Soft-Hair-Enhanced Entanglement Beyond Page Curves in a Black-hole Evaporation Qubit Model

TL;DR

This paper introduces a qubit model simulating black hole evaporation that incorporates soft hair, revealing entanglement behaviors that challenge traditional Page curve predictions and suggest a resolution to the firewall paradox.

Contribution

It presents a novel qubit model including soft hair effects, demonstrating enhanced entanglement beyond the Page curve in black hole evaporation.

Findings

Entanglement entropy exceeds the black hole entropy analogue.

Early radiation is entangled with soft hair.

Late radiation remains highly entangled with black hole degrees of freedom.

Abstract

We propose a model with multiple qubits that reproduces the thermal properties of 4-dimensional (4-dim) Schwarzschild black holes (BHs) by simultaneously taking account of the emission of Hawking particles and the zero-energy soft hair evaporation at horizon. The results verify that the entanglement entropy between a qubit and other subsystems, including emitted radiation, is much larger than the BH entropy analogue of the qubit, as opposed to the Page curve prediction. Our result suggests that early Hawking radiation is entangled with soft hair, and that late Hawking radiation can be highly entangled with the degrees of freedom of BH, avoiding the emergence of a firewall at the horizon.

Equations

$$T_{\mathrm{rad}}=\frac{1}{8\pi G⟨H_N⟩},$$

$$S_{\mathrm{BH}}={\left(16\pi G{T}^2\right)}^{-1}.$$

$$H=\omega |+⟩⟨+|,$$

$$H_{\mathrm{fs}}$$

$$\times I_1\otimes \cdots I_{i-1}\otimes |s_i⟩⟨s_j|\otimes I_{i+1}\cdots$$

$$\otimes I_{j-1}\otimes |s_j⟩⟨s_i|\otimes I_{j+1}\otimes \cdots I_N,$$

$$H_R$$

$$+|+⟩⟨-|\int g^{\ast }(x){\Psi }_R(x)dx.$$

$$U_R|+⟩|\mathrm{vac}⟩=c_{+}(t)|+⟩|\mathrm{vac}⟩+|-⟩\left(\int \phi (x,t){\Psi }_R^{†}(x)dx\right)|\mathrm{vac}⟩,$$

$${\Gamma }_R\left[\rho \right]={\mathrm{Tr}}_{{\Psi }_R}\left[U_R\left(\rho \otimes |\mathrm{vac}⟩⟨\mathrm{vac}|\right)U_R^{†}\right],$$

$${\Gamma }_R\left[|+⟩⟨+|\right]=\left(1-r\right)|+⟩⟨+|+r|-⟩⟨-|,$$

$$H_S$$

$$+|-⟩⟨0|\int g^{\ast }(x){\Psi }_S(x)dx,$$

$${\Gamma }_S\left[\rho \right]={\mathrm{Tr}}_{{\Psi }_S}\left[U_S\left(\rho \otimes |\mathrm{vac}⟩⟨\mathrm{vac}|\right)U_S^{†}\right],$$

$$\Gamma \left[\rho \left(T(0)\right)\right]=\left(1-p\right)|0⟩⟨0|+p\rho \left(T\right),$$

$$p=1-s\left(1+r\exp \left(-\omega /T(0)\right)\right)/\left(1+\exp \left(-\omega /T(0)\right)\right),$$

$$T=\frac{\omega }{\ln \left[\frac{\exp \left(\omega /T(0)\right)+r}{1+r}\right]},$$

$$H_{tot}=\sum _{i=1}^N\left(H_i+H_{R_i}+H_{S_i}\right)+H_{\mathrm{fs}},$$

$$p(\tau )\approx \exp (-\tau ),$$

$$T(\tau )\approx T(0)\exp \tau ,$$

$$E(\tau )=p(\tau )\frac{\omega \exp \left(-\omega /T(\tau )\right)}{1+\exp \left(-\omega /T(\tau )\right)}\approx p(\tau )\frac{\omega }{2}.$$

$$⟨H_N⟩=NE={\left(8\pi GT\right)}^{-1},$$

$$S_{EE}=-\mathrm{Tr}\left[\begin{array}[]{c}\left(\left(1-p(\tau)\right)|0\rangle\langle 0|+p(\tau)\rho\left(T(\tau)\right)\right)\\ \times\ln\left(\left(1-p(\tau)\right)|0\rangle\langle 0|+p(\tau)\rho\left(T(\tau)\right)\right)\end{array}\right].$$

