A Unitary Model of The Black Hole Evaporation

A unitary effective field model of the black hole evaporation is proposed to satisfy almost the four postulates of the black hole complementarity (BHC). In this model, we enlarge a black hole-scalar field system by adding an extra radiation detector that couples with the scalar field. After performing a partial trace over the scalar field space, we obtain an effective entanglement between the black hole and the detector (or radiation in it). As the whole system evolves, the S-matrix formula can be constructed formally step by step. Without local quantum measurements, the paradoxes of the information loss and AMPS's firewall can be resolved. However, the information can be lost due to quantum decoherence, as long as some local measurement has been performed on the detector to acquire the information of the radiation in it. But unlike Hawking's completely thermal spectrum, some residual correlations can be found in the radiations. All these considerations can be simplified in a qubit model that provides a \emph{modified quantum teleportation} to transfer the information via an EPR pairs.


Introduction
In 1976, Hawking pointed out the information loss paradox [1,2], which says that the quantum state of the Hawking radiation emitted from the black hole is not pure, but completely thermal. This means that the black hole evaporation is not a unitary process, and information will be lost when the evaporation finishes. After Hawking's work, various alternatives to restore the black hole unitary have been proposed and studied, one of which is the black hole complementarity (BHC) [3]. It postulates that (i) black hole formation and evaporation are described via unitary quantum evolution; (ii) the region outside the stretched horizon is well described by QFT in curved space; (iii) to a distant observer, the black hole appears to be a quantum system with states given by, for example |M , with M the mass of the black hole; (iv) an in-falling observer can cross the horizon without encountering any trouble, in particular, a field vacuum can always be present in the near horizon region.
obtained, implying that the information is not completely lost in this case.
A qualitative model including the gravitational perturbation is further investigated, in which the gravitons (or gravitational field perturbations) play the role of an intermediate medium for the energy transfer between the interior and exterior of the black hole. Moreover, we show that the effective entanglements between the black hole and the radiations (in the detector) belong to a class, whose members are nonlocal and generated by some other already existing entanglements. This entanglement generation can be well demonstrated by a qubit model, in which correlations between two distant systems can be established through an EPR pairs. The outline of our paper is as follows. In section 2.1, we study the extension of the BHC (ii) to include a postulate about the interior region of a black hole. In sections 2.2 and 2.3, we develop our effective field model and obtain the required entanglement between the black hole and the added radiation detector. In section 3.1, a S-matrix formula for our model is constructed, while in sections 3.2 and 3.3, the situations of the information for our model is discussed in detail. The inclusion of the gravitational perturbation is qualitatively analyzed in section 4.1. Finally, in section 5, we give a brief summary and propose a qubit model of the black hole evaporation. Two appendixes are added. In appendix A, a simple model with a singular evolution operator is studied, and in appendix B, the mathematical detail of the qubit model is given.

Extension of the BHC (ii)
In the physics of a black hole, for example, a Schwarzschild black hole with a mass M , there are mainly two classes of observers: one class consists of the distant observers, or more generally static observers, while the other one is composed of the in-falling observers. For a static observer, the reference frame is given by the global (Schwarzschild) coordinate (t, r, θ, φ) with a metric singularity at r = 2M , leading to a hypersurface called an event horizon. As a result, there are two kinds of static observers which are localized in the exterior and interior of the event horizon (or black hole) respectively. In the view of a static exterior observer, an in-falling particle will never cross the event horizon to enter the interior of the black hole, in the sense of using infinite time because of the singular event horizon at r = 2M . Analogously, in the eyes of a static interior observer, even a beam of light can never escape out of the black hole by crossing the event horizon because of the same singular event horizon. While for an in-falling observer, the chosen coordinate is some locally inertial one ξ α X so that no singularity occurs. Consequently, in his view, the in-falling particle can cross the event horizon in a finite time.
The descriptions of above two classes of observers are apparently in contradiction, which seems to invalidate the principle of general covariance. In the quantum version, this contradiction is in fact expressed by the information loss paradox, in which the information is lost in the view of a static exterior observer. Certainly, the BHC was actually proposed to reconcile the contradiction between the two descriptions, but it seems to violate the monogamy of entanglement of quantum mechanics as argued in [4]. In [8], by adding an ancillary Hilbert space, the authors tried to reconstruct the local effective filed theory observables that probe the black hole interior, and relative to which the state near the horizon looks like a local Minkowski vacuum. However, they treated that ancillary Hilbert space only as a carbon copy of the space of the exterior Hawking radiation. Then whether the ancillary Hilbert space can be regarded as the space of the interior modes?
Let's consider the problem in a different way as follows. We start with the description of an in-falling observer, which is well described by effective field theory, in particular the near horizon vacuum always exists. If the principle of general covariance is also proper in the quantum version, the descriptions of two different observers about the physical world should be consistent with each other. However, this seems not to be the case for the static and in-falling observers. Let's consider a simplified model of the formation of a black hole, a shock wave model [10]. Initially, the space-time is a flat one, and after a while, a black hole is produced by a shock wave. The resulting black hole can be well described by the Vaidya space-time with line element [10] . For a static observer, an initial pure state will evolve to a mixed state due to the Hawking effect [1], i.e. the familiar information loss paradox. For an in-falling observer, however, the final state is pure, while the initial state should be thermal due to the Unruh effect [11], since the in-falling observer is accelerated relative to a flat space-time. These can be expressed formally as where |0 in is the Minkowski vacuum for the static observer relative to the initial flat spacetime, and |0 U ∼ |0 in is the Unruh (or near horizon) vacuum for the in-falling observer relative to the produced black hole. Then how to understand these processes, especially the second one involving an evolution from a mixed state to a pure one? If the first process is physically possible, then the second one should also be the case. The above analysis is based on a somewhat passive view, in the sense that for each observer there is an associative fixed coordinate frame, for example (t, r, θ, φ) for a static observer. There is, however, a physical and active view, in which there is not any paradox like (2.2). Initially, a static observer A is inertial relative to the flat space-time; when a black hole is formed, this static observer will be dragged by the gravity, so that it will become an in-falling one which is still inertial relative to the black hole background. As a result, in the eyes of A, the physical description should not be altered, in particular the unitary can be preserved. It is analogous for an initially accelerated observer B, who can only observe one part of the whole flat space-time, a left or right Rindler wedge. On the background of a black hole, if the initial acceleration just balances the gravity, then B will be treated as a static observer in either interior or exterior of the black hole. Consequently, in the view of B, the unitary property for only one part of the whole space-time background is also preserved. This active view is easy to understand conceptually, but not easy to be described technically. Therefore, in order to clarify the problem, the passive view is more convenient. However, by comparing the passive view with the active one, it implies that the paradoxes like (2.2) for the two observers occurs or not simultaneously. And since no paradox occurs in the active view, so does the passive view.
Let's return to the problem in the passive view again. The processes in (2.2) are both non-unitary, because some local measurements have been performed either initially or finally, indicated by the mixed density operators. Obviously, these local measurements are caused by the space-time causal structures in the views of respective observers. It thus seems that the only way to reconcile the contradiction is extending those local measurements by including some complementary ones, so that both the above two processes would become unitary. This has been achieved in reference [8] by adding an ancillary Hilbert space. By treating this ancillary Hilbert space as the one for the interior modes, the descriptions of the static and in-falling observers will be consistent with each other, and the principle of general covariance can thus be obeyed. In fact, according to general covariance, there should be an equivalence relation between the field observables constructed respectively by the in-falling and static observers This equivalence can be roughly expressed by the coordinate transformations and some additional possible transformations between the two observers, especially when the in-falling observer is crossing the event horizon. When restricted in different space-time backgrounds, the observables have different representations, even contain independent components. For instance, on a black hole background, the observables for the static observer will be split into the exterior and interior ones as similarly for the in-falling observables on a flat space-time background. In a word, a single O ext or O int is not complete enough to give a full physical description. Actually, the equivalence relation (2.3) indicates that the in-falling observer just serves as the "super-observer" of [3], whose description can involve both the interior and exterior degrees of freedom. This resembles the case of an EPR pairs, with the identifications where S t is the total spin of two electrons with respective spins S 1 and S 2 . In other words, the observables O ext and O int are independent and can be measured simultaneously, but the measurement results may be correlated for a pure state of the in-falling observer, for example the Unruh vacuum |0 U It's believed that this correlation or entanglement between the interior and exterior degrees of freedom can be transferred to the space of Hawking radiations, via a mechanism named as "transfer of entanglement" in reference [7]. However, we shall show in section 3.2 that, the correlation in (2.6) is not the one required for the purity of Hawking radiations. And our effective field model will provide another mechanism to account for the correlations among the Hawking radiations.
In conclusion, the above discussions imply that the BHC (ii) should be extended as: (ii') both of the exterior and interior regions of the black hole can be well described by QFT in curved space, with the singularity r = 0 excluded from the interior region. That is to say, there are two effective field theories on both the two sides of the event horizon, which are independent from each other in the sense that the observables are constructed with different modes of the fields. This is such a crucial extension that an effective field model can be proposed to satisfy the extended BHC.

