aainstitutetext:  The Institute of Mathematical Sciences, IV Cross Road,
C.I.T. Campus, Taramani, Chennai, India 600113bbinstitutetext:  Homi Bhabha
National Institute, Training School Complex, Anushakti Nagar, Mumbai, India
400094

# Three way information paradox and its resolution using islands

Manish Ramchander  Sitender Pratap Kashyap  Roji Pius<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Black holes possess finite degrees of freedom and thus cannot fuel unbounded
entanglement growth of any system. Instead of the usual information paradox
where the coupled system is one entity, the Hawking radiation, here we couple
a black hole $\chi_{0}$ with two infinite entities: a thermal bath $\chi_{1}$
and an auxiliary system $\chi_{2}$. This produces a novel information paradox
in the sense that gravitational correction to black hole entropy does not rule
out paradoxical growth of $\chi_{1}$ and $\chi_{2}$ entropies. This
immediately raises what kind of resolution such a paradox has, and we address
this question working in the AdS2 JT gravity model, using the island formula,
and ideas of entanglement monogamy. We find the quantum extremal surface that
cures the black hole entropy growth, argue to the nature of how $\chi_{1}$ and
$\chi_{2}$ entropies must behave using monogamy, and derive an island which
satisfies these expectations. A direct consequence of our results is that
gravitation builds entanglement between $\chi_{1}$ and $\chi_{2}$, even though
they start out independently.

##  1 Introduction

Consider a black hole coupled to a system $\chi$. It is well known that when
$\chi$ is radiation, Hawking used semi-classical analysis of treating matter
as quantum and spacetime as classical to show this entropy grows indefinitely,
which leads to a unitarity paradox Hawking:1976bpgc .

In the recent years it has been motivated that Hawking’s saddle point
calculation for entropy did not take into account another saddle which becomes
dominant at later stage of evaporation: the replica wormhole saddle
Penington:2019npb ; Almheiri:2019qdq ; Almheiri:2020cfm ; Almheiri:2019hni .
This idea of looking for additional semi-classical saddle points, has been
embedded in the now well known island formula, which has been used to resolve
Hawking like paradoxes in toy models in 2 dimensions Almheiri:2019yqk ;
Almheiri:2021jwq ; Hartman:2020afg ; Goto:2020rwe  and also in higher
dimensions Almheiri:2020eihd . Let $\mathbf{S}(\chi)$ be the gravitational
entropy of $\chi$, then

$\mathbf{S}(\chi)=\text{min}_{I}\left[\text{ext}_{I}\left(\frac{A(\partial
I)}{4G_{N}}+S(\chi\cup I)\right)\right]$ (1)

where $I$ is a codimension-1 region in the bulk, called an island, and
$A(\partial I)$ is the area of its boundary. The upshot of using island
formula is that at late times, the outgoing radiation does contain the
information about the black hole interior through protocols like bulk
reconstruction in AdS/CFT.

One thing common in all these problems is that black hole was coupled to just
a single system. The resolution came about by introducing an island for system
$\chi$, which changed the entropy profile to one that respects unitarity. In
Almheiri:2021jwq , authors motivated a more involved problem, of coupling the
black hole to two systems $\chi_{1},\chi_{2}$. If we keep the total state
pure, such that for both $\chi_{1},\chi_{2}$ there is an unbounded
entanglement growth, we end up with a novel information paradox.

The usual information paradox involves black hole entropy growth beyond what
is allowed by the number of degrees of freedom present in the black hole. By
purity, solving the paradox for black hole also solves it for $\chi$. In the
three system case however, solving the paradox for black hole does not
describe the nature of entropies $\mathbf{S}(\chi_{1})$ and
$\mathbf{S}(\chi_{2})$. Therefore there is a possibility that entropy growth
of $\mathbf{S}(\chi_{i})$ may still be violating laws of unitarity.

How can one solve such a paradox for a black hole $\chi_{0}$ coupled to
systems $\chi_{1}$ and $\chi_{2}$? Here is the simplification that solves the
black hole part. If one thinks of $\chi=\chi_{1}\cup\chi_{2}$ being coupled to
$\chi_{0}$ with indefinitely growing entropy, this means giving an island to
$\chi$ will solve the black hole entropy growth problem. It also implies that
at late times $\chi_{1}\cup\chi_{2}$ will have access to the information of
black hole interior, which is in line with usual 2 subsystem information
paradoxes. What still unclear is how much access is with $\chi_{1}$ and
$\chi_{2}$ individually, which requires computation of gravitational entropies
$\mathbf{S}(\chi_{1})$ and $\mathbf{S}(\chi_{2})$.

In this paper, we address this problem for two sided black hole ($\chi_{0}$)
in AdS2/JT gravity coupled to a bath ($\chi_{1}$) and an auxiliary system
called journal ($\chi_{2}$); see figure 1. The systems $\chi_{1},\chi_{2}$
will both have infinite degrees of freedom, which will remove the constraining
upper bound on their entropies. We will describe the model in section 2 and
cover how to resolve the paradox for black hole and the appearance of island
for $\chi_{1}\cup\chi_{2}$ as described above in section 3.

Figure 1: Schematic diagram for our setup: two sided AdS black hole $\chi_{0}$
connected with flat spacetimes acting as baths with a sample island. The black
hole temperature $\beta$ matches with the non gravitating bath $\chi_{1}$,
made together with left and right parts, and ensures that total energy of the
black hole and therefore its coarse grained entropy is a constant. The green
lines represent the trajectories taken by boundary observers. The green ball
represents the auxiliary system journal $\chi_{2}$ abstractly. There are two
CFTs in the spacetime, CFT1 in the entirety, and CFT2 restricted to the
gravitating region.

In section 4 we will start by discussing monogamy constraints on
$\mathbf{S}(\chi_{1})$ and $\mathbf{S}(\chi_{2})$, then use the island formula
for journal and the bath to generalize the islands found in Almheiri:2019yqk ;
Almheiri:2021jwq  to this setup. The involved equations are difficult to
handle in full generality, particularly equations for bath islands are coupled
quintics. We overcome this difficulty by looking at solutions in particular
regions using appropriate limits on the equations. An additional difficulty
comes from the nature of the system journal, which will introduce the need for
using boundary condition changing (BCC) operators which naturallly emerge in
BCFTs recknagel_schomerus_2013 ; Sully:2020bcft ; DiFrancescoYellowBook ;
their correlation function with twist fields will be needed for calculating
entropy.111We will follow the spirit of Almheiri:2021jwq  for these
calculations and for reader’s convenience we present an example calculation
involving BCCs in subsection 2.2 Finally, in subsection 4.4 we discuss the
resolution of paradox and its consequences.

As we will explain, the systems $\chi_{1}$ and $\chi_{2}$ are coupled to black
hole in such a way that initially they cannot “see” each other. Given the
gravitational background one would expect that through gravity they will be
able to sense each other’s presence. Our result is in line with this
intuition, as the island we derive builds entanglement between $\chi_{1}$ and
$\chi_{2}$, which is pictorially represented in figure 2.