$$S_{\mathrm{BH}}/N$$

$$p{S}_{\mathrm{th}}$$

$$S_{\mathrm{EE}}$$

$$i\frac{d}{dt}{c}_{+}(t)=\int g^{\ast }(x)\phi (x,t)dx,$$

$$\left({\partial }_t+{\partial }_x\right)\phi (x,t)=-i{c}_{+}(t)g(x).$$

$${\partial }_{+}\phi =-\frac{i}{2}{c}_{+}\left(\frac{1}{2}\left(x^{+}-x^{-}\right)\right)g\left(\frac{1}{2}\left(x^{+}+x^{-}\right)\right).$$

$$\phi =-\frac{i}{2}{\int }_{x^{-}}^{x^{+}}{c}_{+}\left(\frac{1}{2}\left(x^{\prime }-x^{-}\right)\right)g\left(\frac{1}{2}\left(x^{\prime }+x^{-}\right)\right)d{x}^{\prime }.$$

$$i\frac{d}{dt}{c}_{+}(t)=-\frac{i}{2}\int dx{g}^{\ast }(x){\int }_{x-t}^{x+t}g\left(\frac{1}{2}\left(x^{\prime }+x-t\right)\right)c_{+}\left(\frac{1}{2}\left(x^{\prime }-x+t\right)\right)d{x}^{\prime }.$$

$$t^{\prime }=\frac{1}{2}\left(x^{\prime }-x+t\right),$$

$$\frac{d}{dt}{c}_{+}(t)=-{\int }_0^{t}K(t-t^{\prime })c_{+}(t^{\prime })d{t}^{\prime },$$

Full text

Introduction.—There has been a rapid increase in the importance of the entanglement entropy (EE) of macroscopic systems in quantum gravity and condensed matter physics. After the advent of the Ryu-Takayanagi formula RT , which shows that the EE in a $D$-dimensional ($D$-dim) conformal field theory is equal to the Bekenstein-Hawking entropy in a $(D+1)$-dim gravity theory with a negative cosmological constant, the EE sheds light on unexpected features of spacetimes. The formula indicates the interesting possibility that quantum information generates curved spacetime in a holographic way J . In condensed matter physics, EE plays the role of an exotic order parameter for topological insulators KP ; LW . Recently, the EE was directly measured in an ultracold bosonic atom experiment exp , which indicated that the EE time evolution of many-body systems may also be observed experimentally. In the theory , the first-principles calculation of EE evolution is very complicated for macroscopic systems, and it has not been achieved to date.

In discussions related to black hole (BH) physics, a famous conjecture of EE evolution, i.e., the Page curve, is often adopted. Hawking h showed that BHs evaporate by emitting thermal Hawking radiation. It is possible to assume a thought experiment in which the initial state is a pure state and evolves in a unitary way. AdS/CFT arguments support the unitarity of the process h2 . Page conjectured an EE evolution between an evaporating BH and its radiation page , and the evolution is called a Page curve. It is assumed that the EE is equal to the BH thermal entropy after the so-called Page time. Though the conjecture is based on analyses of general many-body systems without horizons, typicality arguments of statistical mechanics are expected to provide essentially the same behavior in various physical systems including BHs.

In this Letter, we argue a possibility that EE of a BH subsystem shared with other systems including radiation is much larger than its BH thermal entropy after the Page time, using a thermal system of decaying $N$ qubits. This implies the direct breakdown of Page’s conjecture. The key idea is to reproduce the Hawking temperature relation of 4-dim Schwartzschild BHs in the qubit system:

[TABLE]

where $T_{\mathrm{rad}}$ is the temperature of thermal radiation emitted by the system, $\left\langle H_{N}\right\rangle$ is the quantum expectation value of the system Hamiltonian, and $G$ is the gravitational constant. Natural units are adopted ($c=\hbar=k_{B}=1$). In the same way as in BH physics, we are able to introduce a Bekenstein-Hawking entropy $S_{\mathrm{BH}}$ by defining the system temperature $T$ equal to $T_{\mathrm{rad}}$, and integrating the first law; $dS_{\mathrm{BH}}=d\left\langle H_{N}\right\rangle/T=8\pi G\left\langle H_{N}\right\rangle d\left\langle H_{N}\right\rangle$. Then $S_{\mathrm{BH}}$ takes the same value of the area-law entropy given by $4\pi r^{2}/(4G)$ with radius $r=2G\left\langle H_{N}\right\rangle$:

[TABLE]

The EE and the thermal entropy are compared with $S_{\mathrm{BH}}$ in this model.