The Hilbert Space: Introduction of a Radiation Detector
As shown in the above subsection, it is the combination of local observables O ext and O int that can give a consistent description. However, a single O ext or O int seems to not violate quantum mechanics. Then how to deal with the measurement problem? According to quantum mechanics [9], there is a so called generalized measurement that can be realized on the target system by performing an orthogonal measurement on a larger system that contains the target one. In other words, we should make a coupling between the target system with a measurement apparatus. In the black hole evaporation problem, we can also add a radiation detector that couples with the target system, for example, a scalar field. This detector can be located somewhere as a static observer, or fall freely into the black hole as an in-falling observer. In our model, we treat it as a distant static observer.
This radiation detector works in the following way. By coupling with the scalar field in the exterior of the black hole, the detector will be entangled with the exterior modes of the scalar field, so that a local measurement for the field observable O ext can be replaced by another one performed on the space of the detector. This approach, without performing an orthogonal measurement on the combined larger system, is different from the one utilized in quantum measurement theory 1 . In fact, what we need is just an environment that interacts with the scalar field, so that the energy or information can be transferred between them. Unlike the space of the scalar field which has been split into two parts due to the causal structure, the space of the detector is complete enough to give a consistent description, since the detector is located in the exterior in our model .
Assume the extended BHC is proper. The total Hilbert space of the entire system, including a (Schwarzschild) black hole B, a scalar field ψ and an added perfect radiation detector D (without energy loss), can be factorized as where the space of the scalar field H ψ is composed of exterior and interior modes. From the BHC (iii) and (iv), however, we can restrict our considerations within a smaller space in which a state can be expressed as 2 where |M B and |E D are the orthonormal basis states of the black hole and the detector, respectively. The entangled state in the bracket describes partially the correlation 3 between the black hole and the scalar field. |0 M ψ or |0 M for short (with M denoted as the dependence on the black hole's mass 4 ) is the near horizon vacuum of the scalar field [10] with b † ω andb † ω the creators of exterior and interior modes respectively. Here, the formula for the initial vacuum state in the shock wave model is used, since |0 in ∼ |0 U [10]. |0 B in (2.9) only stands for a space-time without black holes, with |0 0 ψ the corresponding scalar field vacuum. The free Hamiltonian of the entire system is with H D chosen simply as where d † ω and dω stand for the raising and lowering operators of the energy levels of the detector. As for the H ψ , it can be derived from a general formula with a time translation Killing field K α on a space-like Cauchy hypersurface Σ t . When Σ t approaches the infinite past I − , i.e. Σ t → I − , the corresponding free Hamiltonian becomes where a ω |0 M ψ = 0. For a general Σ t intersecting the event horizon, it will be split into Σ ext Σ int . When it approaches I + H + , i.e. the infinite future together with the future event horizon, the free Hamiltonian will be given by Notice the state |M B |0 M ψ ⊗|E D in (2.9) is an eigenstate of the free Hamiltonian in (2.11), with H ψ given by H a in (2.14), this property can be utilized to construct the S-matrix, as shown in section 3.1.

The Interior and Exterior Interactions
To verify the consistency of the BHC, we have to evolve the entire system and see whether it can always be descried well within the smaller space H 0 . An interaction term H int (t) is needed. The details of the interaction may involve some unknown quantum gravity effects, but we can still propose a simple model based on effective field theory. The scalar field can diffuse over the whole space-time, including both the exterior and interior of the black hole. Then, for a static observer, according to the extended BHC (ii') given in section 2.1, the full interaction can be chosen as H int (t) = H B,ψ (t) + H ψ,D (t). The term H B,ψ (t) gives the interaction between the black hole and the scalar field, which is localized in the interior of the black hole, while H ψ,D (t) is a local interaction between the scalar field and the detector in the exterior of the black hole. Moreover, these two terms should be independent from each other because of the causal structure of the black hole. However, we shall show below that the entanglements, between the b ω andb ω modes implicit in the vacuum state |0 M ψ , can be applied to correlate the causally disconnected interior and exterior of the black hole, which leads to the evaporation of the black hole.
Under these circumstances, H B,ψ (t) could simply be chosen as a direct coupling between the scalar field and the black hole where V(t) is a space-distribution operator acting on the space of the black hole, and the interaction region t × Σ t int is in the interior of the black hole. This interaction term is different from the one used in [8], where the interaction happens only at the stretched horizon, or in the exterior of the event horizon. Here according to the extended BHC (ii'), the interaction between the black hole and the scalar field are assigned to happen in the interior of the event horizon. We can still expand the operator V(t) in terms of V † ω and V ω that map black hole states |M B to |M ± ω B , together with the field vacua |0 M ψ to |0 M ±ω ψ due to the correlation in (2.9). If concerning only the black hole evaporation, V M,ω ≡ V M −ω,M (ω) = M − ω|V ω |M is the required matrix element, determining the emission rate of particle of frequency ω. To obtain these matrix elements, one can use an approximate completeness relation for the restricted space H B,ψ (2.17) In terms of the modes in the stationary regions, the scalar field can be expanded as with U ω ∼ (ω 1/2 r) −1 e −iωv the ingoing modes at the infinite past I − . The outgoing modes u ω at the infinite future I + and the incoming modesũ ω at the future horizon H + are [10] u Here we still use the formulas for the shock wave model, and consider only the s-wave components without the backscattering effects for simplicity. By substituting (2.19) into (2. 16), and noting that the interaction region is in the interior of the black hole, we have Then, by using of (2.17), we have where the factor ψ 0 M −ω |b ω |0 M ψ vanishes according to (2.10), i.e. out of the space H 0 . To avoid this, the created particle needs to be transported into somewhere else, for example the radiation detector. This can be accomplished by the interaction H ψ,D (t) where g ω (t)d † ω ∼g d 3 xφ D (t)u ω . φ D stands for some (localized) field inside the detector, and the interaction region t × Σ t ext is near the infinite future I + . The unitary evolution operator for the full interaction is where the emission and absorption parts have been grouped separately, while the relevant terms for the evaporation process is Then instead of the vanishing factor in (2.21), we will obtain a non-vanishing one [10] has been used. Considering the operation of (2.24) on |Φ in (2.9), the lowest order term is given as which expresses an entanglement or correlation between the black hole and the radiation in the detector. This entanglement between the causally disconnected interior and exterior of the black hole, is generated by the entanglements between the b ω andb ω modes implicit in the vacuum |0 M ψ . A more detailed discussion about this entanglement will be given in section 3.2. If the black hole continues evaporating, higher order terms will contribute. Without extra matter absorptions, for an initial state |M 0 B |0 M 0 ψ ⊗ |0 D , the black hole may evaporate completely in the end, leading to a final state 6 which is still in the restricted space H 0 . These two states can thus be related to each other by an S-matrix, which will be shown in section 3. (2.28) an analogous result asN (M, ω) = (e 8πM ω − 1) −1 [1,2], but with an ω 2 correction [12].
In [12], the authors treat e −8π(M −ω/2)ω as a semiclassical emission rate, but here it's N (M, ω) ≈ e −4πM ω 7 that is a part of the emission amplitude in (2.26). If the absorption part in (2.23) is also included, some other factors may also be obtained, for example a factor 0 M |b † ωbω |0 M that describes the process of the black hole emitting and re-absorbing. 6 (2.27) is not the only final state, since the interaction H ψ,D (t) can correlate the scalar field with the detector. If the absorption part in (2.23) is included, a state |Mmin B |0M min ψ ⊗ |Emax D may also be obtained, i.e. a dynamic balance with a "remnant" in the black hole. 7 The estimate N (M, ω) ≈ e −4πM ω agrees with another computation in (3.40).
Even the factorN (M, ω) can also be related to the process of the detector absorbing and re-emitting, and not be treated as an expectation value of the particle number. In fact, for some initial state |φ , by using of (2.23), the expectation value of the particle number for our model can be given by Furthermore, we can also calculate the correlation function where a completeness relation in (3.2) has been used. Since our model is proposed in a unitary manner, then all the possible intermediate states will contribute significantly, with some correlations being preserved among the radiations in the detector 8 . These in fact imply that a distant exterior observer can acquire the information of the radiations only by performing local measurements on the detector. A general measurement is given by which leads to a super-operator evolution [9] for the detector. And more discussions will be shown in section 3.3.