Figure 2: Pictorial interpretation of our result: the cyan line represents an
entanglement buildup between $\chi_{1}$ and $\chi_{2}$ which were initially
unentangled through gravity.

##  2 Mathematical review

###  2.1 Setup

In this subsection we describe our physical system mathematically, which is
shown in the figure 1. The black hole $\chi_{0}$ at an inverse temperature
$\beta$ is coupled to a bath $\chi_{1}$ at the same temperature. A bath for us
is a quantum mechanical system where we can neglect gravitational effects; for
simplicity we will assume gravity is absent in this region altogether. We will
introduce two non-interacting CFTs called CFT1 and CFT2 with central charges
$c_{1}$ and $c_{2}$ of which CFT1 lives in the full spacetime, and CFT2 only
in the gravitating AdS2 bulk. This is done by setting transparent boundary
condition for CFT1 and reflecting boundary conditions for CFT2. One special
feature for CFT2 however is that its boundary conditions will be random. In
the dual holographic side this arises from “unknown couplings” whose
information is present in a system called journal $\chi_{2}$ Almheiri:2021jwq
. As we mentioned, our framework is of AdS-JT gravity, which comes next.

####  2.1.1 JT gravity

AdS-JT gravity has been used widely Almheiri:2019psf ; Almheiri:2019yqk ;
Almheiri:2021jwq ; Almheiri:2019qdq ; Goto:2020rwe  as a toy model for
studying black holes. Here we review it in the spirit of setting our
conventions; for detailed reviews see Sarosi:2017syk ; Dmitrii:2020ped ;
Engelsoy:2016ibh ; Almheiri:2014mbh . We will now discuss how an eternal two
sided black hole in AdS2 can be obtained as a solution of JT gravity coupled
to conformal matter with reflecting boundary conditions.

The action for JT gravity coupled to a CFT can be split into two parts
$S=S_{0}+S(\mathcal{M},\partial\mathcal{M})$, where $S_{0}$ is topological and
is given by

$S_{0}=\frac{\phi_{0}}{16\pi
G_{N}}\left[\int_{\cal{M}}d^{2}x\sqrt{-g}\,R+2\int_{\partial\cal{M}}K\right].$
(2)

This term provides a large contribution $\frac{\phi_{0}}{4G_{N}}$ to the
entropy of the black hole (in the limit $G_{N}\rightarrow 0$). The dynamical
piece is given by

$S(\mathcal{M},\mathcal{\partial M})=\frac{1}{16\pi
G_{N}}\left[\int_{\cal{M}}d^{2}x\sqrt{-g}\phi\left(R+\frac{2}{\ell^{2}_{AdS}}\right)+2\int_{\partial\cal{M}}\phi_{b}K\right]+I_{\text{CFT}}[g]$
(3)

where $R$ is the Ricci scalar of the spacetime $\mathcal{M}$, $K$ is the trace
of the extrinsic curvature of the boundary $\partial\mathcal{M}$, $\phi_{b}$
is the boundary value of the dilaton field and $\ell_{AdS}$ is the AdS radius
which we will set to 1. The specific manner in which $\partial\mathcal{M}$ is
carved out of pure AdS2 will be called a cutout. This cutout is responsible
for breaking the diffeomorphism invariance of the $S_{0}$ and is intimately
connected to the AdS2 bulk dynamics. Below we will show what cutout gives rise
to a black hole solution. It is important to note that $I_{\text{CFT}}$ does
not couple to dilaton $\phi$. Consequently its bulk equations of motion due to
$\delta_{\phi}S$ variation give $R=-2$ and hence the geometry is locally that
of AdS2.

To describe the boundary, we use the Poincare patch shown in figure 3, whose
metric is

$ds^{2}=\frac{-dt^{2}+dz^{2}}{z^{2}}=\frac{-4dx^{+}dx^{-}}{(x^{+}-x^{-})^{2}}.$
(4)

where $x^{\pm}=t\mp z$. The standard boundary conditions on the cut out are
imposed to be

$g_{uu}|_{bdy}=\frac{1}{\epsilon^{2}}=\frac{1}{z^{2}}\left(-\left(\frac{dt}{du}\right)^{2}+\left(\frac{dz}{du}\right)^{2}\right)|_{bdy}\qquad\phi|_{bdy}=\phi_{b}=\frac{\phi_{r}}{\epsilon}.$
(5)

Here $u$ is the boundary time.

Figure 3: Poincare patch with the right black hole exterior highlighted in
green. The left exterior in beige is outside the Poincare patch, which when
using $y$ coordinates covering the right exterior can be obtained from a $y\to
y+i\beta/2$. The dashed lines represent cutouts. The infinite range of
Poincare time $t$ maps to a $2\pi$ range of the global time. Note that $z$
increases to the left here, and accordingly we choose $x^{\pm}=t\mp z$. The
inclined boundaries including the left tip are reached for $z\to\infty$.

In these coordinates, the boundary is given by $t=f(u),z=\epsilon
f^{\prime}(u)$, where the gluing function $f(u)$ is obtained through the
energy balance as follows. The ADM mass of the gravitating region is

$\displaystyle M(u)=-\frac{\phi_{r}}{8\pi G_{N}}\\{f(u),u\\}.$ (6)

Energy conservation requires that the change in above ADM mass equal the net
flux of energy across the boundary curve,

$\displaystyle\frac{dM(u)}{du}=-\frac{d}{du}\left(\frac{\phi_{r}}{8\pi
G_{N}}\\{f(u),u\\}\right)=T_{y^{+}y^{+}}-T_{y^{-}y^{-}}$ (7)

which, given the stress tensor profile, can be solved for $f(u)$.

Next we see how the gluing map determines the dilaton profile
$\phi(x^{+},x^{-})$. Varying action with respect to metric in the Poincare
patch gives the equations for dilaton:

$\begin{split}2\partial_{x^{+}}\partial_{x^{-}}\phi+\frac{4}{(x^{+}-x^{-})^{2}}\phi&=16\pi
G_{N}\langle T_{x^{+}x^{-}}\rangle\noindent\\\
-\frac{1}{(x^{+}-x^{-})^{2}}\partial_{x^{+}}\left((x^{+}-x^{-})^{2}\partial_{x^{+}}\phi\right)&=8\pi
G_{N}\langle T_{x^{+}x^{+}}\rangle\noindent\\\
-\frac{1}{(x^{+}-x^{-})^{2}}\partial_{x^{-}}\left((x^{+}-x^{-})^{2}\partial_{x^{-}}\phi\right)&=8\pi
G_{N}\langle T_{x^{-}x^{-}}\rangle\noindent\\\ \end{split}$ (8)