It is worth stressing that the relation in Eq. (1) yields a negative heat capacity, which cannot appear in ordinary systems. In order to incorporate this exotic aspect in our model, we consider a transition from a qubit in a zero-energy state $|-\rangle$ to an escaping zero-energy particle of a field $\Psi_{S}$ which mimics soft hair in BH physics. The BH soft-hair conjecture asserts that zero-energy degrees of freedom emerge at a BH horizon HSS HPS . The soft-hair microstates contribute to the BH entropy HSS HTY . Quantum information of infalling matter behind the horizon is (at least partially) stored in the soft hair by use of conserved Noether charges of would-be diffeomorphism HPS . While the horizon soft hairs evaporate, the quantum information may be transmitted into Bondi-Metzner-Sachs (BMS) soft hairs at future null infinity HPS HPS2 .  In our model, after emission of a zero-energy $\Psi_{S}$ particle, the original system settles down in the vacuum state $|0\rangle$ with zero energy, which stands for a lack of qubit. The sum of the number of surviving qubits and $\Psi_{S}$ particles is conserved. Thus the system in $|0\rangle$ never returns spontaneously into $|-\rangle$ or $|+\rangle$ without ingoing $\Psi_{S}$ particles. Since each qubit decay does not decrease the system energy, a smaller number of qubits in the system carries the same energy. After a fast scrambling for themalization of surviving qubits in a state of a sub-Hilbert space spanned by $|\pm\rangle$, the system temperature $T$ becomes higher. This indeed demonstrates the negative heat capacity behavior. A similar mechanism of negative heat capacity generation was proposed in a D0-brane matrix model hanada . There are earlier studies regarding the EE evolution of qubit models M ; G ; A ; however, the negative heat capacity was not discussed.

In our model, quantum states of a decaying qubit belong to a 3-dim Hilbert space spanned by $|\pm\rangle$ and $|0\rangle$, and thermalization occurs only in its 2-dim $|\pm\rangle$ sector. This enables the system to possess a larger EE than thermal entropy of the surviving qubit.

Page curve: Summary and problems.—For a pure state $|\Phi\rangle$ of a composite system $AB$, the bipartite entanglement between the subsystems $A$ and $B$ is quantified by EE: $S_{\mathrm{EE}}=-\mathrm{Tr}_{A}\left[\rho_{A}\ln{\rho_{A}}\right]$, where $\rho_{A}=\mathrm{Tr}_{B}\left[|\Phi\rangle\langle\Phi|\right]$ is the reduced state of subsystem $A$. The Page curve conjecture is based on a typicality theorem L ; Seth ; page . The theorem implies that the EE values between two finite macroscopic systems $A$ and $B$ in typical states of Hilbert space are very close to its maximum value when $D_{A}\ll D_{B}$, where $D_{A}$ and $D_{B}$ denote the dimensions of the sub-Hilbert space of $A$ and $B$, respectively. This state typicality is defined using the Haar measure in the Hilbert space. If we have a nontrivial Hamiltonian with a small interaction between $A$ and $B$, then the state becomes a Gibbs state $\rho_{A}\left(T\right)$ at a finite temperature $T$, with respect to the energy conservation of the total system due to the Sugita theorem sugita ; sugita2 . Thus, for typical pure states of an $AB$ system, the EE between $A$ and $B$ is very close to the thermal entropy of $A$, which is computed from $\rho_{A}\left(T\right)$. Let us consider $N$ qubits and put them on a line. At the initial time, all the qubits are assigned to $A$ components. The total system is in a typical pure state $|\Psi\rangle_{AB}$, which is randomly chosen from the Hilbert space respecting energy conservation. As depicted in Fig. 1, suppose a boundary separates the system into two parts, and moves from left to right.

In the Page curve conjecture, the decay process of $A\rightarrow B$ is described by assigning the left-hand-side qubits to $B$ components. Plotting the thermal entropy of the smaller subsystem at each time generates a Page curve. The conjecture asserts that this plot is the EE time evolution for the decay process of $A\rightarrow B$ in high precision. The time at which the thermal entropy of $A$ equates to the thermal entropy of $B$ is referred to as the Page time, which corresponds to $n=N/2$. After the Page time, the subsystems of $A$ are maximally entangled with $B$. Because of the entanglement monogamy, the entanglement among the subsystems vanishes in the conjecture.

First, it should be noted that the Page curves have no dependence on the decay dynamics. This appears unusual, since details of realistic decay channels can affect the EE evolution, and the conjecture is not capable of discriminating the following two EE evolutions. In the first case (i), fast scrambling interaction in the thermal equilibrium of $A$ components is switched off before the decay of $A$. After that, each $A$ component decays independently into a $B$ component, and no thermalization interaction appears during the process. In the second case (ii), fast scrambling to the equilibrium of $A$ components continues during the decay of $A\rightarrow B$. In BH physics, fast scrambling for an evaporating BH has been proposed fs and is supposed to justify the random selection of pure states of the BH and its radiation. However, the detailed properties remain elusive.

Another weak point of the conjecture is that this is based on many-body systems with positive heat capacity. Actually, we have no plausible arguments for negative heat capacity cases like BH evaporation so far.