Some General Features of The Interactions
Consider the vacuum (|0 M ψ ) expectation value of the full interaction term with B and D denoted as the black hole and the detector respectively. For an in-falling observer equipped with a detector, this vacuum expectation value is the ordinary one without mode split, according to effective field theory. In other words, the field is expanded in terms of a ω modes, so the detector will also receive radiations of a ω modes. In this case, B actually stands for the perturbation of a flat space-time, i.e. the gravitational perturbation. The first order term in (2.34) vanishes obviously, while the second and higher orders give the exchanges of energy among the components of the entire system. In the view of a static observer, the second order contains the following four processes which still describe the energy exchanges or interactions between the components. The first two terms give the self-interactions of the black hole and the detector themselves, while the last two describe the interactions, or more exactly, nonlocal correlations between them. Since the space-time has been separated into two causally disconnected regions, the first two self-interactions are well described in the framework of local effective field theory; while the last two with independent interaction terms can give non-trivial results only through the entanglement between the b ω andb ω modes implicit in the vacuum state |0 M . The locality makes a static exterior detector always receive radiations only in terms of b ω modes; while for a static interior detector but still far from the singularity at r = 0, it will receive radiations only in terms ofb ω modes. All of those terms in (2.35) involve the scalar field's propagator denoted formally by ψ 2 . For an in-falling observer, it is an ordinary propagator, while for a static observer, it will depend on the relevant modes due to the black hole's causal structure. For instance, the self-interaction terms make use of only b ω orb ω modes; while the correlations between the black hole and the detector should make use of both the b ω andb ω modes. Recalling the quantity N (M, ω) defined in (2.25), except for the little difference of the vacua, it is just part of the propagator with contributions from both the b ω andb ω modes. In addition to the combination in (2.23), there is also another one, in which H B,ψ is expanded as for the evaporation can also be obtained, still with a contribution from the field propagator. At a first glance, it seems to be impossible to have different expressions for V ω (t) and d ω from those used in the model of section 2.2.2. This is indeed possible if the black hole and the detector have their own mode expansions, so that four combinations can be constructed for each of the interaction terms H B,ψ and H ψ,D .
Obviously, the description of a static observer is more complicated than that of an in-falling observer. Nevertheless, the principle of general covariance ensures that the two descriptions should be consistent with each other, in particular, the unitary evolution should be preserved in both of the descriptions, as discussed in section 2.1. In a view of evolution in the Heisenberg picture, it means that there should be a following sequence for the field propagator To some extent, this sequence for the static observer demonstrates the unitary property of the evolution for both the formation and evaporation of the black hole, if no black hole occurs at both the starting point and ending point of the evolution. If a local quantum measurement is performed on the scalar field as in the Hawking's arguments, the sequence in (2.36) will be broken since a quantum measurement is non-unitary. This can be roughly explained as follows. According to quantum mechanics, an observable O ext can be measured by means of a measurement apparatus with a coupling λO ext T app [9], with T app a corresponding operator of the apparatus, while the back-reaction of the scalar field on the black hole can still be described by H B,ψ . Then the vacuum expectation value of the full evolution operator is where the two terms can not be combined to give a scalar field's propagator as in QFT, thus breaking the the sequence in (2.36). This is also the difference between our effective field coupling H ψ,D and the quantum measurement coupling λO ext T app .
In the semiclassical treatment, the back-reaction of the scalar field on the black hole is treated by means of a semiclassical Einstein's equation, with one side the classical Einstein tensor, while the other one an expectation value T µν . According to the extended BHC (ii') in section 2.1, there should be two semiclassical equations in both the interior and exterior of the event horizon respectively. These two semiclassical equations can be well modelled by local interactions according to effective field theory. The fundamental interaction is the coupling between the gravitational field or perturbation and the matter field via with h µν the gravitational perturbation. Because of the causal structure of the black hole, the energy-momentum tensor T µν [ψ] is thus separated into two independent parts, one part T µν [ψ b ] in terms of the b ω modes, while the other one T µν [ψb] with theb ω modes. Analogously, the gravitational perturbation must also be split into two components that couple with the corresponding T µν [ψ b ] and T µν [ψb] respectively. Certainly, the proposed term H B,ψ in our model is just an approximation to the interior interaction. A somewhat precise treatment will be given in section 4.1.

The S-matrix Formula and Its Self-consistency
The S-matrix formula for a curved space QFT is more complicated than that for a flat space QFT because of the background dependence on the metric. An analogous but simple model is studied in appendix A, where the S-matrix formula is constructed in detail. In this subsection, we shall construct the S-matrix formula for our model following appendix A. The BHC (iii) indicates that the black hole's state |M can be treated as a steady state, giving a stationary space-time region. We can thus consider the transitions between various states, which are well defined in two stationary regions with different black hole's masses, for example M 1 and M 2 . In other words, an S-matrix formula between two arbitrary stationary regions can be constructed, at least for the restricted Hilbert space H 0 .
Recall H 0 is spanned by the eigenstates of the free Hamiltonian H 0 in (2.11), with H ψ given by H a in (2.14). Thus near the infinite past I − in each stationary region, we have with α denoted as a collection of the quantum numbers (M, E). For those eigenstates, we also have the following relations where the completeness relation (2.17) is extended by adding E |E D E| = I D . Following appendix A, the S-operator can be constructed as where the evolution operator is defined as Since the free field Hamiltonian H ψ depends on the black hole's mass, we have H 01 = H 02 . Hence there is also a singular initial condition U 21 (t 0 , t 0 ) = e iH 02 t 0 e −iH 01 t 0 like (A.14). This singularity also occurs in Hawking's original arguments [1,2], where near the two stationary regions I − and H + I + , the corresponding free Hamiltonians of the scalar field are Since H a = H b,b , a singular evolution operator like the one in (3.4) can be also constructed. Therefore, except for the possible singularity, the unitary is preserved formally 9 . The evolution in (3.3) can be extended to a general one with (meta-)stable sequences for the states of the black hole and the detector where the energy is roughly balanced between the black hole and detector, i.e. M i +E i ≈ M 0 for every i. As for the scalar field, the corresponding sequence of the vacua is A complete sequence of the Cauchy surfaces along the sequences (3.6) may be chosen as with the process 1 denoted as the formation of a black hole, i.e. the evolution in Hawking's arguments. The corresponding sequence of the free Hamiltonians is then given by 10 with H a and H b,b given by (3.5), and the interaction terms H B,ψ and H ψ,D of our model are contained in the full Hamiltonians H 1 , H 2 and subsequent ones in (3.8). 9 As analyzed in Appendix A, that singularity may be from the semiclassical property of curved space QFT, i.e. the classical metric. In fact, for our model, the black hole and the scalar field can be combined formally as a single component, with the corresponding Hamiltonian HB and state |φM B . Then the state for the entire system can be expressed as M +E=M 0 C(M )|φM B |E D , and the relevant interaction can be constructed as ω (Vωd † ω + h.c.), leading to the correlation between the black hole (together with the scalar field) and the detector. But this treatment is useless for our understanding of the black hole physics. 10 From this sequence of the free Hamiltonians and the subsequent evolution in (3.9), it indicates that the full evolution must have been out of the space H0, since |0M is not an eigenstate of H b,b . Hence we should apply an extended space with H0 just as a subspace. Some more discussions will be shown in section 3.4. Figure 1. The first two steps of a whole evolution, which is described in (3.9) with the Cauchy surfaces given in (3.7).
The first two steps of the above sequences are depicted in figure 1, and the required evolution operator can be constructed formally as The term in the brace in the last line describes the processes of the formation and evaporation of a black hole, which is so complicated that we can not give an exact expression. However, a full Hamiltonian H 20 exists so that since there is always such a time parametrization satisfying Then (3.9) will become which is the required associative relation for the evolution operator. In a functional form, this evolution can also be described as with B and D still denoted as the black hole and the detector, respectively. Analogously, the complete S-operator U (+∞, −∞) can be constructed step by step following the sequences (3.6). Since those sequences are chosen arbitrarily, there may be one in which the mass of an already formed black hole continues decreasing so that the black hole disappears in the end, as depicted in figure 2. In this figure, the long dash line is the event horizon of the formed black hole at r H 0 = 2M 0 , while the region surrounding by the zigzag line is the black hole interior, due to decreasing of the event horizon in size during the evaporation. There are also four typical t-slices (space-like Cauchy surfaces) in figure 2, foliating the space-time. Similarly as reference [3], for the t 2 slice, because it intersects the interior of the black hole, the state on it should be expressed as with the sum of the indexes i, j denoted as correlations between the interior and exterior degrees of freedom due to the interactions H B,ψ and H ψ,D . Notice (3.14) is different from the one used in [3], where the full state was given by a direct tensor product of the interior and exterior components, since there were no interactions that induce correlations between the interior and exterior modes in their article. Moreover, in the evaporation figure of [3], the interior region of the black hole is depicted as if it suddenly disappeared just after the complete evaporation. While in figure 2 the gradual changes of the event horizon is demonstrated apparently, that is, the interior region becomes exterior gradually during the evaporation. On the t 3 slice the evolution continues, but with less interior modes than those on the t 2 one. At last, on the t 4 slice the black hole disappears completely, so the interaction H B,ψ has stopped. And we will obtain a final pure state |Ψ(Σ t 4 ) , with the remaining correlations stored among the scalar field, the weak gravitational field and the detector because of the interaction H ψ,D and the one in (2.38). The evaporation depicted in figure 2 is just a particular one of all the possible complete evolutions that can be accomplished via (3.9) or (3.13) step by step. Therefore, we can conclude that the whole process of formation and evaporation of a black hole is unitary as the entire system evolves.