We further note that $T_{x^{+}x^{-}}$ can be absorbed into the definition of
$\phi_{0}$ and as such can be dropped from the first equation in (8). With
such a choice a general solution to these equations, upto an $SL(2,R)$
transformation, can be written as

$\displaystyle\phi(x^{+},x^{-})$ $\displaystyle=$
$\displaystyle-\frac{2\pi\phi_{r}}{\beta}\frac{x^{+}+x^{-}}{x^{+}-x^{-}}-\frac{8\pi
G_{N}}{x^{+}-x^{-}}\int_{0}^{x^{-}}dt(x^{+}-t)(x^{-}-t)T_{x^{-}x^{-}}$ (9)
$\displaystyle+\frac{8\pi
G_{N}}{x^{+}-x^{-}}\int_{0}^{x^{+}}dt(x^{+}-t)(x^{-}-t)T_{x^{+}x^{+}}$

and for $T_{x^{-}x^{-}}=0$ along the cutout, the dilaton takes form
Hollowood2020:ipc ; Goto:2020rwe

$\displaystyle\phi(x^{+},x^{-})=-\phi_{r}\left(\frac{2f^{\prime}(y^{+})}{x^{+}-x^{-}}-\frac{2f^{\prime\prime}(y^{+})}{f^{\prime}(y^{+})}\right).$
(10)

In the above we have to think of $x^{+}=f(y^{+})$ i.e. the Poincare
coordinates are related to the Minkowski coordinate by the gluing map.

The field $\phi+\phi_{0}$ is related to the area of the transverse directions
that are integrated out when the action $S$ is viewed as a dimensional
reduction from theories in higher dimensions. If for example the AdS2 arises
from the near horizon geometry of an extremal black hole then $\phi_{0}$
corresponds to the area of the extremal black hole, while $\phi$ denotes
deviations from this extremal value. Consequently the entropy formulae will
have $\phi$ in the area term of the entropy formulae. As we move towards
boundary, dilaton value increases to reflect surface area of a 2 sphere, as in
higher dimensions In this spirit, specifying dilaton specifies the geometry of
associated higher dimensional black hole spacetime.

####  2.1.2 Eternal black hole

One can interpret the interior of Rindler patches in AdS2 as the exterior of a
two sided black hole of temperature $\beta^{-1}$. Now we describe how to make
use of the above equations to couple a black hole solution in the AdS2 to the
Minkowski bath. Because the black hole will be coupled to both on both sides,
we use this left-right symmetry to focus on only one side. We demand that the
black hole is in equilibrium with the bath, there is no net flux and hence
$\frac{dT}{du}=0$ and hence

$\displaystyle\partial_{u}\\{f(u),u\\}=0.$ (11)

A solution that corresponds to a temperature $\frac{1}{\beta}$ is given by

$\displaystyle f(u)=e^{\frac{2\pi u}{\beta}}.$ (12)

Having solved for the gluing function, we can use it to extend the coordinates
$y^{\pm}$ that were earlier defined in the bath region to the gravity region
as well via

$x^{\pm}=f(y^{\pm})=\pm\exp\left(\pm\frac{2\pi}{\beta}y^{\pm}\right).$ (13)

Figure 3 shows the right exteriro where $x^{\pm}=f(y^{\pm})$ is valid as
green, and the left exterior, which can be reached by $y\to y+i\beta/2$, in
beige. Given the map (13), the Poincare metric in (4) becomes

$ds^{2}=-\left(\frac{2\pi}{\beta}\right)^{2}\frac{dy^{+}dy^{-}}{\sinh^{2}\frac{\pi}{\beta}(y^{-}-y^{+})}$

and the dilaton profile using (10) is

$\phi=\frac{2\pi\phi_{r}}{\beta}\frac{1}{\tanh\frac{\pi}{\beta}(y^{-}-y^{+})}.$

We will however be mostly working in Kruskal-Szekeres coordinates

$w^{\pm}=\pm e^{\pm\frac{2\pi
y_{R}^{\pm}}{\beta}}=\pm\left(x^{\pm}_{R}\right)^{\pm 1}\qquad\qquad
w^{\pm}=\mp e^{\mp\frac{2\pi
y_{L}^{\pm}}{\beta}}=\mp\left(x^{\pm}_{L}\right)^{\mp 1}.$ (14)

In this coordinate the metric takes the form

$ds^{2}=\frac{4dw^{-}dw^{+}}{\left(1+w^{-}w^{+}\right)^{2}}.$ (15)

The dilaton profile in the gravitating region in the $w$-coordinates takes the
form

$\phi=\phi_{0}+\frac{2\pi\phi_{r}}{\beta}\frac{1-w^{+}w^{-}}{1+w^{+}w^{-}}.$
(16)

Therefore the location of the singularity is given by
$w^{+}w^{-}=\frac{1}{\theta}$, where
$\theta=\frac{2\pi\phi_{r}-\beta\phi_{0}}{2\pi\phi_{r}+\beta\phi_{0}}$.

Figure 4: Kruskal-Szekeres, or the $w$ coordinates, cover this entire diamond
region, which includes the attached baths. The vertical lines are the AdS2
boundaries and the dotted line is the singularity $w^{+}w^{-}=\theta^{-1}$.
The colours of black hole exteriors in figure 3 are retained here for
reference.

Further, the future horizon of the black hole is at $w^{-}=0$ and past horizon
is at $w^{+}=0$. Finally the location of the physical boundary of the black
hole geometry that is being glued to the bath is given by
$w^{+}w^{-}=-e^{\frac{2\pi\epsilon}{\beta}}\simeq-1$.

####  2.1.3 Auxiliary system journal

Consider a new conformal matter field restricted to AdS2 region, which we call
CFT2, that does not interact with CFT1. This CFT2 has random conformal
boundary conditions which is represented through its state on the AdS space
being mixed:

$\rho_{2}=\sum_{J}P(J)|\psi_{2},J\rangle\langle\psi_{2},J|$

where $|\psi_{2},J\rangle$ represents the state of just CFT2 on AdS2, and $J$
labels its boundary condition.

CFT2 can be purified by an auxiliary system as follows. Consider a Hilbert
space $\mathcal{H}_{J}$, and choose an orthonormal basis
$\\{|J_{1}\rangle,|J_{2}\rangle,\ldots\\}$, which may be countably infinite.
By creating an entangled state

$|\Psi\rangle=\sum_{i}\sqrt{P(J_{i})}|\psi_{2},J_{i}\rangle|J_{i}\rangle$

each $|J_{i}\rangle$ corresponds to a choice of boundary condition for
$\text{CFT}_{2}$ at the physical boundary of the eternal black hole spacetime,
which is what we wanted. Thus $|\Psi\rangle$ is a purification of $\rho_{2}$
with the desired interpretation. We call this auxiliary system journal, like
in Almheiri:2021jwq .

The significance of this esoteric model is that it attempts to answer when UV
details of bulk quantum gravity become important. We briefly touch this point
here for the context; see Almheiri:2021jwq  for more details.