Qubit model.—Let us introduce decaying $N$ qubits by simultaneously emitting zero-energy radiation of soft hair and energetic radiation corresponding to Hawking radiation. The free Hamiltonian of a decaying single qubit is given by

[TABLE]

where $\omega$ is a positive constant and $|+\rangle$, $|-\rangle$, and $|0\rangle$ are the eigenstates of the energy eigenvalues $\omega$, [math] and [math], respectively. We have a fast scrambling interaction $H_{\mathrm{fs}}$ for surviving qubits. The scrambling process is assumed to be much faster than the emission of particles. It occurs only among qubits in the $|\pm\rangle$ sector state, which preserves the total energy, and does not affect subsystems in $|0\rangle$. A simple two-body interaction example of $H_{\mathrm{fs}}$ is given by

[TABLE]

where $h\left(s_{i}s_{j}\right)$ are real coupling constants and $I_{i}$ is the unit matrix for the $i$th site subsystem. At $t=-t_{o}<0$, the $N$ qubits are set in a pure state, which is composed of $|\pm\rangle$, with the total energy expectation value fixed as $E_{i}$. By acting the scrambling operator $\exp(-it_{o}H_{\mathrm{fs}})$ onto the initial pure state, we get a typical pure state at $t=0$. In a typical pure state, each qubit is in a Gibbs state $\rho(T(0))$ at temperature $T(0)$. Throughout this Letter, $\rho(T)$ represents the Gibbs state of the two-level system at temperature $T$, i.e., $\rho(T)=\left(|-\rangle\langle-|+\exp{(-\omega/T)}|+\rangle\langle+|\right)/(1+\exp{(-\omega/T)})$. $T(0)$ is uniquely determined by $E_{i}$ via $T(0)=\omega/\ln\left(\frac{N\omega}{E_{i}}-1\right)$.

In our model, a Hawking particle with energy $\omega$ is emitted out of the qubit flipping $|+\rangle$ to $|-\rangle$. The particle escapes from the qubit along a real axis denoted by $x$ and propagates to $x=+\infty$. The state of one particle at some position $x$ is described using a 1-dim bosonic Schrödinger field $\Psi_{R}(x)$, as $\Psi_{R}^{{\dagger}}(x)|\mathrm{vac}\rangle$. Here, $|\mathrm{vac}\rangle$ is the vacuum state of the field that satisfies $\Psi_{R}(x)|\mathrm{vac}\rangle=0$, and the initial state of the field. The effective Hamiltonian that describes the emission is given by

[TABLE]

$g(x)$ is a localized function around the qubit and provides coupling between the qubit and $\Psi_{R}$. The unitary time-evolution operator is given by $U_{R}=\exp\left(-itH_{R}\right)$. The dynamics conserves the excitation number, $|+\rangle\langle+|+\int\Psi_{R}^{{\dagger}}(x)\Psi_{R}(x)dx$. Therefore, when a qubit is in $|+\rangle$, the composite system evolves into an entangled state, such that

[TABLE]

where $\varphi(x,t)$ is the wave function of the created particle. The survival probability of $|+\rangle$  is given by $\left|c_{+}(t)\right|^{2}$. This dynamics can be solved, as shown in Supplemental Material SM . A quantum channel for the qubit is introduced as

[TABLE]

and the evolution of the qubit in $|+\rangle\langle+|$ is given by

[TABLE]

where $r$ is the probability of finding $|-\rangle$ and is given by $r=1-\left|c_{+}(t)\right|^{2}$. The quantum channel satisfies $\Gamma_{R}\left[|0\rangle\langle 0|\right]=|0\rangle\langle 0|$ and $\Gamma_{R}\left[|-\rangle\langle-|\right]=|-\rangle\langle-|$ due to the conservation of the excitation number. Although the field quanta created by the transition trigger reexcitation from $|-\rangle$ into $|+\rangle$ while $\varphi(x,t)$ has nonzero overlap with $g(x)$, this is not essential. Indeed, the probability $r~{}$monotonically decreases with time due to the leakage of $\varphi$ out of the overlap region. The transition from $|-\rangle$ into $|+\rangle$ occurs for a qubit by the fast scrambling with other qubits.

Now let us introduce another channel for the decay of a zero-energy qubit into a zero-energy soft-hair particle. Suppose a similar Hamiltonian for the process:

[TABLE]

where $\Psi_{S}(x)$ mimics the soft hair and satisfies $\Psi_{S}(x)$ $|\mathrm{vac}\rangle=0$. We then obtain a quantum channel for the decay:

[TABLE]

where $U_{S}=\exp\left(-itH_{S}\right)$. This satisfies $\Gamma_{S}\left[|+\rangle\langle+|\right]=|+\rangle\langle+|$, while a qubit in $|-\rangle$ decays into $\Psi_{S}$ with probability $s$: $\Gamma_{S}\left[|-\rangle\langle-|\right]=(1-s)|-\rangle\langle-|+s|0\rangle\langle 0|$. Once the decay occurs, the qubit cannot come back by the next operation of $\Gamma_{S}$, i.e., $\Gamma_{S}\left[|0\rangle\langle 0|\right]=|0\rangle\langle 0|$. Also the fast scrambling does not cause the transition from $|0\rangle$ to $|\pm\rangle$.