Discussions on The Paradoxes of Information Loss and AMPS's Firewall
As shown in the above subsection, the black hole evaporation based on our model is unitary, then how to understand Hawking's information loss arguments? According to [1], a distant exterior observer should construct an operator like O ext , which can act only on the space generated by b † ω . Then, if we calculate the expectation value like theb ω modes cancel automatically, leading to a mixed state. This can also be seen by performing a partial trace over theb ω modes in the initial density operator |0 M 0 M |, giving a super-operator evolution for the exterior b ω modes. Now, let's consider this problem in a different way. From the previous discussions we see that an operator O will evolve in the Heisenberg picture as giving a time parametrization of the expectation values ensures that the other part can also be known at the same time when the measurement is performed. That is, via a single local measurement, one is able to know, at least in principle, both the results about the exterior and interior modes because of the correlation implicit in (3.17), i.e. the lost information is recovered.
In reference [7], a "pull-back-push-forward" strategy is proposed to identify the interior with the exterior degrees of freedom. It's assumed there are two unitary transformations U and V that can evolve an initial full operator O as The exterior operator can thus be related to the interior one simply by [7] O Now let's go into the Schrödinger picture. Since O is a full operator, in order for (3.15) to contain only an "ext" (or "int" ) sector, in general, it should be assumed that then we will have This means the unitary transformation V −1 U would be singular, which is impossible. This inconsistency can also be seen as follows. Consider two non-commutative full operators O 1 and O 2 , then from (3.18) we can get that is, one representation can be found so that two non-commutative operators are commutative, which is also impossible. Therefore, the unitary transformations U and V utilized in the "pull-back-push-forward" strategy can not be constructed consistently. In fact, the observables O ext and O int are independent, and the exterior and interior degrees of freedom can be related to each other only through the correlation implicit in (3.17), or more exactly, the entanglements implicit in the vacuum |0 M . Whether the information stored in the correlation or entanglement implicit in (3.17) or the vacuum |0 M can be physically acquired ? The measurement result (3.15) gives an expectation value of the corresponding observable. Assuming the initial state is replaced by some other states, for example b † ω 0 |0 M , an analogous result can be obtained. To illustrate this problem, we take the particle number as an example. After some calculations, the two expectation values are given by with factor N (ω 0 ) = e 8πM ω 0 (e 8πM ω 0 −1) −1 the normalization constant for the state b † ω 0 |0 M . Obviously, it is the definite value 2+e −8πM ω 0 (e 8πM ω 0 −1) −1 in (3.23b) that may be regarded as a possible physical result. That is, to obtain a physical result, the vacuum expectation value should be subtracted, just like the case of vacuum energy in QFT. Further, the particle number in (3.23b) varies drastically with the particle energy ω 0 , from ∞ for low energy level to 2 for high energy level. This is such a large uncertainty that (3.23b) can not be treated as a well-defined physical result. Instead, it could be treated as a part of some transition amplitude, as shown below (2.29). We can extend this argument to every observable O ext or O int , and thus resolve Hawing's information loss paradox by saying that the measurements he used are not physically practicable. This can also be regarded as the difference between our super-operator evolution (2.33) and that obtained by Hawking [1]. For this reason, a radiation detector has been introduced in our model to receive the radiation, with local measurements performed on the detector, as in (2.32) and (2.33). These measurements are physically practicable, because the space of the detector is complete.
Then where is the physically acquirable information? Without any measurement, the black hole and the detector in our model are effectively correlated or entangled via the entanglement implicit in |0 M , as given roughly by the transition in (2.26). The physical information is thus stored in this new generated correlation or entanglement between them. This does not violate the monogamy of entanglement, since any state in the space H 0 can be both an a ω -vacuum and a d † ω d ω eigenstate, instead of the b † ω b ω as argued in [4] 11 . The physical information of the radiations has been transferred into the detector, via the interaction H ψ,D . Under this circumstance, the arguments of the AMPS's firewall don't hold any longer, because all the radiations, no matter an early or a late part, are entangled not to theb ω modes behind the event horizon, but only to the states of the black hole. In fact, there are four subsystems effectively, the black hole, the detector, and the two disjoint parts of the scalar field described by those b ω andb ω modes respectively. While in the firewall paradox, only three subsystems are present so that the entanglements would be shared by two pairs: one pair is the black hole and the b ω modes, while the other one is the b ω andb ω modes, violating the monogamy of entanglement. In our model, the entanglement between the black hole and the b ω modes is replaced by a new generated one between the black hole and the detector. When this new entanglement is being generated, the old one implicit in |0 M may change into other forms, for example another vacuum |0 M −ω . The analysis of these changes about the vacuum |0 M will be shown in section 4.2.
This generation of entanglement through an old one is different from the "transfer of entanglement" in reference [7], where the author says that as time evolves entanglements shift from the near horizon region to the evaporation products of the black hole. That is, the entanglements between the b ω andb ω modes are all transferred into the whole exterior radiation space, leading to a sort of "conservation of entanglement" [6]. This is impossible by noting that the energy of the radiations should not come from the near horizon region, but from the black hole itself. It is believed the expectation value of the energy-momentum tensor 0 M |T µν |0 M gives the energy extracted from the black hole, via the semiclassical Einstein's equation. But this is not the case. From the Hawking's evolution implied by (3.5), one can see that the initial average energy of the scalar field 0 M |H a |0 M is exactly 0, while the final one 0 M |H b,b |0 M is of the order M −2 . This energy difference should not be treated as the evaporated energy, because a contribution from the interiorb ω modes is also included. Nevertheless, it can be regarded to be the work done by the black hole on the scalar field due to the causal structure, just like the case in which gravitational potential energy is transported into the matter via the work done by the gravity. For large M , this effect is weak, since the range of the event horizon at r = 2M can become so large that even the whole space-time could be contained in the black hole interior when M → ∞. But for small M , that energy difference will become divergent, implying the need of a quantum gravity theory 12 .
In our model, however, the energy transfer between the black hole and the detector is accomplished via the quantum transitions given by (2.26). We can still divide the radiations into an early and a late part, as a whole, these two parts should be entangled. The early part has already been transferred into the detector, while the late part is still stored in the black hole. Since the detector is entangled with the black hole via the new generated entanglement, the early and late parts of the radiations are thus entangled, too. This can be simply expressed as where the late radiation state |ω 2 can be treated as a part of the black hole's state before it is transferred into the detector. That is, the correlations among the radiations are provided by those within the states of the black hole, together with the nonlocal correlations between the black hole and the detector. In this sense, the physical information will not be lost, but mainly stored in the correlations between the black hole and the detector. When the black hole evaporates completely, the information can still be stored in the correlations among the detector, a weak gravitational field and the scalar field.
In conclusion, the physically acquirable information is not stored in the entanglement implicit in the vacuum |0 M , but in a new entanglement between the black hole and the detector generated via the one implicit in |0 M . Thus without any local measurement, this information can not be lost for ever. Moreover, the new generated entanglement does not violate the monogamy of entanglement, thus no AMPS's firewall could emerge.