The Hamiltonian for CFT2 changes as $J$ does; consequnetly the time evolution
for different $J$ produces entanglement growth between CFT2 degrees of freedom
and the journal. In Almheiri:2021jwq  this lead to an unbouded logarithmic
growth of the journal entropy; (23). The resolution was an island for the
journal which implied that uncertainty in couplings prevented access to a
region in the interior in the sense of entanglement wedge reconstruction. In
the broader context, it suggested that not knowing the details of the UV
theory may prevent us from doing bulk reconstruction in the black hole
interior.

###  2.2 Method for calculating entropy using twist operators

Entropy of conformal fields can be calculated using twist operator correlation
functions Calabrese:2004qft ; Calabrese:2009dicft ; Cardy:2007ffb . Here we
will mention the basic points of how this is done. The standard formula for
computing matter entropies in curved spacetime is

$S(\chi)=\lim_{n\to 1}\frac{1}{1-n}\ln\hbox{Tr}\left(\rho^{n}_{\chi}\right).$

This is the gravitational entropy for $\chi$ with an empty island. One can
formally write the partition function
$Z_{n}=\hbox{Tr}\left(\rho_{\chi}^{n}\right)$ as a Euclidean path integral
over $n$ copies of the original manifold in a method called replica trick. For
2d conformal matter $Z_{n}$ can be obtained using twist operator correlation
functions. In particular, for a 1+1 dimensional conformal field theory with
matter in the interval $[a,b]$,

$Z_{n}([a,b])=\left\langle\sigma(a)\widetilde{\sigma}(b)\right\rangle$

where $\sigma,\widetilde{\sigma}$ are primary operators of scaling dimension
$\Delta_{n}=\frac{c(n^{2}-1)}{12n}$ and by $a,b$ we mean coordinates at points
$a,b$. In other words, twist operators sit at the end points of the interval
of interest. They act on the orbifold theory defined by replication and
subsequent $\mathbb{Z}_{n}$ quotienting of the original theory. Similar
formula holds for multiple intervals:
$Z_{n}([a,b]\cup[c,d])=\left\langle\sigma(a)\widetilde{\sigma}(b)\sigma(c)\widetilde{\sigma}(d)\right\rangle$.

Now let us make the curved manifold $\mathcal{M}$ on which CFT lives explicit.
Say it has the metric $ds^{2}=\Omega^{-2}{dzd\bar{z}}$ and $dzd\bar{z}$ is the
metric for the complex plane. Then one can do a Weyl transformation to complex
plane,

$\left\langle\sigma(a)\sigma(b)\right\rangle_{\mathcal{M}}=\Omega^{\Delta_{n}}(a)\Omega^{\Delta_{n}}(b)\cdot\left\langle\sigma(a)\sigma(b)\right\rangle_{\mathbb{C}}.$
(17)

If the conformal field in question is in vacuum state, then we can use global
conformal invariance techniques here onwards to compute entropies.

When studying boundary conformal field theories, such as the free boson theory
on upper half plane or the disk, one has to also worry about boundary
conditions on non-asymptotic boundaries. In particular if the boundary
condition $\phi_{J}$ is a random variable described by distribution $P(J)$,
the replica trick calculaton needs $n$ independent random variables, one for
each sheet. For its path integral to make sense, the boundary conditions must
change continuously, therefore gluing of sheets is done using “boundary
condition changing operators”, which we discuss below using an example.

###  2.3 Example calculation using BCC

Figure 5: Left: interval where entropy of CFT with random boundary conditions
is to be calculated. Right: Euclidean picture of how $J$ is the boundary
condition specified for preparing the state $|\beta,J\rangle$ on $EB$. The
density matrix $\rho^{J}_{AB}=\text{Tr}_{EA}|\beta,J\rangle\langle\beta,J|$
may be represented through this figure. The thick line $AB$ represents two
cuts over which boundary conditions aren’t specified, which makes this figure
a representation of an operator.

Let us do a calculation of CFT2 matter entropy in the interval AB in the right
exterior with $B$ touching the boundary, figure 5, left. This was done in
Almheiri:2021jwq  for the entire exterior region. Let us consider the combined
system of journal and boundary CFTs to be

$|\Psi\rangle=\sum_{J}\sqrt{P(J)}|\beta,J\rangle|J\rangle$

where $|\beta,J\rangle$ is the TFD prepared using boundary Hamiltonians
$H_{L}^{J},H_{R}^{J}$. Tracing over the journal gives

$\rho_{\beta}=\sum_{J}P(J)|\beta,J\rangle\langle\beta,J|$

and further tracing on ${EA}$ gives

$\rho_{AB}=\text{Tr}_{EA}\left(\sum_{J}P(J)|\beta,J\rangle\langle\beta,J|\right)=\sum_{J}P(J)\cdot\rho^{J}_{AB}.$
(18)

Mathematically it follows that,

$Z_{n}=\text{Tr}\rho_{AB}^{n}=\int\prod_{i=1}^{n}dJ_{i}P(J_{i})\cdot\text{Tr}_{AB}\left(\prod_{i=1}^{n}\rho_{AB}^{J_{i}}\right)$
(19)

where we’ve made transition to a continuous probability distribution. It is
more convenient however to work with diagrammatic path integrals. For instance
the density matrix (18) may as a path integral may be represented as in figure
5, where $AB$ is actually representating two cuts where further information
can be specified for computing operator products. The replica manifold’s
partition function calculation utilizes these cuts as shown in 6, where the
upper cut is identified with a lower one on the next sheet in a circular
manner for $n=4$. The fact that boundary condition is $J_{i}$ for the $i$th
sheet is represented through colour variation.

Figure 6: Diagramatic representation of trace part of (19) for $n=4$. The
different boundary colours highlight different boundary conditions involved
for each $\rho_{AB}^{J_{i}}$.

Since we have a 2D conformal field theories, we can now use twist operators to
implement the replica manfold structure in the orbifold theory. Infact, here
one of the twist operators is at the boundary which behaves like an identity
operator, so effectively we have just one twist in the bulk. However, we also
need to implement changes in boundary conditions from one sheet to another. To
capture this, we insert a composite operator
$O_{\\{J\\}}=O_{J_{1},J_{2}}\otimes O_{J_{2},J_{3}}\otimes\cdots\otimes
O_{J_{n},J_{1}}$ where $O_{J_{i},J_{i+1}}$ changes the boundary conditions
from $J_{i}$ to $J_{i+1}$. These are primary operators that are inserted on
the boundary and have conformal dimension $h_{b}$. To determine the locations
of operators one needs to tread the replica maniold along the specified
gluings; this orbifold picture is given in figure 7.

Figure 7: In the orbifold picture, we need to place $\sigma$ at point $A$, and
the BCC at $B$. These placement locations become clear by treading the replica
manifold along the gluings specified. Note that the twist operator on boundary
is redundant here, so only one suffices.

