a ainstitutetext: Department of Physics, Osaka University Toyonaka, Osaka
560-0043, JAPAN b binstitutetext: Institute of Physics, University of Tokyo,
Komaba,
Meguro-ku, Tokyo 153-8902, JAPAN c cinstitutetext: Center for Gravitational
Physics, Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, JAPAN d dinstitutetext: The
Hakubi Center for Advanced Research, Kyoto University, Yoshida Ushinomiyacho,
Sakyo-ku, Kyoto 606-8501, JAPAN

# A comment on a fine-grained description of evaporating black holes with baby
universes

 Norihiro Iizuka b  Akihiro Miyata c,d  Tomonori Ugajin  iizuka (at)
phys.sci.osaka-u.ac.jp,
miyata (at) hep1.c.u-tokyo.ac.jp,
tomonori.ugajin (at) yukawa.kyoto-u.ac.jp

###### Abstract

We study a partially fine-grained description of an evaporating black hole by
introducing an open baby universe with a boundary. Since the Page’s
calculation of the entropy of Hawking radiation involves an ensemble average
over a class of states, one can formally obtain a fine-grained state by
purifying this setup. For AdS black holes with a holographic dual, this
purification amounts to introducing an additional boundary (i.e., baby
universe) and then connecting it to the original black hole through an
Einstein-Rosen bridge. We uncover several details of this setup. As
applications, we briefly discuss how this baby universe modifies the semi-
classical gravitational Gauss law as well as the gravitational dressing of
operators behind the horizon.

###### Keywords:

Black Holes,Conformal Field Theory, AdS-CFT Correspondence

††preprint: OU-HET-1119, UT-Komaba/21-6, YITP-21-133

##  1 Introduction

Ever since Hawking discovered that a black hole has a temperature and emits
thermal radiations Hawking:1975vcx ; Hawking:1976ra , how its time-evolution
is consistent with the principle of quantum mechanics is one of the greatest
problems in theoretical physics. One of the key points of recent developments
in quantum gravity is the role of the Euclidean wormholes, which play a
crucial role in resolving the black hole information loss problem through
their non-perturbative effects (e.g., Penington:2019kki ; Almheiri:2019qdq ;
Marolf:2020xie ).

The von Neumann entropy of the Hawking radiation can be defined by the
entanglement entropy of the bath region $R$, attached to the asymptotic
infinity of the black hole, see figure 1. The island formula Penington:2019npb
; Almheiri:2019psf ; Almheiri:2019hni  tells us that this entropy is given by

$S(\rho_{R})=\underset{I}{\rm MinExt}\left[\frac{A(\partial
I)}{4G_{N}}+S_{{{\rm bulk}}}(R\cup I)\right],$ (1.1)

where $I$ is a region in the bulk gravitating spacetime. The region which
extremizes the above generalized entropy functional is called the island. This
formula can be regarded as a natural extention of the RT/HRT formulae and
their quantum extentions for holographic entanglement entropy in AdS/CFT
Ryu:2006bv ; Ryu:2006ef ; Hubeny:2007xt ; Faulkner:2013ana ;
Engelhardt:2014gca . This island formula is indeed obtained by including so
called Euclidean replica wormholes to the gravitational path integral
Penington:2019kki ; Almheiri:2019qdq . The island formula correctly reproduces
the Page curve Page:1993wv ; Page:2013dx  for the entanglement entropy of
Hawking radiation, thus gives results consistent with the principles of
quantum theory within semi-classical regime of gravity. (See e.g.,
Almheiri:2020cfm ; Chen:2021lnq  for reviews on this topic, and related
discussions on the island formula, e.g., Rozali:2019day ; Chen:2019uhq ;
Bousso:2019ykv ; Almheiri:2019psy ; Chen:2019iro ; Balasubramanian:2020hfs ;
Hollowood:2020cou ; Alishahiha:2020qza ; Chen:2020uac ; Geng:2020qvw ;
Chandrasekaran:2020qtn ; Li:2020ceg ; Bousso:2020kmy ; Dong:2020uxp ;
Hollowood:2020kvk ; Chen:2020jvn ; Chen:2020tes ; Hartman:2020khs ;
Balasubramanian:2020coy ; Balasubramanian:2020xqf ; Ling:2020laa ;
Chen:2020hmv ; Bhattacharya:2020uun ; Harlow:2020bee ; Akal:2020ujg ;
Hernandez:2020nem ; Chen:2020ojn ; Goto:2020wnk ; Matsuo:2020ypv ;
Hsin:2020mfa ; Akal:2020twv ; Numasawa:2020sty ; KumarBasak:2020ams ;
Geng:2020fxl ; Deng:2020ent ; Karananas:2020fwx ; Wang:2021woy ;
Kawabata:2021hac ; Fallows:2021sge ; Bhattacharya:2021jrn ; Kim:2021gzd ;
Anderson:2021vof ; Miyata:2021ncm ; Wang:2021mqq ; Ghosh:2021axl ;
Aalsma:2021bit ; Geng:2021iyq ; Balasubramanian:2021wgd ; Uhlemann:2021nhu ;
Qi:2021sxb ; Kawabata:2021vyo ; Chu:2021gdb ; Langhoff:2021uct ; Lu:2021gmv ;
Akal:2021foz ; Balasubramanian:2021xcm ; Ahn:2021chg ; Miyaji:2021lcq ;
Matsuo:2021mmi ; Goto:2021mbt ; Anegawa:2020lzw .)

Figure 1: The Penrose diagram of the AdS black hole attached to the non-
gravitating heat bath. We take the radiation region $R$ (violet solid line) in
the heat bath. After the Page time, the island region $I$ (blue solid line)
becomes non-empty, and the black hole region $BH$ (orange solid line) is
outside the horizon of the black hole. The entanglement wedge of the Hawking
radiation $R$ is the union of the domain of dependence of the radiation $R$
and island $I$ regions (violet and blue shaded regions), and that of the black
hole $BH$ is the domain of dependence of the black hole region (orange shaded
region).

The island formula suggests that the entanglement wedge of the Hawking
radiation contains not only the radiation region $R$, but also the island
region $I$. On the other hand, the entanglement wedge of the black hole is its
compliment (see figure 1 again). The boundary of the island region $\partial
I$ is located just behind the horizon for an evaporating black hole
Penington:2019npb ; Almheiri:2019psf ; Almheiri:2019hni ; Gautason:2020tmk ;
Hartman:2020swn . However for an eternal black hole, it is located outside the
horizon Almheiri:2019yqk ; Gautason:2020tmk ; Anegawa:2020ezn ;
Hashimoto:2020cas ; Hartman:2020swn .

In a recent interesting paper Geng:2021hlu , such a form of entanglement wedge
disconnected from the asymptotic boundary is argued to be inconsistent with
the long range nature of the gravitational force or equivalently
diffeomorphism invariance. Let us consider a small local operation on the
island $I$ which corresponds to a local excitation of on the radiation Hilbert
space $H_{R}$. In a theory of gravity, such a small operation seems to be
detected on the asymptotic boundary of the spacetime using the gravitational
Gauss law. This is problematic because the asymptotic boundary belongs to the
entanglement wedge of the black hole, see figure 2. This implies that the
local operation on the island region $I$ (which is supposed to be an operation
on $H_{R}$) can actually change the entanglement wedge of the black hole.

Figure 2: The Penrose diagram of the AdS black hole attached to the non-
gravitating heat bath with a small local operation $\phi(P_{I})$ ($P_{I}\in
I$, red dot) in the island region $I$. The entanglement wedge of the black
hole $BH$ (orange shaded region) includes the asymptotic AdS boundary. The
gravitational Wilson line $W(P_{I},P_{R})$ (green line) connecting two points
$P_{I}$ and $P_{R}$ ($P_{R}\in R$, cyan dot) intersects the entanglement wedge
of the black hole (orange shaded region).