Quantum Decoherence Due to Measurement on The Detector
As shown above, the evolution of the entire system is unitary, as long as there are no local measurements. Under this circumstance, the information won't be lost. In other words, if there is any information loss, it must be attributed to some non-unitary local measurement, for instance, the measurement (2.32) on the detector space or Hawking's measurement (3.15) on the exterior modes. These two measurements are both local in the sense that they are performed only on some component of a bigger system. These partial measurements can be described well by the so called positive operator-valued measure (POVM) [9] approach, leading to a super-operator evolution for the measured component, like the one given by (2.33). Since any measurement outcome must be classical, the original quantum coherence among the components of the bigger system disappears after the measurement, i.e. the so called quantum decoherence. The measurement (3.15) used by Hawking has been argued to be not physically practicable in the last subsection, so we consider the measurement (2.32) on the detector in this subsection. And we shall show the information loss due to quantum decoherence during the black hole evaporation.
As shown previously, the black hole and the detector are effectively entangled, so we can consider a black hole-detector system by performing a partial trace over the scalar field space. The coupling between them can be approximated by an linear operator 13 where the lowest order term in (2.26) has been used, and the coupling constant is given by Except for the non-unitary, this coupling resembles the amplitude-damping channel [9], a schematic model of the decay of an excited state of an atom due to spontaneous emission of photons. Assuming that the initial state is ρ B (0) ⊗ |0 D 0|, and that there is no extra matter absorbed by the black hole, the evolution of ρ B in the interaction picture can be described by a first order Lindblad's equation [9] ρ We can further make an operator factorization as V ω = V M,ω c, with c an annihilator-like operator acting on the states of the black hole. Then following [9], we have which is the familiar exponential law for the decay, with a decay rate ω |λ ω V M,ω | 2 . That is, in the view of a black hole alone, the evaporation is like a non-unitary decay process. This can be explained by quantum decoherence. Actually, to obtain the Lindblad's equation (3.27), we have performed a partial trace over the detector spacė with the first order Kraus representation operator given by 14 (3.30) 13 Since we have performed a partial trace over the scalar field space, the operator LB,D can not be unitary, thus leading to some information loss. But the lost information is all about the scalar field which is not the relevant physical information, as analyzed in the last subsection. 14 The general Kraus representation of the super-operator evolution is $(ρB) = µ MµρBM † µ , where the operator Mµ = ψ,D µ|U B,ψ,D |0M ψ |0 D with the |µ ψ,D denoted as the state of both the scalar field and the detector.
Roughly speaking, during the black hole evaporation, we keep performing partial measurements on the detector to determine the back-reactions on the black hole approximatively, i.e. the Markovian approximation [9]. As a consequence, the information will be lost due to quantum decoherence. Now, let's see whether the information is completely lost, in the sense of Hawking's completely thermal spectrum [10] In our model, the correlation is given in (2.31). By using of the approximation in (3.25), the general state of the black hole-detector system can be expressed as (3.32) Then the reduced density operator ρ D in (2.33) will be expressed as i.e. a statistical ensemble. After some calculations, we then get because of the probability relation P (ω 1 , ω 2 ) = P (ω 1 )P (ω 2 ). This implies that the radiations in the detector are not completely independent, instead some correlations or entanglements are remaining. In this sense, the information is not completely lost. This can simply be explained by noting that the entanglements used here are those between the black hole and the detector, not those between the b ω andb ω modes. Therefore, by performing some local non-unitary quantum measurements, we will obtain mixed states inevitably because of quantum decoherence. As a result, the information must be lost, but only partly, giving a non-thermal property of the radiations.

Extensions Including Excited States of The Scalar Field
In the space H 0 given by (2.8), the scalar field is assumed to be always in its vacuum state depending on some particular black hole's mass. In this subsection, we shall make some extensions to include excited states. But first, let's consider another extension about the dependence on the black hole's mass, which has been ignored in the previous analysis. For the vacuum state |0 M in (2.10), in addition to the explicit dependent factor e −4πM ω , the creators b † ω andb † ω can also depend on the black hole's mass, as indicated by the sequence (3.8). In the calculations of the factors 0 M −ω |b ω |0 M in (2.21) and N (M, ω) in (2.25), the mass dependence of the creators and annihilators has been neglected, by assuming these operators belong to one Fock space with some fixed black hole's mass. Then one may ask whether the results would become very different, with the mass dependence of the creators and annihilators included.
As an example, let's consider a general factor 0 M 1 |b 0 with the indexes 0 and 1 denoted as the mass dependence of M 0 and M 1 respectively. The annihilators a 0 ω and a 1 ω are related by a Bogoliubov transformation, and there is also a relation between the two vacua Then we have where another Bogoliubov transformationb 0 ∼ αa 0 + β(a 0 ) † has been used. From (3.37) we see that, as long as the initial state is some vacuum, particles of both the two modes must be created or annihilated in pairs (generally in even numbers), leaving the vacuum almost unaltered. This is also the reason for the non-vanishing of the factor N (M, ω) which contains annihilators of both the two modes. By including the mass dependence of the creators and annihilators, it may be extended to a non-diagonal factor N (M, ω, ω ), which leads to a general correlation ωω f ωω |M − ω B |E + ω D between the black hole and the detector. Now, let's consider the excited states of the scalar field. The previous discussions only demonstrate the possibility of vacua for the scalar field during the evolution. As shown in section 3.1, however, the space H 0 is not enough for a full unitary evolution, that is, excited states can be obtained during the evolution. In the evolution operator (3.9), the completeness relations in terms of b ω andb ω modes instead of a ω modes, should be inserted. It thus seems that the evolution may be described in a direct product Hilbert space H b ⊗ Hb, with the vacuum |0,0 . Then the creation and annihilation of particles of the b ω andb ω modes may be un-related, behaving like two independent dynamical systems. However, given a single system with an initial state |0 M , it should be impossible to obtain two independent subsystems under a unitary evolution. This can be roughly explained as follows. With the interactions of our model being turned on, one would obtain some states which can be expressed as, with finite creators and annihilators acting on the initial vacuum. They can be derived from terms like (a † ) l |0 M , with a relation l = m + n + p + q due to the Bogoliubov transformations. This relation should be preserved under the unitary evolution, meaning that the completeness relation for the Hilbert space H a of the a ω modes can still be used. When expressed in terms of the b ω ,b ω modes, it is identified with the completeness relation for the direct product space H b ⊗ Hb. If further l = 2k, i.e. an even number, then from (3.36), the scalar field can be in some vacuum state, but only with a small probability for large M , indicated by the estimate in (2.29) 15 . Moreover, the relation (3.36) implies that the creators are required to be combined in some coherent manner to obtain another vacuum |0 M 1 from an initial one |0 M 0 , which can not be achieved by using of only a perturbation method. Thus, in terms of only perturbations, states excited from some chosen vacuum could be obtained as long as the radiations have not been absorbed by some other systems.
For convenient of calculation, except the initial vacuum |0 M , in each subsequent stationary region, the states of the scalar field can be assumed to be expressed as 16 |φ = α|0,0 + β|1,1 + · · · + γ|n,ñ + · · · , (3.38) with all the possible ω labels neglected for simplicity. As a coherence state, the vacuum is just a particular one of them. Instead of (2.17), another approximate completeness relation can be utilized for the scalar field space Then some transition rates can be calculated, where the mass dependence of the creators and annihilators can still be neglected for simplicity. For example, the transition rate for |0 M to |0,0 is given by where we have used an approximation This gives a first order black hole mass decreasing rate where c = |g ω V M,ω | 2 is assumed to be a model dependent constant approximatively. L l=0 is the l = 0 luminosity (without backscattering effects), a little smaller than the result (768πM 2 ) −1 [10] based on the Hawking's arguments. The transition rate for |0 M to all of the first excited states |ω 0 ,ω 0 is given by reduced by a factor (8π 2 M 2 ) −1 comparing with |λ 0 ω V M,ω | 2 . This means higher order transitions contribute little as long as the black hole mass is large enough 17 . 16 It should be stressed that the state (3.38) is not proper, especially when there are only finite terms in the expression. Hence the transition rates calculated in (3.40) and (3.43) are not physically meaningful. 17 The transition amplitude can be approximated by 0M |bωbω|0M for small energy transfer, but this term will lead to a divergent l = 0 luminosity (without backscattering effects) − ln 128π 3 M 2 , with → 0. However, these results are not exact and should not be treated seriously, since all of them depend on the model. The analysis above implies that, during the evolution of our model, the scalar field can be excited to higher levels. When the emitted energy of the black hole is still stored in the scalar field and not transferred into the detector, the scalar field must be excited. However, when the energy almost goes to the infinite future and is absorbed by the detector, the scalar field can be in some vacuum state. Although the scalar field vacuum is not necessary during the evolution, it indeed provides the necessary entanglements to generate the new correlations between the black hole and the detector. Then can we destroy the entanglements implicit in the vacuum |0 M by a Hawking's measurement (3.15)? If this was possible, then |0 M would collapse into a state like |α |β , which behaves like two independent systems described alone by b ω orb ω modes respectively. As a consequence, even the space-time may be broken into two disconnected regions along the event horizon. In our model, however, the measurement is performed on the space of the detector, then after the measurement it is the detector's state collapses, so does the state of the black hole due to the entanglements between the black hole and the detector. These have little influences on the vacuum |0 M , without destroying the entanglements implicit in it.