The computation is then given by

$\displaystyle\text{Tr}_{AB}\left(\prod_{i=1}^{n}\rho^{J_{i}}_{AB}\right)=\langle\sigma(A)O_{\mathbf{J}}(B)\rangle_{AdS_{2}}.$
(20)

To compute it, one needs to first compute a two point function on a flat disk
by conformally transforming to half plane and using doubling trick. That makes
it a three point function, which can be determined upto a constant. Thereafter
a Weyl transfromation is needed to get on a Euclidean disk of circumference
$\beta/\epsilon$. In $w$ coordinates, this turns out to be

$\left\langle\sigma(w,\bar{w})O_{\mathbf{J}}(v)\right\rangle_{\text{disk}}=c_{n}(\\{J_{i}\\})\left[\frac{2\pi\epsilon}{\beta}\left(\frac{1-|w|^{2}}{|1-w\bar{v}|^{2}}\right)\right]^{h_{b}}$

where $c_{n}(\\{J_{i}\\})$ is a constant that can be neglected for QES
calculations upto leading order.

#### Cross ratio

We will run into four point computations, for which conformal invariance
leaves a cross ratio dependent function $G(\eta)$ undetermined. Here we
discuss its impact on the QES. The entropy in such cases will be of the form

$S=S^{(CI)}+\log G(\eta).$

For a spacelike intervals such as ones involved in entropy calculation, the
cross ratio varies between $0$ and $1$. While at both of these limits, one can
use OPE to get $G(\eta\to 1)=1$, and boundary operator expansion for
$G(\eta\to 0)=g_{n}^{2}$, that does not tell about its derivatives, which are
needed to tell if $G(\eta)$ can shift the QES.

If $\partial_{\eta}\log G(\eta)$ has a singularity in $(0,1)$, then it will
dominate remaining matter entropy terms and directly compete with the
$O(G_{N}^{-1})$ area term hence it cannot be neglected, making the island hunt
a hopeless task without knowing $G(\eta)$ and therefore details of the theory.
Typically studies assume the matter theory to be holographic
Takayanagi:2011hdbcft ; Myers:2020ebb , which allows one to explicitly give
$G(\eta)$ to be of the form

$H(\eta)=\frac{g^{2}}{(1-\eta)^{c_{1}/6}}\Theta(\eta^{*}-\eta)+\eta^{-c_{1}/6}\Theta(\eta-\eta^{*})$

where $\eta^{*}=(1+g^{12/c_{1}})^{-1}$, for which $\partial_{\eta}\log H$ is
regular throughout $(0,1)$. We will assume our matter theories CFT1 and CFT2
obey the same, that is $\partial_{\eta}\log G(\eta)$ is finite, which means
they belong to a superset of such holographic theories. This will allow us to
drop the cross ratio term for finding the island.

##  3 Black hole information paradox and resolution

###  3.1 Combined system before Page time

The two sided black hole starts in a pure state at $u=0$ but with time it
keeps getting entangled with the bath and the journal through CFT1 and CFT2
respectively. Its initial entropy growth is thus given by the matter entropies
of the bulk, and the black hole just provides the fixed curved background.
Because we assume that CFTs don’t interact, the initial black hole entropy is

$\mathbf{S}(\chi_{0})=S_{1}(\Sigma_{u})+S_{2}(\Sigma_{u}).$ (21)

While the term independence may motivate such a split of entropy to be
natural, we actually are making a definite assumption here: that state of the
CFT matter factorizes as $\rho=\rho_{1}\otimes\rho_{2}$, or in words, the two
are unentangled. The entropies $S_{1}(\Sigma_{u})$ and $S_{2}(\Sigma_{u})$, as
we describe below, grow unboundedly, which leads to an information paradox for
the black hole as it only has a finite number of degrees of freedom.

The equation (21) is that statement that CFT1 degrees of freedom in the bulk
and purified by those in the bath, and CFT2 is purified by the journal alone.
Therefore we can use the results for $S_{1}(\Sigma_{u})$ and
$S_{2}(\Sigma_{u})$ from previous works which dealt with them separately. This
setup is visually described in figure 8. In the case of black hole coupled to
the bath with just the transmitting CFT, Almheiri:2019yqk  entropy of the bath
grows linearly

$S_{1}(\Sigma_{u})=\frac{c_{1}}{3}\log\left[\frac{\pi}{3}\cosh\left(\frac{2\pi
u}{\beta}\right)\right].$ (22)

On the other hand, in the case of black hole coupled to the journal
Almheiri:2021jwq , authors considered the free boson as their CFT with
probability distribution $P(J)$ to be a Gaussian of variance $m^{-2}$. As a
result, the journal has uncountably infinite states $|J\rangle$ for
$J\in\mathbb{R}$, and therefore infinite degrees of freedom. The entropy
calculation gives logarithmic growth

$S_{2}(\Sigma_{u})=\frac{c_{2}}{2}\log\left[\frac{1}{2\pi^{2}\alpha^{\prime}m^{2}}\log\left(\frac{\beta}{\epsilon\pi}\cosh\frac{\pi
u}{\beta}\right)\right].$ (23)

Clearly the black hole entropy increases unboundedly and creates a paradox.
When the total
$\mathbf{S}(\chi_{0})=S_{\text{max}}=\phi_{0}/(2G_{N})+\pi\phi_{r}/(4\beta
G_{N})$, which defines the Page time, then we must search for an island for
the rescue Almheiri:2020cfm , because entanglement is monogamous and all of
the black hole’s degrees of freedom are locked.

Figure 8: Schematic description of how under the independent CFT
approximation, the three body system is effectively acting as a four body
system. The system $\chi_{1}\cup\chi^{(1)}_{0}$ is in a pure state here, and
system $\chi_{0}^{(2)}=\cup\chi_{2}$ is also in a pure state. Such states are
the unique ones to satisfy strong subadditivity condition with an equality in
finite dimensional systems Hayden:2004sss .

###  3.2 Black hole after Page time

The matter CFTs are present in a gravitating spacetime, and therefore one
would expect them to eventually get entangled, and in that sense (21) is an
approximation. The island formula says that this approximation breaks down at
Page time by changing the formula for $\mathbf{S}(\chi_{0})$, and at the same
time, entangles the matter CFTs. That will naturally have consequences for the
journal and bath interaction as well, but first we look at how the black hole
entropy changes. 222There is another way to see this using discrete limits of
the systems as follows. Because of total purity, (21) is actually strong
subadditivity (SSA) condition with an equality:
$\mathbf{S}(\chi_{0})=\mathbf{S}(\chi_{1})+\mathbf{S}(\chi_{2})\quad\text{(before
Page time)}$ After Page time, $\mathbf{S}(\chi_{0})$ becomes a constant, and
therefore if right side continues to increase, then it will revert to an
inequality. The only states that satisfy SSA with an equality are the Markov
chain states Hayden:2004sss , which exhibit such a property. Therefore the
independence of CFTs can be equated with a Markov chain state, which is lost
after Page time, like in figure 8.