The above apparent inconsistency of the gravitational Gauss law is related to
the inconsistency of the gravitational dressing of local operators on the
island region Geng:2021hlu . In a theory with diffeomorphism invariance, a
local operator can not be physical since it is not gauge invariant. Instead,
such a local operator $\phi(P)$ needs to be “gravitationally dressed”
Donnelly:2015hta ; Donnelly:2016rvo  by attaching a Wilson line $W(P,P_{B})$
connecting the point $P$ to a point $P_{B}$ on the asymptotic boundary. Then
the resulting operator $\phi(P)W(P,P_{B})$ is gauge invariant. For the
gravitational dressing for a local operator on the island region $P_{I}\in I$,
which is a part of the entanglement wedge of the Hawking radiation, it is
natural to choose a point on the bath region $R$ as the asymptotic boundary
point $P_{R}\in R$. In such a case, the Wilson line $W(P_{I},P_{R})$
connecting the two points intersects with the entanglement wedge of the black
hole. However the dressed operator $\phi(P_{I})W(P_{I},P_{R})$ should belong
only to the entanglement wedge of the Hawking radiation, thus this is
problematic again. See the figure 2 again.

In this paper, we address the above paradox by carefully examining how the
effects of random fluctuations of an evaporating black hole are geometrized in
a semi-classical description of gravity. In principle the black hole
evaporation process is described by the bipartite system of the Hilbert space
of the black hole $H_{BH}$ and the one for the Hawking radiation $H_{R}$. Of
course, the description of such an entangled state involves a quantum theory
of gravity, therefore it seems impossible to study such a system efficiently.
However, as was first observed by Page Page:1993wv , one can obtain a time
evolution of the radiation entropy consistent with unitarity, by averaging the
entropy over the random fluctuations in the entangled state. This opens up the
possibility of having a partially fine-grained description of the evaporating
black hole while maintaining its semi-classical nature, to the extent of
getting results consistent with the principles of quantum theory. Indeed, in
this way, the island formula makes it possible to recover the Page curve in a
semi-classical way. Specifically, the Euclidean replica wormholes nicely
capture the effects of these random fluctuations and their averaging through a
geometric way.

This paper concerns a description of these random fluctuations in a Lorentzian
spacetime in the semi-classical regime. We argue that the averaging over the
random fluctuations can be purified by introducing an auxiliary system, often
called a baby universe. This new piece of the spacetime is connected to the
original spacetime with the black hole by an Einstein-Rosen bridge, can be
thought of as accommodating partially fine-grained information of the
evaporating black hole (see figure 4). See also Polchinski:1994zs ;
Marolf:2020rpm  for discussions on the role of baby universes in the
information loss paradox.

Motivated by this observation, we then study the gravitational Gauss law in
the presence of the baby universe sector. Such an introduction of the baby
universe significantly modifies the form of the gravitational Gauss law. For
instance, assuming the baby universe part has an asymptotic boundary, the
gravitational Gauss law does not exactly hold within the original spacetime as
there is a contribution from the baby universe sector. This makes sense,
because restricting our attention only to the original black hole spacetime
corresponds to a coarse-graining. This is clearly seen from a Schwarzschild
black hole solution, which has the horizon area and therefore the Bekenstein-
Hawking entropy $S_{BH}$. However it is just one solution, showing no
degeneracy of the states, contradicting the huge degeneracy given by the
entropy $e^{S_{BH}}$. What we expect is that in full quantum gravity, one
obtains microstates of the black hole and by counting its degeneracy, one
obtains $e^{S_{BH}}$. However after coarse-graining, all the details of the
microstates are lost, and one cannot see the microstate differences in the
Schwarzschild solution. Assuming the dynamics of the black hole is
sufficiently chaotic, two distinct energy eigenstates can never have the same
energy, and the minimal value of the difference is of order $e^{-S_{BH}}$.
Thus the geometric description of such a class of microstates by a single
black hole spacetime inevitably involves a coarse-graining, in which the
energy differences of order $e^{-S_{BH}}$ are neglected. This suggests that in
the black hole spacetime, one can only trust the gravitational Gauss law up to
$O(e^{-S_{BH}})$ corrections.

The introduction of the baby universe with an asymptotic boundary naturally
resolves the paradox of the gravitational dressing as well, because the
gravitational Wilson line starting from the island region can now end on the
boundary of the baby universe. This is a kind of an expected result because
the island region corresponds to fine-grained information of the evaporating
black hole, so can not have a simple description within the original black
hole spacetime.

The rest of the paper is organized as follows. In section 2, we study
wormholes, and the baby universe. In section 3, we present our main idea and
explain how we modify the gravitational Gauss law in the presence of the baby
universe. We also comment on how the boundary of the baby universe can resolve
the gravitational dressing paradox. In appendix A, we give the calculation of
the von Neumann entropy of the Hawking radiation and that of the Hawking
radiation plus the baby universe in our formalism.

Note added: During the preparation of this paper, the papers Renner:2021qbe ;
Qi:2021oni ; Almheiri:2021jwq  appeared, and discussed extra information
coming from the ensemble nature of gravity, which is related to the baby
universe degrees of freedom in our paper.

##  2 Baby Universe and Ensemble nature of Semi-Classical Gravity

In this section, we clarify the role of the baby universe in the computation
of the fine-grained entropy of Hawking radiation through the island formula.

To this end, it is appropriate to begin with the fact that there are two
distinct descriptions of a theory of gravity. The first one is the fine-
grained description, and the second one is the coarse-grained one. In the
first full-fledged fine-grained description of quantum gravity, we have a
sufficient number of observables (i.e., the complete set of operators of
quantum gravity) to perfectly distinguish quantum states. Note that, in the
description, we can perform measurements with arbitrary precision. We are
interested in the gravitational system where a black hole keeps emitting
Hawking radiations. In a full-fledged fine-grained microscopic description, an
actual state in such a system has the following form,

$|\Psi_{M}\rangle=\sum_{i=1}^{\mathcal{N}}\sum_{\alpha=1}^{k}{F}_{i\alpha}^{M}|\psi_{i}\rangle_{BH}|\alpha\rangle_{R}\,,$
(2.1)

where ${F}_{i\alpha}^{M}$ takes a fixed number. Here we define the orthonormal
bases $|\psi_{i}\rangle_{BH}$ and $|\alpha\rangle_{R}$ of the Hilbert space
$H_{BH}$ for microstates for the black hole and the similar Hilbert space
$H_{R}$ for the Hawking quanta participating in the entanglement.
$\mathcal{N}$ and $k$ are their dimensions.

The second description of the system is the coarse-grained one in terms of a
semi-classical theory, where we have a restricted number of observables, i.e.,
a subset of the complete set of observables of quantum gravity, or coarse-
grained observables like thermodynamical quantities. The spatial and time
resolution of such observables is much larger than the Planck scale. In this
description, even by measuring coarse-grained observables precisely, we cannot
completely distinguish the underlying full quantum states of the full theory,
but at best a set of states with the same expectation values of the coarse-
grained observables and the same semi-classical geometries.

Owing to the restricted number of observables and also to the fact that the
resolution is much larger than the Planck scale, one is forced to describe the
system in a coarse-grained way, in terms of a mixed state, i.e., an ensemble
of states $\\{p_{M},|\Psi_{M}\rangle\\}_{M}$111See Balasubramanian:2020lux
for a similar discussion.. This ensemble consists of the class of the states
$|\Psi_{M}\rangle$ with the random coefficient matrix $C_{i\alpha}^{M}$

$|\Psi_{M}\rangle=\sum_{i=1}^{\mathcal{N}}\sum_{\alpha=1}^{k}C_{i\alpha}^{M}|\psi_{i}\rangle_{BH}|\alpha\rangle_{R}\,.$
(2.2)

From the semi-classical gravity point of view, two such states
$|\Psi_{M}\rangle,|\Psi_{N}\rangle$ with different random coefficients
$C^{M},C^{N}$ can not be distinguished. This corresponds that a coarse-grained
observer describes the state in terms of the following mixed state,

$\rho_{BH\cup R}=\sum_{M}p_{M}|\Psi_{M}\rangle\langle\Psi_{M}|\,,$ (2.3)

where $p_{M}$ is the Gaussian probability distribution determined by the
ensemble of states or random coefficient matrix $C_{i\alpha}^{M}$ as

$\displaystyle p_{M}$
$\displaystyle=\left(\frac{\mathcal{N}k}{\pi}\right)^{\mathcal{N}k}\exp\left(-\mathcal{N}k\operatorname{tr}(C^{M}C^{M\dagger})\right),$
(2.4)

and satisfies $\sum_{M}p_{M}=1$. See (A.1)-(A.3) in appendix A. We also note
that the coefficients $C_{i\alpha}$ are satisfying the following relationship,