Graviton as An Intermediate Medium For Energy Transfer
The model studied in section 2.2 provides a nonlocal correlation between the causally disconnected interior and exterior of the black hole. According to local QFT, the interactions H B,ψ and H ψ,D in (2.16) and (2.22) should decouple from each other. However, because of the entanglements between the b ω andb ω modes implicit in the vacuum state |0 M , it is possible to combine the two interaction terms in a way like (2.23) to obtain a nonvanishing transition (2.26), leading to correlations between those two disconnected regions. In this subsection, we shall make some further investigations in the framework of the curved space QFT, where unlike the traditional semiclassical treatment, the gravity is treated by including the gravitons or the perturbations of the black hole background.
For a flat space-time background, a useful expansion is given by g µν η µν + h µν , with the perturbation h µν treated as the quantum gravitational field or graviton. Here, we make an analogous expansion based on some black hole background with g B µν the classical metric of a black hole with mass M 0 , and the perturbation h µν treated still as the quantum gravitational field based on this black hole. Analogous to the mode expansion (2.18) for the scalar field, we have 18 with µν the polarizations of the gravitons. The full interaction is given by which occurs in both of the interior and exterior of the black hole according to the extended BHC (ii'). For simplicity, the interactions will be studied only in the momentum space labeled by the energy ω, with all the other items such as the polarizations being neglected. According to local QFT, the terms in (4.3) will contain the following interaction patterns with the combinations of the mode functions such as u * ∂u∂u being ignored for simplicity. Let's then evolve the entire system, and a Hartree-Fock-like method should be used. For an initial state |i g of the graviton, we can evolve it and obtain a final state |f g via the evolution operator for the interaction in (4.3), where the fields are expanded based on an initial background metric g B µν as in (2.18) and (4.2). Then the new background metric is given by g B µν + g f |h µν |f g , and the procedure continues. Let's first consider the black hole formation. The initial state |i g of the graviton must be chosen to satisfy g i|h µν |i g 0, so that the black hole is initially stationary. To form a larger black hole with mass M 0 + ∆M = M , the scalar field must be in its excited state initially, for example 19 Because of the Bogoliubov transformations, this state can be expressed as a superposition of various states, among which two particular ones are After the formation of the larger black hole, the state of the scalar field can be chosen to be the vacuum |0 M ψ . That is, the energy ∆M stored in ψ φ|T µν [ψ b,b ]|φ ψ have been almost transferred into the black hole (interior), keeping the energy balance 20 The graviton's final state can be chosen as, for example (according to BHC (iii)) since the energy of the black hole must be stored almost in the interior degrees of freedom. Then there would be some transitions due to the interactions in (4.3) Here ω a † ω ≡ a † ω 1 a † ω 2 · · · , ω ≡ ω1 + ω2 + · · · , similarly for other equations below, such as (4.6), (4.8) and (4.10), etc. 20 Here the first order Einstein's equation has been used, since Rµν [g B ] = 0. The energy balance is a general condition satisfied by two components of a closed system.
with energies ∆M 1 and ∆M 2 for |f 1 g and |f 2 g , respectively.
The transition 2 in (4.9) can simply be induced by an interactionc † ωbω 1b ω 2 in (4.4), which is local in the interior of the black hole. While for the transition 1, it seems that an interactionc † ω b ω 1 b ω 2 can do the job, which is impossible since it contains operators from causally disconnected regions. But following the effective field model in section 2.2, the transition 1 can be induced by some combinations of some local interaction terms 21 That is, the energy of the scalar field is first transferred into the exterior gravitational field via some local interactions, then into the interior gravitational field through the entanglements implicit in |0 M 0 ψ . These can be verified by noting the following actions where the creators and annihilators of the same mode has been considered to be approximately cancelled 22 , while those of different modes are in pairs so that the entanglements between the b ω andb ω modes are still retained. There are two interaction terms which do not appear in the ordinary patterns (4.4) In the framework of flat space QFT, these two terms are related to a factor δ(ω 1 + ω 2 + ω), and thus should vanish. In the black hole physics, however, the whole space-time is divided into an interior and an exterior regions by the event horizon, with each one incomplete. As a consequence, the integral intervals of the space-time integrals in each region are also incomplete, leading to non-vanishing results. Therefore, with the help of the combinations in (4.10), the transition 1 in (4.9) can be induced step by step. The other states in the superposition of |φ ψ can be treated in an analogous way. These transitions can be illustrated in the following diagram We have neglected the mass dependence of these operators for simplicity. 22 Since b,b can not annihilate |0M 0 ψ , thus bωb † ω |0M 0 ψ = |0M 0 ψ + b † ω bω|0M 0 ψ . The first term |0M 0 ψ is already in (4.11a) and (4.11b), while the second term can be considered to be from another procedure involving a state |φ 1 ψ ∼ ω 1 bω 1 |0M 0 ψ , which can also be derived from the general state in (4.5).
with E(b,b) denoted as the entanglements implicit in the vacuum |0 M 0 ψ . The transferred energy flows along the arrows. First, it is stored in the scalar field as matter's energy. For the energy stored in theb ω modes, it can be transferred directly into thec ω modes as the black hole's mass via the local interaction (bbc † + h.c.). While for the energy stored in the b ω modes, it firstly has to be transferred into the c ω modes via the local interaction (bbc † + h.c.), then into thec ω modes as the black hole's mass via the nonlocal correlation (c †c + h.c.) that is generated by E(b,b). This generated nonlocal correlation is just like the one discussed in section 3.2, by notingc † ω c ω ( ψ 0 M 0 |b ω 1b ω 2 b ω 1 b ω 2 |0 M 0 ψ ) from (4.10). The notation (×) in the above diagram is used to indicate the impossibility to transfer energy in the corresponding direction 23 , similarly for another diagram below.
For the black hole evaporation, we can reverse the above formation process, with the initial and final states given by |0 M ψ |f g and |φ ψ |i g , respectively 24 . The required transitions can be induced by means of an local interactionc ωb † ω 1b † ω 2 and the following combinations of local interactions Then the energy stored in the interior of the black hole can be transferred back into the scalar field, and the scalar field will be excited. In order for the black hole to evaporate further, the energy stored in the scalar field must be transferred into somewhere else so that the entanglements E(b,b) can still be utilized. The distance detector in the model of section 2.2 just does this job. These can be illustrated in an analogous diagram where we have reversed the direction for the energy flowing between theb ω andc ω modes so that the energy can almost be transferred into the distant detector. The role played by the gravitons as an intermediate medium is thus demonstrated in the above diagram, or more apparently in the following expression (4.14) 23 In fact, there is an analogous way to transfer energy between thebω and bω modes via the entanglements between thecω and cω modes of the graviton's states. However, this seems to be impossible here since we have to make an average h to give a classical background in each stage. 24 This procedure for the evaporation, by directly reversing the previous formation process, is a little like the stimulated radiation, because the state of the gravitons has been excited after the previous formation.
The grouped contribution from the scalar and gravitational field serves as the propagators in the exterior region, while the terms in the two ends just give the model in section 2.2 with an identificationc ω → V ω . Roughly speaking, the energy (or information) is not transferred instantaneously from the interior of the black hole into the distant detector, but transferred step by step via the intermediate gravitational and scalar field. Therefore, the correlations between the black hole and the detector are physically practicable. As discussed in section 2.1, in the (classical) black hole physics, there is a contradiction between the descriptions of a static observer and an in-falling observer because of the metric singularity at r = 2M . In a quantum version, as illustrated in the above two diagrams, it is impossible for an exterior or interior particle to cross the event horizon directly, since the exterior particle is described by the b ω modes, while the interior one is described in terms of theb ω modes. This causally disconnectedness agrees with the classical one in the view of a static observer, since the chosen reference frame is still given by (t, r, θ, φ). For an in-falling observer, the chosen reference frame is regular so that there is no mode split. Then the physical world in his eyes can be described well according to flat space QFT. The scalar field and the gravitational perturbation are expanded in terms of the a ω and e ω modes respectively, in particular, the vacuum for the scalar field is |0 M . The full interaction is given by the term (2.38), which can induce various quantum transitions, including those induced by nonlocal correlations when expressed in terms of the b ω andb ω modes. For example, by means of the Bogoliubov transformations, an interaction pattern e † ω a ω 1 a ω 2 can be expressed as where the term c † ω b ω 1 b ω 2 is an local interaction in the exterior region, while the other onẽ c † ω b ω 1 b ω 2 is a nonlocal one which is required to induce the transition 1 in (4.9).
For a static observer, however, the termc † ω b ω 1 b ω 2 in (4.15) can not be constructed directly since it contains modes from disconnected regions, violating the causality. But the quantum transitions are physical so that they should not depend on the observers. These quantum transitions, in the view of a static observer, are given by (4.10) and (4.13) via the entanglements implicit in the vacuum |0 M . That is, they are induced by some nonlocal correlations between the interior and exterior degrees of freedom of the black hole, leading to some quantum tunneling effects across the event horizon. Thus, the energy or information can be tunneled across the event horizon gradually through those quantum transitions. Consequently, the static exterior or interior observer can acquire the information stored in the other region, as long as the entanglements implicit in |0 M ψ will never be destroyed. The remaining problem is whether the different descriptions are consistent with each other, one is based on the static observer, the other is based on the in-falling observer. For instance, on one hand,c † ω b ω 1 b ω 2 , as one part of the full interaction in (4.15), can not be constructed directly for a static observer; on the other hand, terms like e ω a ω 1 a ω 2 should be vanishing for an in-falling observer, while terms like c ω b ω 1 b ω 2 orc ωbω 1b ω 2 can be constructed for a static observer. This consistency problem has been answered in section 2.1 only in principle by means of the principle of general covariance, without a detailed inspection.