We saw that $\mathbf{S}(\chi_{0})$, the entanglement of two sided black hole
with remaining system, was increasing linearly. By the Page time, the
$O(G_{N}^{-1})$ entanglement between left and right black holes gets traded
for $O(G^{-1}_{N})$ entanglement between individual black holes and coupled
systems, so that $\mathbf{S}(\chi_{0})\sim\mathbf{S}(L)+\mathbf{S}(R).$ In the
island framework, one can see the same by using island formula for
$\mathbf{S}(\chi_{1}\cup\chi_{2})$ and demanding that island gets bigger with
time, so that matter entropy part can be factorized into left and right.
Therefore, to resolve the paradox, we can directly look at entropy of just
one, say the right black hole, which at the same time will give us
entanglement wedge of the right CFT.

The entanglement wedge of boundary CFT is given by the QES formula. For the
right CFT,

$\mathbf{S}(R)=\text{min}_{Q}\left[\text{ext}_{Q}\left(\frac{A(Q)}{4G_{N}}+S(\Sigma^{R}_{Q})\right)\right]$
(24)

where $\Sigma_{Q}^{R}$ is the codimension-1 surface homologous to the boundary
$R$ such that its boundary is $Q\cup R$. Because of the left-right symmetry
we’ll only need to look at one CFT’s entanglement wedge.

Figure 9: The red interval is $\Sigma_{Q}^{R}$. Here $R$ is the right
observer, moving with coordinates $v^{\pm}=\pm e^{\pm 2\pi u/\beta}$.

Here $A(Q)$ is given by the dilaton (16), and the main task is calculating the
$S(\Sigma_{Q}^{R})$ where $\Sigma_{Q}^{R}$ is the region connecting $R$ and
$Q$, see figure 9. As there is both CFT1 and CFT2 is this region, the matter
entropy on fixed background still splits:

$S(\Sigma_{Q}^{R})=S_{1}(QR)+S_{2}(QR)$

where the associated partition functions are

$Z_{1}(QR)=\left\langle\sigma^{(1)}_{Q}\sigma^{(1)}_{R}(u)\right\rangle,\quad
Z_{2}(QR)=\int
P(J_{i})dJ_{i}\cdot\left\langle\sigma_{Q}^{(2)}O_{R}(u)\right\rangle_{\mathbf{J}}.$

The correlation function for CFT1 is computed easily by Weyl transforming to
the flat Minkowski space that includes the bath and using conformal
invariance. The CFT2 correlation function is computed on the AdS2 disk, which
on doubling becomes a three point function, as was explained in subsection
2.2. Let $w^{\pm}$ be the coordinates for $Q$ and $v^{\pm}$ for $R$. Then, the
entropies are

$S_{1}(QR)=\frac{c_{1}}{6}\log\left(\frac{(v^{-}-w^{-})(w^{+}-v^{+})}{1+w^{-}w^{+}}\right),$
(25)
$S_{2}(QR)=\frac{c_{2}}{2}\log\left[\log\left(\frac{(1+v^{-}w^{+})(1+w^{-}v^{+})\beta}{(1+w^{-}w^{+})\cdot
2\pi\epsilon}\right)\right].$ (26)

Taking derivatives after including the dilaton, with the approximation

$\log\left(\frac{(1+v^{-}w^{+})(1+w^{-}v^{+})\beta}{(1+w^{-}w^{+})\cdot
2\pi\epsilon}\right)\sim\log\frac{\beta}{2\pi\epsilon},$

we get coupled quadratic equations for $w^{\pm}$, after substituting
$v^{\pm}=\pm e^{\pm 2\pi u/\beta}$ with the solution

$w^{\pm}=\pm e^{2\pi u/\beta}\cdot\frac{\beta
G_{N}}{6\pi\phi_{r}}\left({c_{1}+\frac{3c_{2}}{\log\frac{\beta}{2\pi\epsilon}}}\right).$
(27)

Since $w^{\pm}=A_{\pm}e^{\pm 2\pi u/\beta}$ for constants $A_{\pm}$, it is
trivial to see that $S_{1}(QR),S_{2}(QR)$ and (16) are independent of time.
Let us now point out that in (27) the coeffecient of $v^{\pm}$ is just the sum
of coeffecients for QES in Almheiri:2021jwq  and in Almheiri:2019yqk . Now,
taking the limit that $|A_{\pm}|\ll 1$ because $G_{N}\to 0$, one gets

$\mathbf{S}(R)=\frac{\phi_{0}}{4G_{N}}+\frac{2\pi\phi_{r}}{4\beta
G_{N}}+O(1).$

Thus, we find that this QES for right side leads to a saturated black hole
entropy, the saturation value being its thermal entropy. By symmetry the same
follows for the left side. Thus, we conclude that this QES resolves the black
hole paradox satisfactorily.

#### Island for $\chi_{1}\cup\chi_{2}$

Preceding discussion tells what region is reconstructible using $R,L$ and
$\chi_{1}\cup\chi_{2}$. The causal diamond for the red portion in figure 10 is
the entanglement wedge of left and right boundary CFTs. The complement portion
by purity is the island of $\chi_{1}\cup\chi_{2}$, which covers the black hole
interior.

Figure 10: The island for $\chi_{1}\cup\chi_{2}$ is in cyan, and the causal
diamonds of red intervals give the entanglement wedges for left and right
CFTs.

The knowledge of entanglement wedge of $\chi_{1}\cup\chi_{2}$ does not say
anything particular about regions reconstructible using just $\chi_{1}$ or
$\chi_{2}$ alone. To find that we need to compute $\mathbf{S}(\chi_{1})$ and
$\mathbf{S}(\chi_{2})$.

##  4 Islands for bath and journal

###  4.1 Constraints on $\mathbf{S}(\chi_{1})$ and $\mathbf{S}(\chi_{2})$

To proceed, we make two observations. As we’ve seen, the black hole’s energy
is fixed, and as a result its coarse grained entropy is constant in time.
Obligingly, $\mathbf{S}(\chi_{1}\cup\chi_{2})$ has become constant after
fixing the paradox for black hole. We’ll now argue that this can only happen
if $\mathbf{S}(\chi_{1})$ and $\mathbf{S}(\chi_{2})$ are either constant, or
grow at the same rate.

#### Monogamy of Entanglement

Figure 11 schematically describes the situation at Page time. To identify the
entangled parts we’ve split the bath, journal and black hole into two parts
each: $\chi_{i}^{(1)}$ and $\chi_{i}^{(2)}$ for $i=0,1,2$. Think of the
collection of degrees of freedom of $\chi_{1}$ entangled with those of
$\chi_{0}$ as $\chi_{1}^{(2)}$, and likewise for other cases. Note that
subsystems $\chi_{1}^{(1)}$ and $\chi_{2}^{(2)}$, whose degrees of freedom are
not entangled with those of the bath, necessarily have infinite degrees of
freedom.