By combining $\Gamma_{S}$ and $\Gamma_{R}$, we define a channel $\Gamma=\Gamma_{S}\Gamma_{R}$. A schematic diagram of this model is given in Fig. 2.

A qubit in a Gibbs state at initial temperature $T(0)$ evolves by $\Gamma$ as

[TABLE]

where the survival probability $p$ is given by

[TABLE]

and $T=\omega/\ln\left[\frac{1-s}{1-r}\left(\exp\left(\omega/T(0)\right)+r\right)\right]$. Although the state in Eq. (9) is not a Gibbs state of the 3-dim Hilbert space, we are able to identify $T$ as the temperature of the decaying qubit system, because the thermal flux of $\Psi_{R}$ divided by the number of surviving qubits is determined by $T$. Let us impose a natural condition that $T$ is infinity when $T(0)$ is infinity. This fixes a nontrivial relation between the two probability parameters $r$ and $s$ as $s=2r/(1+r)$. First, let us consider case (i). The temperature $T$ rises when the radiation emission probability $r$ increases as

[TABLE]

although the particle extracts energy from the system. This implies that the model realizes a negative heat capacity. However, the temperature does not go to infinity at $r=1$. This feature is different from that of BH evaporation, where the BH temperature goes to infinity at the last burst.

Continuous scrambling.— On the other hand, continuous fast scrambling (case (ii)) makes the final temperature infinity. As shown in Supplemental Material SM , the one-qubit Gibbs state remains unchanged after each fast scrambling and qubit free evolution. Every fast scrambling loses the correlation between a single qubit and the fields. Let us consider $t$ times operation of $\Gamma$ and take the $r\rightarrow 0$ limit with $rt=\tau$ fixed, $\Gamma^{t}\left[\rho\left(T(0)\right)\right]=\left(1-p(\tau)\right)|0\rangle\langle 0|+p(\tau)\rho\left(T(\tau)\right)$. This provides the dynamics of a single decaying qubit reduced from the full dynamics of $N$ decaying qubits, $\Psi_{R}$ and $\Psi_{S}$ with the total Hamiltonian:

[TABLE]

where $H_{i}$ is the $i$ th subsystem free Hamiltonian of Eq. (3), $H_{R_{i}}$ and $H_{S_{i}}$ are the $i$ th field Hamiltonians of Eqs. (4) and  (8), and $H_{\mathrm{fs}}$ is the fast scrambling Hamiltonian for the ($|+\rangle$, $|-\rangle$) sector. Here, as usual, we assume that the Hamiltonian of evaporating qubits can be approximated by the free Hamiltonian contribution: $H_{N}=\sum_{i}H_{i}$.

In order to reproduce the BH thermal properties in this model, it is noted that $N$ depends on $T(0)$, and $N=\left(4\pi G\omega T(0)\right)^{-1}$ should hold. The energy $\omega$ is also assumed to be much smaller than $T(0)$. The constraint $\omega\ll T(0)$ yields

[TABLE]

and since the temperature is given by

[TABLE]

the temperature at $\tau=\infty$ is infinity. Eqs. (10), (12) and (13) are derived in Supplemental Material SM . One qubit energy $E=\mathrm{Tr}\left[H\rho\right]$ is computed as

[TABLE]

Therefore, we have the relation $E=\omega T(0)/(2T)$ for a single qubit. The expectation value of the total energy $\left\langle H_{N}\right\rangle$ of $N$ qubits is given by

[TABLE]

and this is the precise relation in Eq. (1) because $T_{\mathrm{rad}}=T$. Thus the Bekenstein-Hawking entropy $S_{\mathrm{BH}}$ is defined by Eq. (2) in this model. By introducing a new time coordinate $t^{\prime}$, such that $\tau=\ln{(1-t^{\prime}/t_{\mathrm{BH}})^{-1/3}}$, the qubit system decays completely at finite lifetime $t^{\prime}=t_{\mathrm{BH}}$, just as 4-dim Schwarzschild BHs do. The temperature at the Page time is evaluated as $T_{\mathrm{Page}}\approx 2T(0)$.