Three Classes of Entanglements or Correlations
Up to now, we have obtained three kinds of entanglements: (1) Entanglements that are always implicit in some steady pure state, such as the vacuum state |0 M ; (2) Entanglements that can be established through some ordinary local interactions, like those (bbc † + h.c.), (bbc † + h.c.) and (bd † + h.c.); (3) Entanglements which are nonlocal in the sense that they are established between two causally disjoint regions, by means of the entanglements of class (1), like (c †c + h.c.) and (d † V + h.c.).
From (3.23) and the discussions below, one can see that the entanglements of class (1) have some intrinsical properties: they can not be observed physically by local measurements, so the information stored in them can not be acquired; they should not be destroyed by local measurements, and the information will not be lost; they can not be transferred into other systems, but may be changed into other forms. While the entanglements in the other two classes can be destroyed by local measurements, and the information stored within them will be lost due to quantum decoherence. These agree with the discussions in section 3.2 about the distinctions between the generation of entanglements in our model and the transfer of entanglements in reference [7]. The author of [7] tried to transfer the entanglements of class (1) into the radiations so that they can be observed physically. This is impossible according to the intrinsical properties of class (1). In our model, the physical information is stored in the entanglements of class (3) between the causally disconnected interior and exterior of the black hole. According to the properties of class (3), that information can be acquired by local measurements and be lost due to quantum decoherence.
Let's now consider the changes of the vacuum |0 M during the evolution. In section 2.2, a restrict space H 0 is assigned, in which the scalar field are assumed to be in the vacua depending on different black hole's masses. As the entire system evolves, a sequence of those vacua (3.6c) is assigned to describe the changes of the vacuum during the evolution. It seems to be impossible to decide which vacuum has more entanglements, since each vacuum can be expressed by (2.10) in terms of their respective b ω ,b ω modes, i.e. maximum entangled in each Fock space. If |0 M 0 is a maximum entangled state, then |0 M 1 is away from maximum by including excited states from the relation (3.36). But this realtion can also be rewritten in a reversed order so that |0 M 1 is maximum entangled while |0 M 0 is away from maximum. One may say that their entanglements can be compared by calculating the Von Neumann entropy for each vacuum. This should be forbidden, since local measurements like 0 M |O ext |0 M will be involved, which are non-physical according to the discussions below (3.23). In fact, those vacua are just meta-stable states, and should be treated as some auxiliary states. For a full evolution, free states in the real t → ±∞ regions should be utilized, since the causal structures of the space-time there should be regular without black holes so that no mode split will occur 25 . In this sense, it's meaningless to count the amount of entanglements implicit in those vacua.
For local interactions in terms of a perturbation expansion, we can obtain states with finite creators and annihilators acting on the vacuum, which are still entangled states, for example a state like (b) m (b † ) n (b) p (b † ) q |0 M . That is, the entanglements between the b ω and b ω modes are still remaining, since |0 M can not be destroyed by those actions composing of only finite creators and annihilators. This is not the case for a global action, for example the inverse of the operator e ω e −4πM ω b † ωb † ω in (2.10) that can be used to act on |0 M to obtain another vacuum |0,0 , i.e. a direct product state. Fortunately, this kind of operators with global properties seldom occurs in local effective field theory. This can also be explained by noting that the local interactions H B,ψ and H ψ,D are independent so that there exist no prior correlations for them to produce an operator with global properties. Therefore, without destroying the entanglements implicit in the vacuum |0 M , the quantum transitions given in sections 2.2 and 4.1 can still continue, leading to the black hole evaporation. It thus implies that the space H 0 should be extended to a more general one to include all the possible entangled states of the scalar field, generated by those operations composing of finite creators and annihilators acting on each vacuum state |0 M , analogous to the case of the four Bell states. This can be explicitly seen in a qubit model proposed in appendix B, where all the four Bell states appear during the unitary evolution.

Summary
Some essential points for our model of the black hole evaporation are summarized as follows: (1) Black hole complementarity (BHC) can almost be satisfied, but with an extended postulate (ii'): both the exterior and interior regions of the event horizon are well described by QFT in curved space, with the singularity r = 0 excluded from the interior region. (3) In the case of black hole evaporation, a Hawking-like measurement 0 M |O ext |0 M should be forbidden, for it is not physically practicable. Instead, a physically practicable quantum measurement can be operated on an added detector that receives radiations. In this way, a non-thermal spectrum of the radiations will be obtained, instead of Hawking's thermal one. (Sections 2.2.2, 3.2 and 3.3) (4) Information will be lost when some local measurements have been performed. This is caused by quantum decoherence that destroys the entanglements among the components of a closed system. Hence the black hole evaporation may be treated as a decay process approximately. (Section 3.3) (5) In the framework of curved space QFT, a unitary evolution including both the black hole formation and evaporation can be constructed formally, although the constructed evolution operators have singularities. Those singularities can not be easily avoided, indicating that a quantum gravity theory is still needed. (Section 3.1 and Appendix A) Here, we give a qubit model which behaves in a similar manner as the above essential points. Suppose that Alice and Bob own an EPR pairs, Alice takes one qubit denoted by a, while Bob carries the other one named by b. Now there is a task for them to establish correlations between two independent systems A and B, which are far away from each other. Usually, for two systems to correlate with each other, one direct method is to couple them via some local interactions, for example an EPR pairs can be produced in this way. However, if the systems A and B can not be moved to close to each other, or even they may be located in two causally disconnected regions, for instance the interior and exterior of a black hole, then how to correlate them? The EPR pairs owned by Alice and Bob can be applied to accomplish this task in the following way. Let Alice and Bob travel to the locations of A and B respectively, together with their own qubits of the EPR pairs. After arriving, let them carry some local unitary operations U aA and U bB on the combined systems aA and bB respectively, then A and B will be correlated with each other by carefully controlling the operations. The mathematical detail of this model is given in appendix B. Obviously, the systems A and B can be regarded as the black hole and the detector respectively, while the EPR pairs serves as the vacuum state |0 M .
We then conclude that, the black hole formation and evaporation processes can be modelled in a unitary manner according to effective field theory, so that the paradoxes of the information loss and the firewall can be resolved. A more general case including the charges and angular momenta has to be further investigated. Since our model is unitary, and a non-thermal spectrum is also obtained indicated by (3.34), it thus seems that there may not be a thermodynamic character for the pure gravity. Certainly, this needs to be investigated further. In conclusion, although our model is just an approximation, it indeed implies that quantum mechanics and gravity can be combined in a consistent way.