Now we use monogamy of entanglement. Since $\chi_{1}^{(2)}$ is highly
entangled with $\chi_{0}^{(1)}$, it means that $\chi_{1}^{(1)}$ has low or
zero entanglement with $\chi_{1}^{(2)}$ and if it is not interacting with
$\chi_{2}$ at all, then $\chi_{1}^{(1)}$ by itself is almost pure. Likewise
holds for $\chi_{2}^{(2)}$. Therefore, if through gravity the systems
$\chi_{1}$ and $\chi_{2}$ are able to sense each other, it would mean
$\mathbf{S}(\chi_{1}^{(1)})=\mathbf{S}(\chi_{2}^{(2)})$. As we argued,
$\chi_{i}^{(j)}$ for $j=1,2$ have negligible entanglement, thus it follows
easily that $\mathbf{S}(\chi_{1})-\mathbf{S}(\chi_{2})$ is a constant after
Page time.

Figure 11: Before Page time, we noted that black hole got partly entangled
with both $\chi_{1}$ and $\chi_{2}$, and afterwards the black hole QES has
shown that this entanglement persists. Here we’ve split the system $\chi_{i}$
into $\chi_{i}^{(1)}\cup\chi_{i}^{(2)}$ for $i=0,1,2$ to highlight that by
Page time, $\chi_{0}^{(1)}$ is highly entangled with $\chi_{1}^{(2)}$ and
$\chi_{0}^{(2)}$ with $\chi_{2}^{(1)}$, which leaves $\chi_{1}^{(1)}$ and
$\chi_{2}^{(2)}$ in almost pure states. Therefore if through gravity
$\chi_{1}^{(1)}$ and $\chi_{2}^{(2)}$ were to start entangling, their
entropies will be equal.

The above argument assumes that degrees of freedom that were entangled earlier
stay entangled later as well. This is in part supported by black hole entropy
being constant even after Page time, but not fully, because one can ask the
question - whether the journal can “steal” the qubits that were entangled with
the bath, in which case black hole entropy would still remain constant. Such
stealing however requires disentangling of $\chi_{1}^{(2)}$ and
$\chi_{0}^{(1)}$ degrees of freedom so that latter may be available to
$\chi_{2}^{(1)}$ which in turn requires unitaries that act non-locally on the
degrees of freedom in the black hole $\chi_{0}$ and those far outside in the
bath $\chi_{1}$. We don’t know whether such a protocol’s physicality can be
explained via the island programme. Nevertheless, we will continue with this
assumption.

#### Connected island

The mouths of replica wormhole cover the gravitational region entirely on each
sheet, but as $n\to 1$, they reduce to an island in the interior. In all the
studies thus far, this island has been a connected one, but in principle the
island may be disconnected. For instance, if an external system such as bath
gets entangled with two separated regions of the bulk, then this can
happen.333We want to thank Thomas Hartman for this point. Here we explore this
possibility in our setup.

We have seen that as quantum fields on AdS spacetime of fixed curvature, CFT1
and CFT2 do not interact, and that they are purified by CFT1 in the bath and
the journal respectively. When these systems were studied individually, both
the bath and the journal got entangled with the interior degrees of freedom of
respective CFT, which was a connected island. If either of them were to
entangle with degrees of freedom present outside the black hole, then we could
have seen a disconnected island. Following this for our combined but
independent case, we expect their islands to be connected.

Using the left-right symmetry and the no cloning theorem then allows us to
conclude that quantum gravity correction is made either in
$\mathbf{S}(\chi_{1})$ or in $\mathbf{S}(\chi_{2})$ but not in both. In other
words, either $\chi_{1}$ gets an island, or $\chi_{2}$ gets an island.

To conclude this subsection, the paradox resolution requires saturation of the
black hole entropy and same rate of entropy growth for $\chi_{1}$ and
$\chi_{2}$. If the island either belongs to $\chi_{1}$ or to $\chi_{2}$ , this
can happen in only two ways: either $\mathbf{S}(\chi_{1})\sim\log u$ like
$S(\chi_{2})$, or $\mathbf{S}(\chi_{2})\sim u$ like $S(\chi_{1})$, where
proportionality constants ensure same rate of growth.

###  4.2 Bath island

We now look at non-trivial islands for the bath at late times. Consider the
left-right symmetric island $AB$, where $B^{\prime}$s coordinates are
$w^{\pm}$ and let the right boundary observer move with coordinates
$v^{\pm}=\pm e^{\pm 2\pi u/\beta}$. First let us look at the matter entropy
for CFT1 with island; its partition function is given by the four point
function
$Z_{1}=\left\langle\sigma^{(1)}_{L}\sigma^{(1)}_{A}\sigma^{(1)}_{B}\sigma^{(1)}_{R}\right\rangle$
(see figure 12). Here the $(1)$ superscript denotes this twist operator is for
CFT1.

Figure 12: For CFT1, the twist operators placed at the spacelike infinity in
the bath are topological. In the region $AB$, CFT2 is also present, and its
contribution in the matter entropy for fixed background is just added to the
CFT1 contribution.

This four point function can be computed upto the conformally invariant piece
$G_{1}(\eta_{1})$ which we may neglect for QES calclulation as per discussion
in subsection 2.2, but the ensuing extremization of generalized entropy gives
coupled quintic equations for $w^{\pm}$. Instead we take the usual approach
Almheiri:2021jwq ; Almheiri:2019yqk  of assuming the island is large at late
times, which will allow for the factorization
$Z_{1}\simeq\left\langle\sigma^{(1)}_{L}\sigma^{(1)}_{A}\right\rangle\left\langle\sigma^{(1)}_{B}\sigma^{(1)}_{R}\right\rangle$.

As the fixed curvature entropy splits over the two CFTs, the total matter
entropy will be sum of the above and the entropy for CFT2 in the interval $AB$

$S_{2}(AB)=\lim_{n\to
1}\frac{1}{1-n}\ln\int\prod_{i=1}^{n}dJ_{i}P(J_{i})\left\langle\sigma_{A}\widetilde{\sigma}_{B}\right\rangle_{\mathbf{J}}$

where the correlation function is evaluated on Euclidean AdS2 disk with
circumference $\beta/\epsilon$; the associated orbifold theory has $n$ CFTs
each with different boundary condition $J_{i}$. The two point function on disk
becomes a four point function on the plane, and after using the global
conformal invariance, we’re left with an undetermined function
$G_{\mathbf{J}}(\eta)$ where $\eta$ is the cross ratio of the four points,
which carries the theory dependence, including the $\mathbf{J}$ dependence. In
other words, the integral is only over $G_{J}(\eta)$, and entropy takes the
form