$\displaystyle\langle 1\rangle$ $\displaystyle=1$ (2.5) $\displaystyle\langle
C_{i\alpha}C^{\dagger}_{\beta j}\rangle$
$\displaystyle=\frac{1}{k\mathcal{N}}\delta_{ij}\delta_{\alpha\beta}$
$\displaystyle\langle C_{i\alpha}C^{\dagger}_{\beta
j}C_{k\gamma}C^{\dagger}_{\delta l}\rangle$
$\displaystyle=\frac{1}{(k\mathcal{N})^{2}}\left(\delta_{ij}\delta_{\alpha\beta}\cdot\delta_{kl}\delta_{\gamma\delta}+\delta_{il}\delta_{\alpha\delta}\cdot\delta_{jk}\delta_{\beta\gamma}\right)$
$\displaystyle\langle(\Pi_{a=1}^{n}C_{i_{a}\alpha_{a}})(\Pi_{b=1}^{n}C^{\dagger}_{\beta_{b}j_{b}})\rangle$
$\displaystyle=\frac{1}{(k\mathcal{N})^{n}}(\text{all possible contractions of
indices})$
$\displaystyle\langle(\Pi_{a=1}^{n}C_{i_{a}\alpha_{a}})(\Pi_{b=1}^{m}C^{\dagger}_{\beta_{b}j_{b}})\rangle$
$\displaystyle=0\quad\text{for }m\neq n$

where $\langle\cdot\rangle$ means the average over the random coefficient
matrix $C_{i\alpha}^{M}$. The randomness of the coefficient in (2.2) is due to
the fact that the dynamics of a black hole is highly chaotic. These can be
understood as follows: Suppose that an observer tries to experimentally
specify the fine-grained state (2.1). Then the observer needs to perform a
measurement with the Planck scale precision. However, for coarse-grained
observers, the resolution of the measurement is much larger than the Planck
scale. Note that during the measurement time-scale, the microscopic state can
evolve. Therefore, if the measurement time-scale is much larger than the
Planck scale, the microscopic state can evolve to almost all states of the
form (2.2). In this way, coarse-grained observers see the black hole state as
(2.2). This provides an intuitive way to understand the reason why the random
matrices appear in the semi-classical description of the black hole dynamics.

Once we coarse-grain the system, the state is reduced from the pure state
(2.1) to the mixed state (2.2), and apparently we lose the microscopic details
of the states. However we nevertheless can compute some aspects of the fine-
grained entropy of Hawking radiation by purifying this mixed state by
introducing an auxiliary system $H_{BU}$, which we often call the baby
universe. For instance, recent progress in understanding the island formula
suggests that the purification enables us to capture some part of the fine-
grained information of Hawking radiation while maintaining the semi-classical
description. Discussions on the relevance of random fluctuations for the
physics of black holes can be found for example in Verlinde:2012cy ;
Langhoff:2020jqa ; Nomura:2020ska . We also note that Gaussian random
fluctuations have a geometric interpretation in terms of end of the world
branes in two-dimensional JT gravity Penington:2019kki .

Note that to purify the original system with the mixed state (2.3), we need an
auxiliary system $H_{BU}$ whose dimension is at least equal to or greater than
that of the original system. The dimension of the baby universe Hilbert space
depends in particular on the coarse-graining procedure. On this new Hilbert
space, the simplest purified state is given by

$|\Phi\rangle_{BH\cup R\cup BU}=\sum_{M}\sqrt{p_{M}}|\Psi_{M}\rangle_{BH\cup
R}|M\rangle_{BU},$ (2.6)

where $\\{|M\rangle_{BU}\\}$ are orthonormal baby universe states. A fine-
grained observer can access this auxiliary system, but coarse-grain observers
can not. Let us emphasize that the description using the auxiliary system is
not a full fledged fine-grained description of the system. This is because we
are artificially adding degrees of freedom, which do not show up in the
original Hilbert space $H_{BH}\otimes H_{R}$. More concretely, in the quantum
gravity description, the actual fine-grained state realized in the system is
one of the states in the ensemble, not the one with the baby universe. We
nevertheless consider the purified state (2.6), because it has an effective
semi-classical description, on the contrary to the full fledged fine-grained
state in quantum gravity. Furthermore, as we will show later, if we are only
concerned with the averaged property of the fine-grained entropy, such as the
Page curve, considering this purified state is good enough.

Note that by tracing out the black hole degrees of freedom $BH$ in the mixed
state (2.3), the reduced density matrix of the Hawking radiation $\rho_{R}$
gives an approximately thermal mixed state, and the von Neumann entropy
$S_{\rm{vN}}[\rho_{R}]$ gives the Hawking’s result

$\displaystyle S_{\rm{vN}}[\rho_{R}]$
$\displaystyle=S_{\mathrm{vN}}[\;\langle\rho_{(M)R}\rangle_{M}\;]$ (2.7)
$\displaystyle=\log k,$

where we have defined,

$\rho_{(M)R}=\operatorname{tr}_{BH}\left[\,|\Psi_{M}\rangle\langle\Psi_{M}|_{BH\cup
R}\,\right],\quad\langle\rho_{(M)R}\rangle_{M}=\sum_{M}p_{M}\;\rho_{(M)R}.$
(2.8)

See the appendix A.1 for detailed derivation.

Now let us consider the same entropy of Hawking radiation in the fine-grained
description. To do so, let us first figure out a geometric description of the
purified state (2.6). In this state, the Hawking radiation $H_{R}$ and the
black hole $H_{BH}$ are entangled with the auxiliary baby universe $H_{BU}$.
From the viewpoint of ER=EPR Maldacena:2013xja , we expect that this is
realized geometrically by an Einstein-Rosen bridge connecting the baby
universe and the original system (see figure 4). The property of the ER bridge
depends highly on the choice of the ensemble. If we realize this system within
the framework of the AdS/CFT correspondence, the auxiliary universe can be
modeled by an additional boundary and its gravity dual involves an Einstein-
Rosen bridge connecting the new boundary222See Heckman:2021vzx  for a similar
discussion by using sting theory.. This purification process is the key in
recent studies, especially in the finding of the island formula which captures
some aspects of fine-grained information of the quantum gravity states, in the
semi-classical description through a non-perturbative way. For instance, in
describing an evaporation process of a black hole semi-classically, such non-
perturbative contributions are required to get a consistent result. In such a
process discreteness of the energy spectrum of the black hole microstates is a
crucial ingredient to ensure unitarity of the process. However, in the coarse-
grained description, energy differences between black hole micro-states are
invisible, since they are typically of order $\mathcal{O}(e^{-S_{BH}})$, where
$S_{BH}$ is the Bekenstein-Hawking entropy Balasubramanian:2006iw . A discrete
energy spectrum is only after including non-perturbatively small contributions
which are provided by Euclidean wormholes Maldacena:2001kr ; Saad:2019lba .

What the island formula implies is that one should identify the fine-grained
Hilbert space of the Hawking radiation $H_{\mathbf{R}}$ with the tensor
product of two Hilbert spaces $H_{R}\otimes H_{BU}$, after the Page time. On
the other hand, before the Page time $H_{\mathbf{R}}$ should be identified
with just that of the Hawking radiation $H_{R}$, and correspondingly the
Hilbert space of the black hole should coincide with the tensor product of the
black hole and the baby universe $H_{BH}\otimes H_{BU}$. This difference
between the radiation Hilbert spaces before and after the Page time comes from
the fact that the inequality for the dimensions of the Hilbert spaces of the
Hawking radiation and that of the black hole changes. In fact, before the Page
time, since the total state (2.6) is pure, the von Neumann entropy of the
union of the black hole and the baby universe $BH\cup BU$ is equal to the
previous von Neumann entropy (2.7) of $R$, i.e., $S_{\mathrm{vN}}[\rho_{BH\cup
BU}]=S_{\mathrm{vN}}[\rho_{R}]=\log k$, which is consistent with the island
formula before the Page time.