A A Simple Model With a Singular Evolution Operator
A simple model with a singular evolution operator is studied in detail here. The action is that for a harmonic oscillator with a prescribed, time-dependent spring "constant" [13] where The function ω 2 (t) becomes constant in the remote past and the remote future By solving the equations of motion, the in and out mode functions can be obtained [13]. As in curved space QFT, these two modes are related by a Bogoliubov transformation. From (A.3),the free Hamiltonians in the t → ±∞ limits are with the eigenstates given by The "out"(+) and "in"(−) states are defined as which satisfy the condition [14] exp(−iHt) Then the "out" and "in" states can be expressed as where The S-matrix is thus given by 10) or equivalently the S-operator is where the evolution operator is defined as Differentiating U +− (t 2 , t 1 ) with respect to t 2 and t 1 respectively, we have Since H + = H − , then V + (t) = V − (t) generally, meaning that the first differential of U +− (t 2 , t 1 ) is discontinuous. This can also bee seen from a singular initial condition indicating that U +− (t 2 , t 1 ) works well only if t 2 = t 1 . U +− (t 2 , t 1 ) occurs in the two-point function 0 + |q(t)q(t )|0 − , while for 0 + |q(t)q(t )|0 + and 0 − |q(t)q(t )|0 − , the evolution operators U ++ and U −− are Easily to see, U ++ (t 2 , t 1 ) is ill-defined in the neighborhood of −∞, so is U −− (t 2 , t 1 ) in the neighborhood of +∞. Thus, we have three classes of evolution operators that cover the whole parameter space [−∞, +∞], and each class has its own singularity. The S-operator in (A.11) can be constructed as where U ±± are in their well-defined domains respectively, but the singular U +− (t, t) occurs inevitably. To avoid this singularity, another singular operator U −+ (t, t) should be included, so that they cancel with each other Notice that it is the asymptotic condition H We consider only the time translation. According to QFT [14], a theory is invariant under time translation, it means the same operator e −iHt 0 acts on both the "out" and "in" states It's convenient to work with the S-operator formula. For an ordinary asymptotic condition H t→±∞ −→ H 0 , an operator e −iH 0 t 0 can be applied to act on the free state space where |φ and |χ are two arbitrary states in the free space determined by H 0 . Thus (A.19) will hold if [14] e +iH 0 t 0 Se −iH 0 t 0 = S . However, all these don't hold for S +− or U +− (+∞, −∞). In the two stationary regions t → ±∞, the harmonic oscillator are free, and can be quantized as follows with the Bogoliubov transformations [10] These relations can lead to some unusual effects, for example, the particle creation The two vacua are related as [10] Consider the S-matrix element 0 + |S +− |0 − . A time translation operator e −iH + t 0 will act on both the future and past Hilbert spaces because of the relation (A.27). Following the same steps from (A.20) to (A.23), we will obtain the condition For Ω + (+∞), this condition is satisfied, since H In fact, the model (A.1) is only semiclassical due to a classical potential term ω 2 (t)q 2 (t), with ω 2 (t) treated as an external classical field satisfying some classical equations of motion. Besides, this potential breaks the classical symmetries of the time, the reason for the quantum symmetry breaking.
The above broken symmetry may be recovered in the following way. The condition e +iH + t 0 S +− e −iH − t 0 = S +− can be satisfied, if we enlarge the free space by combining H + with H − constrained by a relation [H + , H − ] = 0. This can be achieved by replacing the parameter t in ω 2 (t) with another independent one µ, i.e. ω 2 (µ). There will thus be a collection of oscillator states parameterized by µ, with the two in (A.24) corresponding to the limits µ → ±∞. Then instead of (A.25), we have [a + , a − ] = 0 . (A.29) The whole collection of states in the free space will then becomes {a † µ , a µ , |0 }(ω µ ) . (A.30) In this case, [a µ 1 , a µ 2 ] = 0, and only a single vacuum |0 is needed, since the Hilbert space of the harmonic oscillator has been enlarged, with the free Hamiltonian H 0 = dµH µ , and an ordinary asymptotic condition H t→±∞ −→ H 0 .
From the above analysis, we can conclude that, for a semiclassical system such as the model (A.1), a formal S-matrix or evolution operator can be constructed, implying that the evolution of the system is unitary. But the evolution operator has some singularities which can not be easily avoided. Moreover, the formal S-matrix may lose some meaningful symmetries, for example the time translation symmetry, so it can not be used to construct well-defined physical quantities for describing the system. This conclusion can be extended to curved space QFT, where the S-matrix may lose some important Lorentz invariance. Certainly, a full and nonsingular description may be given by a quantum gravity theory.

B A Qubit Model of The Black Hole Evaporation
In section 5, a qubit model of the black hole evaporation is proposed, in which an EPR pairs can be utilized to correlate two distant systems without direct couplings. In this appendix, we give some mathematical details of this model. To resemble the black hole evaporation, the EPR pairs are chosen to be in the Bell state |β 00 ab ≡ 2 −1/2 (|00 ab + |11 ab ) . (B.1) The initial state of the systems A and B are chosen to be |1 A |0 B , meaning that A is excited while B is in the ground state. With A regarded as the black hole and B as the detector, a state |0 A |1 B is expected to occur during the evolution, as if the energy of the black hole was transferred into the detector. But this analogy should not be taken seriously, since no energy is transferred in this qubit model, as will be shown below. The unitary operation U is the one that generates the Bell states, realized by a Hadamard gate and a subsequent controlled-NOT (or CNOT) gate. The Bell states are generated as follows [15]  The initial state of the entire system abAB is then |β 00 ab |1 A |0 B . We evolve it via a combined unitary operation U aA U bB following the actions in (B.2). The first action gives which will lead to the initial state |β 00 ab |1 A |0 B , by using of (B.4) and (B.5). Therefore, we obtain a cyclic procedure expressed as where the second evolution can be regarded as the formation process of a black hole. Thus we prove the BHC (i) in a not rigorous way.
Notice that this qubit model is completely different from the one in reference [16], where the author proved a theorem saying that the formation and evaporation of a black hole will lead to mixed states or remnants. The proof was based on an argument stating that the vacuum state |0 M is stable during the evaporation, in the sense that the entanglements implicit in it will not be changed. In our model, however, this condition is relaxed by stressing that the entanglements should not be destroyed, but can be changed into other forms as discussed in section 4.2. This can also be seen in the above qubit model, where the initial |β 00 ab can be changed into other three Bell states without destroying the entanglements. Moreover, any measurement performed on the qubits of the EPR pairs should be forbidden, or else the evolution (B.8) will be destroyed and the established correlations between A and B will be lost, just like the model in section 2.2.
Let's now consider the issues about the relativity violation, that is, whether there is energy or information transfer between A and B in the following way i.e. transitions between |0 A |0 B and |1 A |1 B . Since the eight-action procedure give a cycle, then the best choice to realize (B.9) is the four-action. But the result is without any information transfer. This is a general result that prevents the above qubit model from transferring information faster than light, thus preserving the relativity. For the qubit model, a cyclic property is obtained in (B.8) and (B.10), in which the initial state of the EPR pairs |β 00 ab is recovered for the four-action and eight-action. This is related to the group structure of the unitary operation as follows. By grouping the four computational basis states as a column vector (|00 , |01 , |10 , |11 ) T , the unitary operation given in (B.2) can then be rewritten in a matrix form with I 2×2 the unit matrix, and σ 1 the first Pauli matrix. It is easy to verify that the unitary matrix U given in (B.12) satisfies which give the cyclic property in (B.8) and (B.10). There are also some other unitary operations with different periods, for example both of which have a period of 2, not 8. Although U 1 and U 2 can not give the opposite terms as those in (B.8) and (B.10), they can still be utilized to correlate the systems A and B. For the real black hole evaporation, the vacuum state |0 M is variable during the evolution as analyzed in section 4.2, in particular it can be acted by some operations composing of finite creators and annihilators, as shown in section 3.4. As a consequence, there is not an intermediate stage corresponding to the four-action of the qubit model. But for a closed system, there must be an upper bound for the mass of the formed black hole, so the time for the formation and evaporation will be finite. In this sense, the black hole formation and evaporation for a closed system can also be regarded to be "cyclic", corresponding to the eight-action in the qubit model.