In the Page curve conjecture, one decaying qubit is almost maximally entangled with emitted matter after the Page time and has no correlation with other qubits due to entanglement monogamy. Therefore, $S_{\mathrm{EE}}$ between one decaying qubit and other subsystems (other $N-1$ qubits+$\Psi_{R}$ +$\Psi_{S}$) must be equal to $S_{\mathrm{BH}}/N$ after the Page time. $S_{\mathrm{EE}}$ at time $\tau$ is computed as

[TABLE]

As shown in Supplemental Material SM , the average thermal entropy of the total system is computed as $NpS_{\mathrm{th}}$, where $S_{\mathrm{th}}$ is the qubit thermal entropy at temperature $T$, which is given by $S_{\mathrm{th}}=\ln\left(1+\exp\left(-\omega/T\right)\right)+\omega/(T\exp\left(\omega/T\right)+T)$. Thus, the average of one-qubit thermal entropy is defined as $pS_{\mathrm{th}}$. In Fig. 3, these three entropies are plotted as a function of $1-T(0)/T$. At the initial time ($T=T(0)$), $S_{\mathrm{EE}}$ of one qubit does not vanish, because the qubit is entangled with other qubits, although $S_{\mathrm{EE}}$ of $N$qubits vanishes at that time. The entropies are conjectured to be equal to each other after the Page time. Actually, they behave very differently ($S_{\mathrm{BH}}/N\ll pS_{\mathrm{th}}\ll S_{\mathrm{EE}}$). In a high-temperature regime with $T\gg T(0)$, they are analytically evaluated as

[TABLE]

The difference between $S_{\mathrm{BH}}/N$ and $pS_{\mathrm{th}}$ originates from the extra factor $p$ in the energy $E=pE_{\mathrm{th}}$. The discrepancy is derived in Supplemental Material SM . The difference between $pS_{\mathrm{th}}$ and $S_{\mathrm{EE}}$ is the binary entropy of $p$ that reflects the non-trivial contribution of the vacuum states, $|0\rangle$. Thus, in this model, the Page curve conjecture does not work.

*Conclusion and discussion. *— In a model of decaying qubits into zero-energy degrees of freedom, the thermal properties of 4-dim Schwarzschild BH evaporation are precisely reproduced. The EE is much larger than the average thermal entropy and the Bekenstein-Hawking entropy analogue for each qubit. This is the first result of a breakdown of the Page curve ansatz in a model which satisfies the Hawking temperature relation in Eq. (1).

The result provides a new feature for a resolution of the information loss problem. In our model, the emission of a Hawking particle of $\Psi_{R}$ at an early stage makes a transition from $|+\rangle$ into $|-\rangle$ of a qubit. After the Page time, the qubit in $|-\rangle$ almost decays into a zero-energy particle of $\Psi_{S}$, which may be interpreted as BMS soft hair propagating to future null infinity. This suggests that in the BH firewall paradox AMPS ; B ; HS , early Hawking radiation is entangled with zero-energy BMS soft hair and that late Hawking radiation can be highly entangled with the degrees of freedom of the BH (surviving qubits in $|-\rangle$), avoiding the emergence of a firewall at the horizon.

The soft hair influence for black holes with positive heat capacity like large AdS black holes remains elusive.

Acknowlegements. —The authors thank Masanori Hanada and Hal Tasaki for useful discussions. This research was partially supported by Kakenhi Grants-in-Aid No. 16K05311 (M.H.) and No. 16H01094 (Y.N.) from the Japan Society for the Promotion of Science (JSPS) and by the Tohoku University Graduate Program on Physics for the Universe (K.Y.).

Appendix A SUPPLEMENTAL MATERIAL

Decay Channel

Substituting Eq. (5) into the Schrödinger equation with Hamiltonian Eq. (4) yields the following equations of motion:

[TABLE]

[TABLE]

Using $x^{\pm}=x\pm t$, Eq. (S15)

is rewritten as

[TABLE]

Taking account of the retarded boundary condition, integration of the above equation is achieved as

[TABLE]

Substituting Eq. (S16) into Eq. (S14) yields

[TABLE]

By changing the integral variable from $x^{\prime}$ to $t^{\prime}$, such that

[TABLE]

the following equation is derived.

[TABLE]

where $K(t)=\int g^{\ast}(x)g\left(x-t\right)dx$. This is generally solved using the Laplace transformation. As a simple example, let us take $g(x)=\lambda\delta\left(x\right)$ with positive $\lambda$. The survival probability of the up state is computed as $\left|c_{+}(t)\right|^{2}=\exp\left(-\lambda^{2}t\right)$. The wave function $\varphi(x,t)$ is given by

[TABLE]

If we take a large $\lambda$, then we have a localized wave packet out of a qubit.

The feature of zero energy states comes from gravitational physics. The zero-enery qubit decay happens due to a large phase space volume for the soft hairs. In ordinally physical systems, the phase space is narrow and the zero-energy decay can be omitted.