After the Page time, the reduced density matrix of the Hawking radiation and
the baby universe $\rho_{R\cup BU}$ in (2.6) gives the the fined grained
entropy of the Hawking radiation, which deviates from the entropy (2.7) of the
naive density matrix (2.3),

$\displaystyle S_{\mathrm{vN}}[\rho_{\mathbf{R}}]$
$\displaystyle=S_{\mathrm{vN}}[\rho_{R\cup BU}]$ (2.9)
$\displaystyle=\log\mathcal{N}$ $\displaystyle=S_{BH}.$

In appendix A.2 we provide details of this calculation. The result reproduces
the behaviour of the Page curve after the Page time, giving the Bekenstein-
Hawking entropy $S_{BH}$. Therefore by appropriately dividing the total system
$BH\cup R\cup BU$, we can get the von Neumann entropy which obeys the Page
curve (see table 1)333 One may also consider the possibility of dividing the
baby universe Hilbert space $H_{BU}$ into two parts $H_{BU_{BH}}\otimes
H_{BU_{R}}$, and then define the radiation Hilbert space as $H_{\bf
R}=H_{BU_{R}}\otimes H_{R}$, instead of $H_{\bf R}=H_{BU}\otimes H_{R}$ which
we do in the body of the paper. In such a case, the states of the baby
universe are given by $|M\rangle_{BU_{BH}}\otimes|M\rangle_{BU_{R}}$. In this
case, assuming the orthogonality of the basis of $H_{BU_{BH}}$, we see that
the entropy of $\rho_{BU_{R}\cup R}$ is given by
$S_{\mathrm{vN}}[\rho_{BU_{R}\cup R}]=-\sum_{M}p_{M}\log
p_{M}+\sum_{M}p_{M}S_{\mathrm{vN}}[\rho_{(M)R}],$ (2.10) where $\rho_{(M)R}$
is the reduced density matrix given by (A.5). Then it is natural to define the
fine grained entropy of Hawking radiation $S(\rho_{{\bf R}})$ as a conditional
entropy of knowing the probability distribution ${p_{M}}$ by subtracting the
classical Shannon piece $H(p_{M})=-\sum_{M}p_{M}\log p_{M}$ $S(\rho_{{\bf
R}})=S_{\mathrm{vN}}[\rho_{BU_{R}\cup
R}]-H(p_{M})=\sum_{M}p_{M}\;S_{\mathrm{vN}}[\rho_{(M)R}].$ (2.11) However we
do not know the natural choice for such a splitting of the baby universe
Hilbert space $H_{BU}$..

| Black Hole | Hawking Radiation | von Neumann Entropy
---|---|---|---
Before the Page time | $BH\cup BU$ | $R$ | $S_{\mathrm{vN}}[\rho_{BH\cup BU}]=S_{\mathrm{vN}}[\rho_{R}]=\log k$
After the Page time | $BH$ | $R\cup BU$ | $S_{\mathrm{vN}}[\rho_{BH}]=S_{\mathrm{vN}}[\rho_{R\cup BU}]=S_{BH}$
Table 1: How to divide the total system $BH\cup R\cup BU$ into two sub systems
before and after the Page time, and the corresponding von Neumann entropies.

At the same time, we know that the fine-grained entropy
$S_{\mathrm{vN}}[\rho_{\mathbf{R}}]$ of Hawking radiation is computed by the
island formula (1.1) too. In the entropy calculations using this formula, it
was crucial to include the contribution of the island, which typically
occupies the region behind the horizon of the black holes. Therefore it is
natural to identify the island region behind the horizon with the Einstein-
Rosen bridge of the purified state (2.6) connecting the original spacetime and
the baby universe, which stores fine-grained information of the original
spacetime.

These states $\\{|M\rangle_{BU}\\}$ in the fine-grained Hilbert space can be
naturally identified with so called $\alpha$ states Coleman:1988cy ;
Giddings:1988cx ; Giddings:1988wv  in the baby universe Hilbert space which
diagonalizes the boundary creation operators Marolf:2020rpm . Then each fine-
grained state $|\Psi_{M}\rangle|M\rangle$ belongs to different superselection
sector, because each $\alpha$ state does. In particular, this means that off
diagonal element of matrix $\langle\Psi_{M}|\langle M|(\mathcal{O}\otimes
I)|\Psi_{N}\rangle|N\rangle$ for any local operator $\mathcal{O}$ on the black
hole $BH$ and the Hawking radiation $R$ vanishes, therefore any local
measurement on them can not distinguish the entangled pure state (2.6) with
the mixed state only with classical correlation of the following form

$\rho=\sum_{M}p_{M}\;|\Psi_{M}\rangle\langle\Psi_{M}|\otimes|M\rangle\langle
M|,$ (2.12)

in the sense that

$\operatorname{tr}\left[|\Phi\rangle\langle\Phi|\,(\mathcal{O}\otimes
I)\right]=\operatorname{tr}\left[\rho\,(\mathcal{O}\otimes
I)\right]=\sum_{M}p_{M}\langle\Psi_{M}|\mathcal{O}|\Psi_{M}\rangle.$ (2.13)

In other words, LOCCs acting only on the black hole $BH$ and the Hawking
radiation $R$, which can be available to coarse-grained observers, can not
distinguish the classically and quantum mechanically correlated states (2.6),
(2.12). However one can easily see the entanglement entropy of these two
states on $\mathbf{R}=R\cup BU$ are different. Indeed, the entropy of $\rho$
contains a classical Shannon term, whereas the same entropy of (2.6) does not.
From another point of view, LOCCs on the sub-system $BH\cup R$ and the baby
universe $BU$, which can only be available to fine-grained observers, can
distinguish the classically and quantum mechanically correlated states, since
the equalities in (2.13) do not necessarily hold for operators on $BH\cup
R\cup BU$.

In the next section, we discuss several properties of the baby universe and
the wormhole connecting the baby universe and the original spacetime. The
wormholes may be dependent on the actual geometry of the baby universe. We
cannot fully specify the geometry of the baby universe from the first
principles of quantum gravity. There is a canonical and minimal choice for
such a baby universe; starting from the original system $|{\Psi}_{M}\rangle$,
we prepare a copy of it $|\tilde{\Psi}_{M}\rangle$, and regard it as a
purifier $|M\rangle_{BU}=|\tilde{\Psi}_{M}\rangle_{\text{Puri.}}$. Then the
expression (2.6) becomes

$\sum_{M}\sqrt{p_{M}}|\Psi_{M}\rangle_{BH\cup
R}|\tilde{\Psi}_{M}\rangle_{\text{Puri.}}.$ (2.14)

The existence of the boundaries in the original system $|{\Psi}_{M}\rangle$
implies that purifier $|M\rangle_{BU}=|\tilde{\Psi}_{M}\rangle_{\text{Puri.}}$
should also have boundaries. More generally, there is a possibility that we
may choose the multiple copies of the original system as the baby universe
$|M\rangle_{BU}=|\tilde{\Psi}_{M}\rangle_{\text{Puri.}}^{\otimes n}$ and
further choose their linear combinations as that. Again from ER=EPR this
entanglement between the two spacetimes implies the existence of the wormhole
connecting two island regions for two spacetimes. This wormhole will affect
the non-perturbative physics of this system. Note that the more the number of
copies of the original spacetime increase, the more the effects from wormholes
are topologically suppressed.

##  3 Gauss Law modified by the Baby Universe

In this section, we discuss the physical consequences of the existence of the
baby universe sector introduced in the last section, which accommodates fine-
grained information of the system. We are mainly interested in how the baby
universe helps to recover information of the black hole interior from Hawking
radiation. We will also briefly mention the relation between our discussion
and the paradox raised in the recent paper Geng:2021hlu .

Before doing so, let us present a remark. In the light of AdS/CFT
correspondence, the introduction of an additional boundary, i.e., the boundary
of the baby universe sounds puzzling, because AdS/CFT is the correspondence
between a theory of full quantum gravity in the bulk and a (non gravitating)
CFT on the boundary. This means that in principle, all the details of the bulk
quantum gravity Hilbert space can be read off from the single CFT Hilbert
space. Therefore, we do not need the second copy of the CFT, as we did in the
previous section, which results in the baby universe sector.

Nevertheless, we are forced to do so, because we are sticking to a semi-
classical description of the system. Then, to restore fine-grained information
within the semi-classical regime, we need to introduce an auxiliary system and
regard the new degrees of freedom as a part of the radiation degrees of
freedom after the Page time. If we do not do this, this restriction amounts to
that on the boundary, we are only accessible to a sub-Hilbert space $H_{{\rm
coarse}}$ which characterizes coarse-grained degrees of freedom. To
incorporate the rest of the CFT Hilbert space, which we term $H_{{\rm fine}}$
just because it describes fine-grained degrees of freedom, we need to
introduce a second copy of the CFT Hilbert space, and accommodate $H_{{\rm
fine}}$ to it.

