evolution of B and R is described essentially by fast scramblingÛ B ⊗Î R of black holes and nonrandom processÛ (emission) BR of the Hawking radiation emission. U B ⊗Î R does not change entanglement between B and R due to its locality. No fast scrambling for R happens. This nonchaotic setup does not ensure that the relaxation of BR system finishes before the last burst. Therefore, in realistic evaporations, the validity of the typical state postulate becomes dubious. The system may always evolve among nontypical states as depicted Figure 7: an actual setup for black hole evaporation, in which energy is transferred only from the black hole to radiation. The energy transportation from outgoing Hawking radiation to the black hole does not take place.
in figure 8. Therefore the quantum state of the old black hole is able to be far from Gibbs states after the Page time. Hence S EE can be totally different from thermal entropy, as opposed to Page's proposition (II).
Taking account of the possibility without (II), it is of significant to discuss modification of the Page curve. Moving mirror models, which mimic gravitational collapse and Hawking radiation emission out of black holes, may be a good device to see the possibilities. Let us consider a massless scalar field in 1+1 dimensions. Adopting light cone coordinates x ± = t ± x, the mirror trajectory is expressed as x where f is a monotonically increasing function of x − . Even if the vacuum state is set as the initial state of the field, the mirror excites the field and emits radiation, whose expectation value of the outgoing energy flux is computed [23] as  A trajectory given by describes a mirror which is at rest in the past, and accelerates with constant acceleration κ in the future. This trajectory is related to a realistic 1+3 dimensional gravitational collapse, which makes an eternal black hole without the back reaction of radiation emission [15]. Eventually the mirror emits constant thermal flux with temperature T = κ/(2π). In fact, substitution of eq. (10) into eq. (9) yields the correct thermal flux, for x − ≫ 1/κ. Now let us think mirror trajectories which may approximately describe black hole evaporation with its back reaction. The first candidate is the following: where h is a very large real constant, and controls the lifetime of the corresponding black hole. The trajectory is depicted in figure 9. Due to the trajectory deformation, the mirror stops in the future. The time evolution of radiation emission is given by the plot in figure 10 for κ = 1 and h = 500. During the evaporation, almost constant flux is emitted, though real black holes increase the temperature and flux of radiation. It is interesting to compute entanglement entropy S EE between the field degrees of freedom inside There exists a ultraviolet divergence in S EE due to infinite number of degrees of freedom of the quantum field [24]. To remove the divergence, a renormalized entanglement entropy ∆S EE is introduced by substituting the vacuum contribution [25]. ∆S EE is given by .
In figure 11, a plot of ∆S EE as a function of x − 2 is provided for κ = 1, h = 500, and x − 1 = −2. The curve is almost symmetric and looks like the Page curve [26]. This aspect comes from a fact that the order of the mirror deceleration, which is expected due to back reaction of the radiation emission, is equal to κ of the acceleration in eq. (11). However, this means locality breaking of dynamics for the evaporation and seems unlikely. κ corresponds to the scale of surface gravity of the 1+3 dimensional black hole, and is considered as the order of M 2 pl /M BH , where M pl is Planck mass and M BH is black hole mass. Thus κ is very small compared to M pl and the inverse of κ provides a cosmologically long time scale. The macroscopic black hole, whose evolution is described by eq. (11), needs to estimate by itself its destiny, how much time remains before its death, and when the deceleration must start. At the half of lifetime (x − ∼ 250), the black hole decides to emit its quantum information, which is stored inside the horizon, so as to finish leaking all the information before the last burst. In order to achieve this, the black hole has to slightly change its evolution at its Page time, which is much before its death, in a different way from those of other black holes with the same mass. For example, let us think two black holes with the same mass M BH , as depicted in figure 12. The left black hole in the figure is just born and very young. It does not begin the emission of the Hawking radiation yet and is almost in a pure state. The right black hole in figure 12 was born with its mass 1.3M BH (∼ M BH /0.77), and has decreased the mass to M BH via the radiation emission. Thus it is an old black hole around the Page time. It is worth stressing that classical geometries of the two macroscopic black holes are the same. Nevertheless, only the old black hole begins to change its evolution at x − ∼ 250. The young black hole will change in a similar way in a further future. To perform such a cooperative motion, all quantum microscopic ingredients of the black hole must watch each other carefully and preserve the long-term memory. Each part must estimate the black hole age by use of the memory. This dynamics is non-Markovian and non-local at least in time. If their dynamics can be approximated by the semi-classical general relativity, which is Markovian, such nonlocal evolution does not emerge. One might expect that small back reaction of the Hawking radiation to the black hole geometry becomes a trigger to leak the information for the old black hole. However, if the black hole has such a sensitivity, the geometrical perturbation induced by a small amount of infalling matter must also drastically change the time schedule of information leakage and completely modify the Page curve. This implies that Page curve is unstable and useless for realistic situations.