Derivation of Temperature Evolution

In this section, we derive Eqs. (10), (12), and (13). By using $\Gamma_{R}$ and $\Gamma_{S}$, we obtain the following result:

[TABLE]

The state is then expressed as

[TABLE]

where the temperature $T^{\prime}$ after the operation is given by

[TABLE]

and the probability $p$ is given by

[TABLE]

Imposing $1/T=1/T^{\prime}=0$ in Eq. (S17) yields

[TABLE]

Substituting Eq. (S18) into Eq. (S17) and replacing T\rightarrow T(0)\,\and $T^{\prime}\rightarrow T$ provides Eq. (10).  By performing $t$ times $\Gamma,$ the temperature after the operation is computed as

[TABLE]

Similarly, the survival probability of a qubit after the operation is given by

[TABLE]

By taking the large $t$ limit with $rt=\tau\,$ fixed in Eq. (S19), the following relation holds:

[TABLE]

Assuming $\omega\ll T(0)$, we get Eq. (13). Similarly, using the same limit, Eq. (12) is derived by substituting Eq. (13) into Eq. (S20).

Finally we add a comment regarding another channel, $\Gamma^{\prime}=\Gamma_{R}\Gamma_{S}$. This yields a different evolution, such that

[TABLE]

In this case, imposing $1/T=1/T^{\prime}=0$ yields $s=2r$. Thus, $\Gamma^{\prime}$ becomes unphysical in the case with $1/2<r\leq 1$ since the probability $s$ exceeds $1$, which implies that we have noncommutativity of $\Gamma_{R}$ and $\Gamma_{S}$. However, if we consider a small $r$ limit in case ii) to derive the 4-dim BH’s thermal properties, the commutator merely gives $O(r^{2})$ correction terms which do not contribute to the final results.

Qubit Fast Scrambling

Here we show the invariance of the thermal state of a decaying qubit in Eq. (9) under fast scrambling and qubit free evolution. We have $N$ three-level identical subsystems.

Let us first comment about the unitary evolution $U$ corresponding to $\Gamma$, which describes the dynamics of a single decaying qubit, a radiation field $\Psi_{R}$, and a soft hair field $\Psi_{S}$. For $|\psi\rangle=|+\rangle,\,|-\rangle,\,|0\rangle$, it provides

[TABLE]

where

[TABLE]

and

[TABLE]

since the unitary time evolution $U$ conserves the excitation numbers. Let us assume that the coupling functions $g(x)$ in Eq. (4) and Eq. (8) are almost localized around $x=0$ with large $\left|g(0)\right|$ as in the above SM section, and that the wave functions $\varphi_{R}(x)$ and $\varphi_{S}(x)$ of the emitted particles already have no overlap with the support of $g(x)$. This ensures that

[TABLE]

Then we are able to neglect the correlation of the state in Eq. (S22) in the evaluation of the time evolution of the decaing qubit system due to the $\left(|+\rangle,|-\rangle\right)$ sector fast scrambling, which causes random phase factors $\exp\left(i\delta_{\pm}\right)=$ $c_{\pm}/\,\left|c_{\pm}\right|~{}$such that

[TABLE]

Here each bar stands for the ensemble avarage of the fast scrambling. Theses properties and Eq. (S22) yield the following decohered state:

[TABLE]

In the evaluation of the single decaying qubit dynamics, it is possible to replace $|\varphi_{R}^{(\psi)}\rangle_{R}\langle\varphi_{R}^{(\psi)}|_{R}$ into $|\mathrm{vac}\rangle_{R}\langle\mathrm{vac}|_{R}$ and $|\varphi_{S}^{(\psi)}\rangle_{S}\langle\varphi_{S}^{(\psi)}|_{S}$ into $|\mathrm{vac}\rangle_{S}\langle\mathrm{vac}|_{S}$ because the emitted particles already leave the interaction region as seen in Eq. (S23), and the field quantum states are local vacuum states around the decaying qubit. Hence the reduced state evolution of the decaying qubit after the fast scrambling can be described again by using the same channel $\Gamma$ as follows.

[TABLE]

This justifies the use of $t$-times successive operations of the channels $\Gamma^{t}\left[\rho\left(T(0)\right)\right]$ in our time evolution analysis.