The full Hilbert space on the single boundary is obtained by gluing two
asymptotic boundaries of the spacetime. In the resulting bulk spacetime, there
are two homologically inequivalent paths, both of which connect a point in the
interior of the black hole (and belong to the island region) to the boundary
of the spacetime (see figure 3). The first path is the trivial one (the blue
line in figure 3 ), which entirely lies within the original spacetime. This
path necessarily intersects with the entanglement wedge of the black hole.
However, in the presence of the baby universe, there is a second path which
does not cross the entanglement wedge of the black hole. Instead, it crosses
the Einstein-Rosen bridge connecting the original spacetime to the baby
universe, and reaches the second asymptotic boundary which accommodates fine-
grained degrees of freedom as in the green line in figure 3. Since these two
boundaries are in the end glued together, it connects the island region and
the conformal boundary, without passing through the entanglement wedge of the
black hole.

Figure 3: Schematic picture of the geometry of the AdS black hole coupled the
bath CFT (left Penrose diagram) and the baby universe geometry (right red
Penrose diagram) connected by the Einstein-Rosen bridge (transparent green
shaded region), corresponding to the state (2.14). After the Page time, the
fine-grained Hawking radiation $\mathbf{R}$ is the union of the Hawking
radiation $R$ (violet region) and the baby universe $BU$ (red region). We
regard the above spacetime describing this union by gluing two distinct
asymptotic boundary regions $BU$ and $R$ . The island region $I$ is connected
to the fine-grained Hawking radiation $R\cup BU$ through two paths, path 1 and
2. The path 1 (thick blue dotted line) intersects with the entanglement wedge
of the black hole $BH$ (orange shaded region), but the path 2 (thick green
dotted line) does not intersect with that.

###  3.1 The modification of Gauss Law

In the presence of the baby universe sector which has its own asymptotic
boundary, the gravitational Gauss law is inevitably modified. Let $\Sigma$ be
a time slice of the spacetime, then the gravitational Gauss law relates the
expectation value of the bulk stress energy tensor $\langle T_{bulk}\rangle$
to the boundary energy $H_{\partial}[h]$ (holographic stress energy tensor)

$\langle T_{bulk}\rangle=H_{\partial}[h].$ (3.1)

Here the boundary energy $H_{\partial}[h]$ is explicitly given by the
integration of the ADM current $J^{i}$ over the conformal boundary
$\partial\Sigma$ Chowdhury:2021nxw ,

$\displaystyle H_{\partial}[h]$
$\displaystyle\equiv\frac{1}{2\kappa^{2}}\int_{\partial\Sigma}d^{d-1}x\sqrt{g}\;n_{i}J^{i}\qquad(\kappa=\sqrt{8\pi
G_{N}}),$ (3.2)

where $n_{i}$ is the normal vector to the boundary $\partial\Sigma$, and the
ADM current $J^{i}$ is defined by

$J_{i}\equiv
N\nabla^{j}\left(h_{ij}-hg^{0}_{ij}\right)-\nabla^{j}N\left(h_{ij}-hg^{0}_{ij}\right)$
(3.3)

under the ADM decomposition

$ds^{2}=-N^{2}dt^{2}+g_{ij}\left(dx^{i}+N^{i}dt\right)\left(dx^{j}+N^{j}dt\right),$
(3.4)

and the expansion from the background metric $g_{ij}=g^{0}_{ij}+\kappa
h_{ij}$. More precisely, (3.1) is a perturbative version of the gravitational
Gauss law which can be derived from the full Hamiltonian constraint

$\displaystyle\mathcal{H}[\pi_{ij},g_{ij}]=2\kappa^{2}g^{-1}\left(g_{ij}g_{kl}\pi^{ik}\pi^{jl}-\frac{1}{d-1}\left(g_{ij}\pi^{ij}\right)^{2}\right)-\frac{1}{2\kappa^{2}}(R-2\Lambda)+\mathcal{H}^{\text{matter}}=0,$
(3.5)

where $g_{ij}$ is the metric on the Cauchy slice, $\pi_{ij}$ is the conjugate
momentum, and $\mathcal{H}^{\text{matter}}$ is the matter Hamiltonian density.
Expanding (3.5) from the background metric, $g_{ij}=g^{0}_{ij}+\kappa h_{ij}$,
then look at the second order of the expansion gives (3.1). Details of the
derivation can be found, for example in Chowdhury:2021nxw . $H_{\partial}[h]$
should be understood as the change of the mass of the black hole,
$H_{\partial}[h]=M_{BH}[g+h]-M_{BH}[g]$ due to the back reaction from the bulk
stress energy tensor, $\langle T_{bulk}\rangle$.

In the paper Geng:2021hlu , it was argued that the gravitational Gauss law
provides an interesting puzzle on the island formula. Suppose we act a local
operation on a state on the island region. Since the information of the island
region is encoded in the Hilbert space of Hawking radiation $H_{R}$, this
operation can be regarded as a local operation on $H_{R}$. This operation
changes the expectation value of the bulk stress energy tensor. Then the
gravitational Gauss law relates this change of $\langle T_{bulk}\rangle$ on
the island region behind the horizon to the change of the boundary energy
$H_{\partial}$. This means that any change on the island region, no matter how
it is small, is always detectable from the conformal boundary
$\partial\Sigma$. However, this sounds troublesome because $\partial\Sigma$
belongs to the entanglement wedge of the black hole. For instance, this
implies that in the bipartite system $H_{R}\otimes H_{BH}$, a local operation
on $H_{R}$ can change the state of $H_{BH}$.

The above paradox is naturally resolved, once we take into account the effects
of the baby universe sector which admits the new boundary (see figure 3). In
the presence of this new part of the spacetime, the gravitational Gauss law
must be modified as

$\langle T_{bulk}\rangle=H_{\partial BH}[h]+H_{\partial BU}[h],$ (3.6)

where we denote $H_{\partial BH}[h]$ by the boundary energy of the original
spacetime with the black hole, and similarly $H_{\partial BU}[h]$ is the
boundary energy of the baby universe.

This form of the gravitational Gauss law immediately implies that, in the
presence of the baby universe, operations on the island region need not to be
detected on the conformal boundary of the black hole. In other words, $\langle
T_{bulk}\rangle\neq 0$ does not necessarily imply $H_{\partial BH}[h]\neq 0$.
Rather, it is natural to relate $\langle T_{bulk}\rangle$ on the island region
to the boundary energy of the baby universe $H_{\partial BU}[h]$ because the
island region is encoded to the Hilbert space of fine-grained Hawking
radiation $H_{\mathbf{R}}=H_{R}\otimes H_{BU}$. Indeed, the island region
encodes fine-grained information of Hawking radiation after the Page time, so
from the boundary point of view such bulk operations on this region should be
encoded in the fine-grained part of the CFT Hilbert space, which coincides
with the boundary Hilbert space of the baby universe.

Another way to put this is the following. Let us consider putting a local
operator in the spacetime. The gravitational Gauss law implies that by
measuring the total flux for an appropriate closed surface we can know the
“mass” of the particle within the closed surface. The non-perturbative
gravitational effect from the wormhole makes the measurement of the flux
highly non-trivial. The wormhole can release some part of the flux of the
original spacetime into the purifier (see figure 5). Here we note that since
in our setup the baby universe has boundaries, flux lines can end on the
boundaries of the baby universe as figure 5. Namely, in measuring the total
flux, we also need to consider the purifier (right spacetime of figure 5) or
equivalently the baby universe in addition to the original spacetime (left
spacetime of figure 5). By the usual gravitational Gauss law, if we just
measure the flux of the original spacetime only (left spacetime of figure 5),
then we cannot specify the exact mass. The modification is not visible within
the coarse-grained precision. However, without the modification, we may
encounter many problems, e.g., violation of the conservation law.