Of course, we are able to think a more natural mirror trajectory, which is consistent with the semi-classical general relativity. One of the examples is given by Here two different scale parameters κ, λ are introduced. κ is the acceleration parameter for the emission of Hawking radiation. λ is the deceleration parameter, which is of Planck scale order M pl and describes the sudden stop of the mirror due to the last burst of the black hole. Thus λ ≫ κ holds. The schematic behavior is given in figure 13. Clearly the last burst region should be described by quantum gravity. In figure 14, ∆S EE is plotted as a function of x − 2 for κ = 1, λ = 100, h = 500, and x − 1 = −2. Note that, until just before the last burst, ∆S EE is estimated by the semi-classical results of the outside Hawking radiation without use of knowledge about quantum black holes. This aspect circumvents uncertainty of quantum gravity, and strengthens the plausibility of this scenario. All the information is retrieved by the last burst. It is often commented that a huge amount of energy is required for such a information leakage However, it may be possible to attain it by use of entanglement with zero-point fluctuating of quantum fields [15] [9][21] as already emphasized above. Hence the energy of Planck mass order is enough to leak the information. In order to see the possibility, let us think an entangled particle pair in the vacuum state [21], as depicted in figure 15. One of the particles becomes a Hawking particle with positive energy after scattering by the mirror. However the partner particle is scattered by the mirror at rest and has no energy even after the scattering. Thus an enormous amount of quantum information, which makes the system state pure, is shared among the Hawking radiation and the outgoing zero-point fluctuation flow of quantum fields in the future null infinity. When this scenario is applied to realistic black hole evaporation, we would expect that quantum gravity generates higher derivative corrections to Einstein equation and the smeared would-be singularity inside the black hole horizon becomes timelike and preserves the information of falling matters.
Here we add some comments. The famous qualitative bound for the information leakage time of Carlitz and Willey [12] is unable to be applied to the sudden stop case in figure 15, because their bound is derived by assuming slow change of acceleration of the mirror and ignoring emission of negative energy flux, which is generated with the informational zero-point fluctuaion. Recently Bianchi and Smerlak found an interesting identity between the energy flux emitted from moving mirrors and the entanglement entropy of the radiation [27]. They also argue that when applied to two-dimensional models of black hole evaporation, this identity implies that unitarity is incompatible with monotonic mass loss. However, it should be stressed that they assume the Page curve as a unitary model of black hole evaporation for the argument. Thus the outgoing zero-point fluctuation flow scenario, as well as the long-lived remnant scenario, is consistence with their claim. It seems natural that the singluar sudden stop of mirror trajectory in figure 15 may be smoothed around Planck length scale by quantum gravity, as depicted in figure 16. Then a gravitational shock wave induced by the last burst ray may trap the entangled partner particle with zero energy for a while, just like in a high-energy gravitational scattering in a Minkowski vacuum. Even after reemission of the particle out of the shock wave, the energy of the partner particle may remain zero [28]. Thus the possibility is still alive that the information leakage by the last burst does not need a huge amount of energy in quantum gravity. This totally differs from the Page curve hypothesis, but is one of interesting possibilities.
Before closing this section, we add an interesting comment about a thermal equilibrium for BR system. Instead of black hole evaporation, let us suppose a pure state which describes a static and stable thermal equilibrium of the composite system in a coarse-grained meaning [29]. In asymptotically anti-de Sitter spacetimes, there exist thermal equilbriums for a large black hole B and its Hawking radiation R. They exchange energy bi-directionally. By adiabatically slowly changing system parameters includ-  ing external forces, and position and pressure of a mirror surrounding R (if we have the mirror), various sizes of the black hole may appear. From the general results in section 2, it turns out that each equilibrium state is typical, and the reduced state for the smaller subsystem among B and R is a Gibbs state with finite temperature, though no firewall emerges. Plotting entanglement entropy as a function of the inverse of black hole size generates a Page-like curve, in which entanglement entropy equals thermal entropy for the smaller subsystem. This may become relevant in the future research of quantum black holes, though it is merely a side story for the original information loss problem.  Figure 16: gravitational shock wave induced by the last burst ray and spacetime shift of zero-point fluctuation.

Summary
In this paper, we revisit the Page curve hypothesis. Adopting a general formulation of canonical typicality with nondegenerate Hamiltonian, it is proven that Page's proposition (I) is not actually satisfied for ordinary systems. The typical states are exponentially close to Gibbs states with finite temperatures. The entanglement between subsystems becomes nonmaximal and removes firewalls. The microcanonical state, which is proportional tô I E , is far from typical for the entanglement between a black hole and its Hawking radiation. In the dynamical situation of black hole evaporation, proposition (II) is also unlikely. We have no strong reason to expect that the entanglement entropy equals the thermal entropy for the smaller subsystem in the evaporation. Taking account of the semi-classical general relativity, a conservative scenario becomes more fascinating in which all the information inside the black hole is emitted by the last burst of the black hole. Finally, it is pointed out, using the general results in section 2, that for static thermal pure states of BR system in the sense of canonical typicality, entanglement entropy between B and R certainly coincides with thermal entropy of the smaller system. This holds for large AdS black holes.
The ensemble average state for S 1 is given bŷ The mean square error is computed as Therefore we can manipulate it as follows.