Next let us explain the entanglement structure of the total system. The two states $|\pm\rangle$ among the three states describe the qubit states. Assuming that emitted particles do not have spatial overlaps, the entanglement among decaying qubits and the particles can be computed by use of an extended model, in which each paricle is treated as a quantum of  an independent field. Each channel operation $\Gamma_{R}$ of $\Gamma$ requires a fresh vacuum state $|\mathrm{vac}\rangle$ of a Hawking radiation field $\Psi_{R}(x)$. Similarly each $\Gamma_{S}$ of $\Gamma$  requires a fresh vacuum of a soft hair field $\Psi_{S}(x)$. Therefore, $2N$ fields are required for a one-time operation of $\Gamma^{\otimes N}$, which means that the $t$-times iteration of $\Gamma^{\otimes N}$ requires $2Nt$ fields. For instance, the original single field $\Psi_{R}$ with $Nt$ particles is described by a set of $Nt$ different $\Psi_{R}$s with one particle excitation in each field. The initial state of the fields is a tensor product of $2Nt$ vacuum states, such that

[TABLE]

where the subscripts $(jk)$ discriminate the fields, and $|\mathrm{vac};R\rangle_{\left(jk\right)}$ ($|\mathrm{vac};S\rangle_{\left(jk\right)}$) is the vacuum state of the $jk$-th radiation field (soft-hair field).

The total dynamics generated by the full Hamiltonian in Eq. (11) yields the following complicated entangled state:

[TABLE]

where $|\phi(t);R\rangle$ is a one-particle state of the radiation field such that

[TABLE]

and $|\phi(t);S\rangle$ is a similar one-particle state of the soft hair field. Interference among the terms of multi-particle emissions can be neglected in this model. The entanglement between a decaying qubit and other systems can be evaluated using $|\Phi\rangle$. The reduced state of the $N$-qubit system is

[TABLE]

for some constants $\{\alpha\}$. Note that the fast scrambling with free evolution of each qubit makes off-diagonal contributions of $\mathrm{Tr}_{\Psi_{R},\Psi_{S}}\left[|\Phi\rangle\langle\Phi|\right]$ such as $|+--\cdots-\rangle\langle-+-\cdots-|$ to vanish when we take a partial trace and compute a one-qubit reduced state. Thus, the fast scrambling ensemble average of $\rho_{1}=\mathrm{Tr}_{2,3,\cdots,N}\mathrm{Tr}_{\Psi_{R},\Psi_{S}}\left[|\Phi\rangle\langle\Phi|\right]$ takes the diagonalized form of

[TABLE]

By taking $q=\frac{\exp\left(-\omega/T\right)}{1+\exp\left(-\omega/T\right)}$, this is rewritten as

[TABLE]

with the temperature $T$ of the decaying qubit. Since the expectation value of the number of surviving qubits is conserved in the fast scrambling, $p$ is unchanged during this process. Moreover, the conservation of the total energy ensures that $T$ is equal to the temperature before the scrambling. Thus, the state of a decaying qubit in Eq. (9) does not change during the fast scrambling.

Average Thermal Entropy of the Total System

In this section, we evaluate the average thermal entropy of $N$ decaying qubits. For one decaying qubit, the state is given by $\left(1-p\right)|0\rangle\langle 0|+p\rho\left(T\right)$, due to the state typicality. With probability $p$, a surviving qubit with temperature $T$ is observed. The entropy is then given by $S_{\mathrm{th}}=-\mathrm{Tr}\left[\rho\left(T\right)\ln\rho\left(T\right)\right]$. The vacuum state is observed with probability $1-p$. Then, no thermal entropy appears. We have  $N$ identical systems. The probability of finding $n$ surviving qubits is given by \left(\begin{array}[]{c}N\\ n\end{array}\right)p^{n}\left(1-p\right)^{N-n}. Therefore, the average of the total thermal entropy is computed as

[TABLE]

Discrepancy between $S_{\mathrm{BH}}/N$ and $pS_{\mathrm{th}}$

In this section, we derive the large discrepancy between $S_{\mathrm{BH}}/N$ and $pS_{\mathrm{th}}$. First of all, $S_{\mathrm{BH}}/N$ is defined using $d\left(S_{\mathrm{BH}}/N\right)=dE/T$. Due to $E=pE_{\mathrm{th}}$ and $dE_{\mathrm{th}}=TdS_{\mathrm{th}}$,

[TABLE]

holds. The second term on the right-hand side appears due to the time dependence of $p$, and causes the large deviation of $S_{\mathrm{BH}}/N$ from $pS_{\mathrm{th}}$. Let $1/T~{}$ be denoted by $\beta$. Using

[TABLE]

and

[TABLE]

assuming $\omega\ll T(0)$, the integration of Eq. (S25) provides

[TABLE]

By taking a small $\beta$ in Eq. (S26), the term of $O\left(T(0)/T\right)$ in the right-hand side vanishes because $S_{\mathrm{th}}$ tends to $\ln 2$. Thus, the relation $S_{\mathrm{BH}}/N=O\left(T(0)^{2}/T^{2}\right)$ is reproduced, even though $pS_{\mathrm{th}}=$ $O\left(T(0)/T\right)$.