There are several other implications of the generalized Gauss law (3.6) as
well. First, the existence of the baby universe boundary energy term indicates
that the gravitational Gauss law does not precisely hold within the original
black hole spacetime, $\langle T_{bulk}\rangle\neq H_{\partial BH}[h]$ in
general. For instance, one way to think about the generalized Gauss law of the
form (3.6) is, it relates the spectrum of the fine-grained part $H_{\partial
BU}[h]$ to the coarse-grained part $H_{\partial BH}[h]$. We expect that the
fine-grained part is discrete, and the typical differences between two nearest
energy eigenvalues are of order $e^{-S_{BH}}$. This forces the coarse-grained
part also discrete, which is necessary for unitary time evolution.

Let us estimate the magnitude of the violation of the gravitational Gauss law
in the black hole spacetime. In order to obtain a unitary time evolution of an
evaporating black hole, we need non-perturbative effects of order
$e^{-S_{BH}}$, where $S_{BH}$ is the entropy of the black hole. This means
that we need fine-grained states in a small energy window of order
$e^{-S_{BH}}$ , thus $H_{\partial BU}$ is of the same order. This leads us to
the conclusion that

$\langle T_{bulk}\rangle-H_{\partial BH}[h]=O(e^{-S_{BH}}),$ (3.7)

i.e., the gravitational Gauss law is violated only non-perturbatively.

Figure 4: Schematic picture of the geometry of the AdS black hole coupled the
bath CFT (left Penrose diagram) and their copy (right Penrose diagram)
connected to the original spacetime through the wormhole (blue region),
corresponding to the state (2.14). The local operator $\phi$ in the island
(cyan dot) can be gravitationally dressed with a gravitational Wilson line
$W_{\text{gravity}}(P,P_{\text{puri.}})$ (green line) which ends on the baby
universe (right Penrose diagram) without intersecting the entanglement wedge
of the original black hole degrees of freedom (orange shaded region).  Figure
5: Schematic picture of flux lines on the geometry corresponding to (2.14) (,
similar to the figure discussed in Akers:2019nfi ). The dotted lines are
horizons. A local operator $\phi$ (red dot) is put at the original spacetime
(left region). The two spacetimes are connected by the Einstein-Rosen bridge
(blue region). Some of flux lines (orange lines) escape into the other
spacetime (right red region) through the Einstein-Rosen bridge .

We should emphasize that such a baby universe is different from those
appearing by cutting Euclidean wormholes into half, in the semi-classical
gravitational path integral. Such a baby universe is always closed and does
not have any asymptotic boundary. Such a closed universe, corresponds to an
additional factor of the von Neumann algebra of the CFT Gesteau:2020wrk . On
the contrary to this, our baby universe has an asymptotic boundary to encode
the fine-grained information of the state. It would be interesting to further
investigate the relation between the two.

We also speculate the realization of fine-grained degrees of freedom in terms
of a baby universe with a boundary has an interesting application to the
physic of a closed universe. Sometimes it is argued that the Hilbert space of
such a closed universe is one-dimensional McNamara:2020uza ; Gesteau:2020wrk
444Sometimes this problem in $d$($\geq 4$)-dimensional spacetime is called the
baby universe hypothesis McNamara:2020uza ., because in the absence of
boundary, the left hand of the gravitational Gauss law (3.1) is always zero,
so any operations are not allowed at all. However, as we saw above, one way to
obtain its fine-grained description is to connect it to an open baby universe
with a boundary. Then the generalized Gauss law (3.6) does allow operations on
the baby universe boundary only. It would be interesting to explore further
implications of the observation.

###  3.2 Comment on gravitational dressing

In a theory with dynamical gravity, a local operator is not physical, since it
is not diffeomorphism invariant. One way to make it diffeomorphism invariant
is to connect the local point $P$ to a point $P_{\partial}$ on the boundary,
via a gravitational Wilson line, i.e.,
$\phi(P)\rightarrow\phi(P)W_{\text{gravity}}(P,P_{\partial})$. This
prescription is called gravitational dressing. In Geng:2021hlu  it was argued
that such a gravitational dressing of a local operator on the island region
leads to an inconsistency of the island prescription. This is because the
relevant gravitational Wilson line connects a point on the island to a point
on the conformal boundary of the AdS black hole. However, this sounds
puzzling, because whereas the island prescription asserts an operator on the
island region locally acts on the radiation Hilbert space, the gravitational
Wilson line attached to it enters the entanglement wedge of the black hole,
thus it does change the state of $H_{BH}$.

In our point of view, the above paradox is naturally resolved, since in the
presence of the baby universe with a boundary, the gravitational Wilson line
can end on this (see figure 4). Furthermore since this new boundary belongs to
the radiation degrees of freedom after the Page time, it is still an operator
on the radiation Hilbert space, even after the gravitational dressing.

##  4 Discussion

In this paper, we studied a partially fine-grained description of semi-
classical evaporating black holes, by introducing an auxiliary system called a
baby universe. We argued that in usual consistent long-range gravitational
theories, the gravitational Gauss law must be modified by the baby universe
connected to the original spacetime, and when there is an island, this
modification is crucial to get results consistent with the idea of the
entanglement wedge reconstruction.

It would be interesting to study concrete geometric models, to understand
further detailed properties of this system. A class of the candidate
geometries is the multi-boundary wormhole solution of three-dimensional
Einstein gravity with a negative cosmological constant. It is convenient to
use the coordinates in which the metric of $AdS_{3}$ takes the following form

$ds^{2}=-dt^{2}+\cos^{2}td\Sigma_{2}^{2},$ (4.1)

where $d\Sigma_{2}^{2}$ denotes the metric of two-dimensional hyperbolic
space. These multi boundary wormholes are constructed by taking appropriate
quotient of the hyperbolic space by the isometry group $SL(2,R)\times
SL(2,R)$. Such a geometry has multiple conformal boundaries, on each of which
we can define a CFT Hilbert space. In each asymptotic region, there is a
horizon, whose area counts the number of degrees of freedom in the boundary
Hilbert space. For simplicity, below let’s consider such a geometry with three
asymptotic boundaries. These three boundaries represent the Hilbert space of
Hawking radiation $H_{R}$, the black hole $H_{{\rm BH}}$, and the baby
universe $H_{{\rm BU}}$. Thus, one can identify the horizon area of each
asymptotic region with the entanglement entropy of each Hilbert space computed
in (A.8), (A.12), (A.16) in Appendix A. The region behind these horizons is
identified with the Einstein-Rosen bridge which connects the original black
hole with the baby universe, discussed in the body of this paper. The
geometric description manifests the following entanglement structure of (2.6).
When $k={\rm dim}H_{R}$ is small, which models the beginning of the black hole
evaporation, this system is almost a bipartite in which $H_{{\rm BH}}$ and
$H_{{\rm BU}}$ are entangled. As we increase $k$, the cross section of the ER
bridge gets larger, and at sufficiently late times $k\gg 1$, $H_{{\rm BU}}$
becomes mostly entangled with the radiation Hilbert space $H_{{\rm R}}$. This
means that the state (2.6) is reconstructable from the two Hilbert spaces
$H_{R}$ and $H_{{\rm BU}}$.

## Acknowledgement

TU thanks Kanato Goto,Yuka Kusuki, Yasunori Nomura, Kotaro Tamaoka and Zixia
Wei for useful discussions in the related projects. The work of NI was
supported in part by JSPS KAKENHI Grant Number 18K03619. TU was supported by
JSPS Grant-in-Aid for Young Scientists 19K14716. NI and TU were also supported
by MEXT KAKENHI Grant-in-Aid for Transformative Research Areas A “Extreme
Universe” No.21H05184.

##  Appendix A The von Neumann Entropy of the naive Hawking Radiation, the
Black hole and the Baby Universe

In this appendix, we give von Neumann entropies of various subsystems for the
states (2.6), (2.3) by using the relationship (2.5). In particular, we
calculate the von Neumann entropies of three cases: (i) the naive Hawking
radiation or the union of the black hole and the baby universe,
$S_{\mathrm{vN}}[\rho_{R}]=S_{\mathrm{vN}}[\rho_{BH\cup BU}]$; (ii) the naive
Hawking radiation and the baby universe or the black hole
$S_{\mathrm{vN}}[\rho_{R\cup BU}]=S_{\mathrm{vN}}[\rho_{BH}]$; (iii) the baby
universe or the union of the black hole and the naive Hawking radiation
$S_{\mathrm{vN}}[\rho_{BU}]=S_{\mathrm{vN}}[\rho_{BH\cup R}]$.

Before starting the calculations, we note that the pure state (2.6) and the
mixed state (2.3) are related by tracing out the baby universe degrees of
freedom

$\displaystyle\rho_{BH\cup R}$
$\displaystyle=\operatorname{tr}_{BU}\left[|\Phi\rangle\langle\Phi|_{BH\cup
R\cup BU}\right]$ (A.1)
$\displaystyle=\sum_{M}p_{M}|\Psi_{M}\rangle\langle\Psi_{M}|_{BH\cup R}.$

For accuracy, we explicitly give the probability distribution $p_{M}$ by
(e.g., Kudler-Flam:2021alo ; Kudler-Flam:2021rpr )

$\displaystyle p_{M}$
$\displaystyle=\left(\frac{\mathcal{N}k}{\pi}\right)^{\mathcal{N}k}\exp\left(-\mathcal{N}k\operatorname{tr}(C^{M}C^{M\dagger})\right),$
(A.2)

and this probability distribution is normalized

$\displaystyle\sum_{M}p_{M}\to$
$\displaystyle\left(\frac{\mathcal{N}k}{\pi}\right)^{\mathcal{N}k}\int\prod_{i,j=1}^{\mathcal{N}}\prod_{\alpha,\beta=1}^{k}dC^{M}_{i\alpha}dC^{\dagger
M}_{\beta
j}\;\exp\left(-\mathcal{N}k\operatorname{tr}(C^{M}C^{M\dagger})\right)$ (A.3)
$\displaystyle=1$

and gives the relationship (2.5). Although we explicitly give the probability
distribution, in calculating the entropies below, we do not use the explicit
form (A.2), but the relationship (2.5).

###  A.1 The entropy of the naive Hawking radiation
$S_{\mathrm{vN}}[\rho_{R}]=S_{\mathrm{vN}}[\rho_{BH\cup BU}]$

To get the von Neumann entropy of the naive Hawking radiation $R$, we consider
the reduced density matrix for the naive Hawking radiation. It is given by

$\displaystyle\rho_{R}$
$\displaystyle=\sum_{M}p_{M}\operatorname{tr}_{BH}\left[|\Psi_{M}\rangle\langle\Psi_{M}|_{BH\cup
R}\right]$ (A.4) $\displaystyle=\sum_{M}p_{M}\,\rho_{(M)R}$
$\displaystyle\equiv\langle\rho_{(M)R}\rangle_{M},$

where in the second line we defined the reduced density matrix

$\displaystyle\rho_{(M)R}$
$\displaystyle=\operatorname{tr}_{BH}\left[|\Psi_{M}\rangle\langle\Psi_{M}|_{BH\cup
R}\right]$ (A.5)
$\displaystyle=\sum_{i=1}^{\mathcal{N}}\sum_{\alpha,\beta=1}^{k}C^{M}_{i\alpha}C^{\dagger
M}_{\beta i}|\alpha\rangle\langle\beta|_{R}$

and in the last line to emphasize the ensemble average of the reduced density
matrix $\rho_{(M)R}$ we introduced the notation
$\langle\rho_{(M)R}\rangle_{M}$ defined by the second line. We note that the
average operation is given by the relationship (2.5) explicitly.

Next we consider the Rényi entropy for the reduced density matrix

$\displaystyle\operatorname{tr}_{R}\rho_{R}^{n}$
$\displaystyle=\sum_{M_{1},M_{2},\cdots,M_{n}}p_{M_{1}}p_{M_{2}}\cdots
p_{M_{n}}\operatorname{tr}_{R}[\rho_{(M_{1})R}\;\rho_{(M_{2})R}\cdots\rho_{(M_{n})R}]$
(A.6) $\displaystyle=\sum_{M_{1},M_{2},\cdots,M_{n}}p_{M_{1}}p_{M_{2}}\cdots
p_{M_{n}}\sum_{i_{1},i_{2},\cdots,i_{n}=1}^{\mathcal{N}}\sum_{\alpha_{1},\alpha_{2},\cdots,\alpha_{n}=1}^{k}C^{M_{1}}_{i_{1}\alpha_{1}}C^{\dagger
M_{1}}_{\alpha_{2}i_{1}}C^{M_{2}}_{i_{2}\alpha_{2}}C^{\dagger
M_{2}}_{\alpha_{3}i_{2}}\cdots C^{M_{n}}_{i_{n}\alpha_{n}}C^{\dagger
M_{n}}_{\alpha_{1}i_{n}}$
$\displaystyle=\sum_{i_{1},i_{2},\cdots,i_{n}=1}^{\mathcal{N}}\sum_{\alpha_{1},\alpha_{2},\cdots,\alpha_{n}=1}^{k}\langle
C^{M_{1}}_{i_{1}\alpha_{1}}C^{\dagger
M_{1}}_{\alpha_{2}i_{1}}\rangle_{M_{1}}\langle
C^{M_{2}}_{i_{2}\alpha_{2}}C^{\dagger
M_{2}}_{\alpha_{3}i_{2}}\rangle_{M_{2}}\cdots\langle
C^{M_{n}}_{i_{n}\alpha_{n}}C^{\dagger M_{n}}_{\alpha_{1}i_{n}}\rangle$
$\displaystyle=\sum_{i_{1},i_{2},\cdots,i_{n}=1}^{\mathcal{N}}\sum_{\alpha_{1},\alpha_{2},\cdots,\alpha_{n}=1}^{k}\frac{1}{(k\mathcal{N})^{n}}\delta_{i_{1}i_{1}}\delta_{\alpha_{1}\alpha_{2}}\delta_{i_{2}i_{2}}\delta_{\alpha_{2}\alpha_{3}}\cdots\delta_{i_{n}i_{n}}\delta_{\alpha_{n}\alpha_{1}}$
$\displaystyle=\frac{1}{k^{n-1}},$

where in the third line we distinguished the labels $M_{1},\cdots,M_{n}$, take
the ensemble averages for factors, which have the same label $M_{s}$, and in
the forth line we used the relationship (2.5).

Therefore we obtain the von Neumann entropy (2.7)

$\displaystyle S_{\mathrm{vN}}[\rho_{R}]$
$\displaystyle=S_{\mathrm{vN}}[\langle\rho_{(M)R}\rangle_{M}]$ (A.7)
$\displaystyle=-\lim_{n\to 1}\partial_{n}\operatorname{tr}_{R}\rho_{R}^{n}$
$\displaystyle=\log k.$

This von Neumann entropy is coincides with the Hawking’s result, and it
implies the information paradox. We note that this von Neumann entropy is
equal to that of the the union of the black hole and the baby universe $BH\cup
BU$

$\displaystyle S_{\mathrm{vN}}[\rho_{BH\cup BU}]$
$\displaystyle=S_{\mathrm{vN}}[\rho_{R}]$ (A.8) $\displaystyle=\log k.$

###  A.2 The entropy of the naive Hawking radiation and the baby universe
$S_{\mathrm{vN}}[\rho_{R\cup BU}]=S_{\mathrm{vN}}[\rho_{BH}]$

To get the von Neumann entropy of the naive Hawking radiation and the baby
universe $R\cup BU(=\mathbf{R})$, we consider the following reduced density
matrix

$\displaystyle\rho_{R\cup BU}$
$\displaystyle=\operatorname{tr}_{BH}\left[|\Phi\rangle\langle\Phi|_{BH\cup
R\cup BU}\right]$ (A.9)
$\displaystyle=\sum_{MN}\sqrt{p_{M}p_{N}}\left(\operatorname{tr}_{BH}|\Psi_{M}\rangle\langle\Psi_{N}|_{BH\cup
R}\right)\otimes|M\rangle\langle N|_{BU}$
$\displaystyle=\sum_{MN}\sqrt{p_{M}p_{N}}\sum_{i=1}^{\mathcal{N}}\sum_{\alpha,\beta=1}^{k}C^{M}_{i\alpha}C^{\dagger
N}_{\beta i}|\alpha\rangle\langle\beta|_{R}\otimes|M\rangle\langle N|_{BU}.$

As in the previous case, we consider the Rényi entropy

$\displaystyle\operatorname{tr}_{R\cup BU}\rho_{R\cup BU}^{n}$
$\displaystyle=\sum_{M_{1},M_{2},\cdots,M_{n}}p_{M_{1}}p_{M_{2}}\cdots
p_{M_{n}}$ (A.10) $\displaystyle\hskip
56.9055pt\times\operatorname{tr}_{R}\left[(\operatorname{tr}_{BH}|\Psi_{M_{1}}\rangle\langle\Psi_{M_{2}}|)(\operatorname{tr}_{BH}|\Psi_{M_{2}}\rangle\langle\Psi_{M_{3}}|)\cdots(\operatorname{tr}_{BH}|\Psi_{M_{n}}\rangle\langle\Psi_{M_{1}}|)\right]$
$\displaystyle=\sum_{M_{1},M_{2},\cdots,M_{n}}p_{M_{1}}p_{M_{2}}\cdots
p_{M_{n}}\sum_{i_{1},i_{2},\cdots,i_{n}=1}^{\mathcal{N}}\sum_{\alpha_{1},\alpha_{2},\cdots,\alpha_{n}=1}^{k}C^{M_{1}}_{i_{1}\alpha_{1}}C^{\dagger
M_{2}}_{\alpha_{2}i_{1}}C^{M_{2}}_{i_{2}\alpha_{2}}C^{\dagger
M_{3}}_{\alpha_{3}i_{2}}\cdots C^{M_{n}}_{i_{n}\alpha_{n}}C^{\dagger
M_{1}}_{\alpha_{1}i_{n}}$
$\displaystyle=\sum_{i_{1},i_{2},\cdots,i_{n}=1}^{\mathcal{N}}\sum_{\alpha_{1},\alpha_{2},\cdots,\alpha_{n}=1}^{k}\langle
C^{\dagger
M_{1}}_{\alpha_{1}i_{n}}C^{M_{1}}_{i_{1}\alpha_{1}}\rangle_{M_{1}}\langle
C^{\dagger
M_{2}}_{\alpha_{2}i_{1}}C^{M_{2}}_{i_{2}\alpha_{2}}\rangle_{M_{2}}\langle
C^{\dagger
M_{3}}_{\alpha_{3}i_{2}}C_{i_{3}\alpha_{3}}^{M_{3}}\rangle_{M_{3}}\cdots\langle
C^{\dagger
M_{n}}_{\alpha_{n}i_{n-1}}C^{M_{n}}_{i_{n}\alpha_{n}}\rangle_{M_{n}}$
$\displaystyle=\sum_{i_{1},i_{2},\cdots,i_{n}=1}^{\mathcal{N}}\sum_{\alpha_{1},\alpha_{2},\cdots,\alpha_{n}=1}^{k}\frac{1}{(k\mathcal{N})^{n}}\delta_{i_{n}i_{1}}\delta_{\alpha_{1}\alpha_{1}}\delta_{i_{1}i_{2}}\delta_{\alpha_{2}\alpha_{2}}\cdots\delta_{i_{n-1}i_{n}}\delta_{\alpha_{n}\alpha_{n}}$
$\displaystyle=\frac{1}{\mathcal{N}^{n-1}},$

where in the fourth line we used the relationship (2.5), and we note that
$\mathcal{N}=e^{S_{BH}}$.

From this Rényi entropy, we get the von Neumann entropy of the union of the
naive Hawking radiation and the baby universe (2.9)

$\displaystyle S_{\mathrm{vN}}[\rho_{R\cup BU}]$ $\displaystyle=-\lim_{n\to
1}\partial_{n}\operatorname{tr}_{R\cup BU}\rho_{R\cup BU}^{n}$ (A.11)
$\displaystyle=\log\mathcal{N}$ $\displaystyle=S_{BH}.$

This von Neumann entropy is also equal to that of the black hole $BH$

$\displaystyle S_{\mathrm{vN}}[\rho_{BH}]$
$\displaystyle=S_{\mathrm{vN}}[\rho_{R\cup BU}]$ (A.12)
$\displaystyle=S_{BH}.$

###  A.3 The entropy of the baby universe
$S_{\mathrm{vN}}[\rho_{BU}]=S_{\mathrm{vN}}[\rho_{BH\cup R}]$

To get the von Neumann entropy of the baby universe $BU$, we consider the
following reduced density matrix,

$\displaystyle\rho_{BU}$ $\displaystyle=\operatorname{tr}_{BH\cup
R}\left[|\Phi\rangle\langle\Phi|_{BH\cup R\cup BU}\right]$ (A.13)
$\displaystyle=\sum_{MN}\sqrt{p_{M}p_{N}}\left(\operatorname{tr}_{BH\cup
R}|\Psi_{M}\rangle\langle\Psi_{N}|_{BH\cup R}\right)\otimes|M\rangle\langle
N|_{BU}$
$\displaystyle=\sum_{MN}\sqrt{p_{M}p_{N}}\sum_{i=1}^{\mathcal{N}}\sum_{\alpha=1}^{k}C^{M}_{i\alpha}C^{\dagger
N}_{\alpha i}|M\rangle\langle N|_{BU}.$

Then we get the Rényi entropy

$\displaystyle\operatorname{tr}_{BU}\rho_{BU}^{n}$
$\displaystyle=\sum_{M_{1},M_{2},\cdots,M_{n}}p_{M_{1}}p_{M_{2}}\cdots
p_{M_{n}}\sum_{i_{1},i_{2},\cdots,i_{n}=1}^{\mathcal{N}}\sum_{\alpha_{1},\alpha_{2},\cdots,\alpha_{n}=1}^{k}C^{M_{1}}_{i_{1}\alpha_{1}}C^{\dagger
M_{2}}_{\alpha_{1}i_{1}}C^{M_{2}}_{i_{2}\alpha_{2}}C^{\dagger
M_{3}}_{\alpha_{2}i_{2}}\cdots C^{M_{n}}_{i_{n}\alpha_{n}}C^{\dagger
M_{1}}_{\alpha_{n}i_{n}}$ (A.14)
$\displaystyle=\sum_{i_{1},i_{2},\cdots,i_{n}=1}^{\mathcal{N}}\sum_{\alpha_{1},\alpha_{2},\cdots,\alpha_{n}=1}^{k}\langle
C^{\dagger
M_{1}}_{\alpha_{n}i_{n}}C^{M_{1}}_{i_{1}\alpha_{1}}\rangle_{M_{1}}\langle
C^{\dagger
M_{2}}_{\alpha_{1}i_{1}}C^{M_{2}}_{i_{2}\alpha_{2}}\rangle_{M_{2}}\langle
C^{\dagger
M_{3}}_{\alpha_{2}i_{2}}C_{i_{3}\alpha_{3}}^{M_{3}}\rangle_{M_{3}}\cdots\langle
C^{\dagger
M_{n}}_{\alpha_{n-1}i_{n-1}}C^{M_{n}}_{i_{n}\alpha_{n}}\rangle_{M_{n}}$
$\displaystyle=\sum_{i_{1},i_{2},\cdots,i_{n}=1}^{\mathcal{N}}\sum_{\alpha_{1},\alpha_{2},\cdots,\alpha_{n}=1}^{k}\frac{1}{(k\mathcal{N})^{n}}\delta_{i_{n}i_{1}}\delta_{\alpha_{n}\alpha_{1}}\delta_{i_{1}i_{2}}\delta_{\alpha_{1}\alpha_{2}}\cdots\delta_{i_{n-1}i_{n}}\delta_{\alpha_{n-1}\alpha_{n}}$
$\displaystyle=\frac{1}{(k\mathcal{N})^{n-1}},$

where in the third line we used the rule (2.5).

From the Rényi entropy, we get the von Neumann entropy of the baby universe

$\displaystyle S_{\mathrm{vN}}[\rho_{BU}]$ $\displaystyle=-\lim_{n\to
1}\partial_{n}\operatorname{tr}_{BU}\rho_{BU}^{n}$ (A.15)
$\displaystyle=\log(k\mathcal{N})$ $\displaystyle=S_{BH}+\log k.$

It is equal to the entropy of the union of the black hole and the Hawking
radiation $BH\cup R$

$\displaystyle S_{\mathrm{vN}}[\rho_{BH\cup R}]$
$\displaystyle=S_{\mathrm{vN}}[\rho_{BU}]$ (A.16) $\displaystyle=S_{BH}+\log
k.$

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