$S_{2}(AB)=\frac{c_{2}}{6}\log\left(\frac{(w^{+}-w^{-})^{2}}{(1+(w^{-})^{2})(1+(w^{+})^{2})}\right)+\ln\left\langle\mathcal{G}(\eta)\right\rangle$
(28) $=S_{2}^{(CI)}+\ln\left\langle\mathcal{G}(\eta)\right\rangle$

where we’ve already used the left-right symmetry and

$\eta=\frac{(1+w^{-}w^{+})^{2}}{(1+(w^{+})^{2})(1+(w^{-})^{2})}.$

The detailed calclulation is in appendix A. Even though we know the theory for
CFT2 explicitly, we don’t know how444The different CFTs on $M/\mathbb{Z}^{n}$
all have different boundary conditions $J_{i}$ which makes this a non-standard
calculation. We’ll comment more on this in subsection 4.4. to find its
$G_{\mathbf{J}}(\eta)$, but as we mentioned before, we’ll assume
$\partial_{\eta}\log G_{\mathbf{J}}(\eta)$ is finite. Later we’ll show that
$\eta$ term does not shift the QES in that case.

After adding the dilaton the generalized entropy becomes

$\frac{1}{3}c_{1}\log\left(\frac{\left(v^{-}-w^{-}\right)\left(w^{+}-v^{+}\right)}{\left(v^{-}v^{+}+1\right)\left(w^{-}w^{+}+1\right)}\right)+\frac{1}{6}c_{2}\log\left(\frac{\left(w^{-}-w^{+}\right)^{2}}{\left(\left(w^{-}\right)^{2}+1\right)\left(\left(w^{+}\right)^{2}+1\right)}\right)+\frac{\pi\left(1-w^{-}w^{+}\right)\phi_{r}}{\beta\left(w^{-}w^{+}+1\right)G_{N}}+\frac{\phi_{0}}{2G_{N}}.$
(29)

Differentiating with respect to $w^{-}$ gives the equation

$-\frac{c_{1}\left(w^{-}w^{+}+1\right)\left(v^{-}w^{+}+1\right)}{v^{-}-w^{-}}+\frac{c_{2}\left(w^{-}w^{+}+1\right)^{3}}{\left(\left(w^{-}\right)^{2}+1\right)\left(w^{-}-w^{+}\right)}-\frac{6\pi
w^{+}\phi_{r}}{\beta G_{N}}=0$ (30)

while for $w^{+}$ the equation is

$\frac{1}{3}c_{1}\left(\frac{1}{w^{+}-v^{+}}-\frac{w^{-}}{w^{-}w^{+}+1}\right)+\frac{1}{3}c_{2}\left(\frac{1}{w^{+}-w^{-}}-\frac{w^{+}}{\left(w^{+}\right)^{2}+1}\right)-\frac{2\pi
w^{-}\phi_{r}}{\beta\left(w^{-}w^{+}+1\right)^{2}G_{N}}=0.$ (31)

Instead of directly attacking (30), (31) which are still hard to solve, we
make the following observation. Almheiri et al in Almheiri:2019yqk  worked
with $c_{2}=0$ without journal, assuming that island $AB$ gets larger with
time, and showed that point $B$ is at

$w^{\pm}=\frac{\beta c_{1}G_{N}}{6\pi\phi_{r}}v^{\pm}$ (32)

which is outside the horizon. In our case, we have $S_{2}(AB)$ as the
additional term. Consider the derivatives of $S_{2}$,

$\partial_{\pm}S_{2}^{(CI)}=\mp\frac{c_{2}\left(w^{-}w^{+}+1\right)}{3\left(w^{-}-w^{+}\right)\left(\left(w^{\pm}\right)^{2}+1\right)},$
$\partial_{\pm}\eta=\pm\frac{2\left(w^{-}-w^{+}\right)\left(w^{-}w^{+}+1\right)}{\left(\left(w^{\mp}\right)^{2}+1\right)\left(\left(w^{\pm}\right)^{2}+1\right)^{2}}.$

From here it takes a minute to see that $\partial_{\pm}S_{2}\sim 0$ and
$\partial_{\pm}\eta\sim 0$ at the earlier solution (32) by using the relations
$w^{+}w^{-}+1\sim 1$ and because of late times, $w^{+}-w^{-}\sim w^{+}$. Thus
we see that does not change much because of CFT2. Therefore, we can make the
ansatz that $w^{\pm}=\alpha(u)v^{\pm}$ for $\alpha(u)\ll 1$ in (30) and (31)
and solve for $\alpha(u)$ to get

$\alpha(u)=\frac{\beta c_{1}G_{N}}{{12\pi\phi_{r}}}\left(1+\sqrt{1-e^{-4\pi
u/\beta}\frac{24\pi c_{2}\phi_{r}}{\beta G_{N}c_{1}^{2}}}\right).$ (33)

Now this island exists only after a small time

$u>\frac{\beta}{4\pi}\ln\frac{24\pi c_{2}\phi_{r}}{\beta
G_{N}c_{1}^{2}}\sim\log\left(\frac{1}{G_{N}}\right).$

and after this time it slightly shrinks the bath island for a while, but at
large $u$, the island tends to the one in Almheiri:2019yqk .

#### Entropy growth for this island

The dilaton and CFT1 contributions to entropy for this island are clearly
constants, whereas $S_{2}^{(CI)}$ grows for a while and then becomes constant.
However, it is possible that $G_{\mathbf{J}}$ may provide some appreciable
growth; we will discuss in subsection 4.4 how $G_{\mathbf{J}}$ providing a
logarithmic growth can resolve the paradox.

###  4.3 Journal island

One can try to use the same strategy to check how the island in
Almheiri:2021jwq  changes here, but will find that the new entropy derivatives
aren’t negligible at the old solution. Here we describe how the $S_{1}(AB)$
term will shift the original journal island into the interior.

Again consider a symmetric island $AB$ with $B$ having coordinates
$w^{+},w^{-}$ like in figure 12. The CFT1 is in the vacuum state over the full
$w$ plane, and like discussed in subsection 2.2. $S_{1}(AB)$ calculation
requires the correlation function
$(\Omega_{A}\Omega_{B})^{\Delta_{n}}\left\langle\sigma^{(1)}_{A}\widetilde{\sigma}^{(1)}_{B}\right\rangle_{\mathbb{C}}$
where $\Omega$’s are the Weyl factors for moving from the curved AdS2 interior
to flat space. This time however interval is symmetrically placed, which
allows expression using just right point’s coordinates $w^{\pm}$.
$\mathbf{S}(\chi_{2})=\text{ext}(S_{1}+S_{2}+\phi_{J})$, where

$S_{1}(AB)=\frac{c_{1}}{6}\log\left(\frac{(w^{+}-w^{-})^{2}}{(1+w^{+}w^{-})^{2}}\right)$
$S_{2}=c_{2}\log\log\left(\frac{\beta}{2\pi\epsilon}\left(\frac{(1+w^{+}v^{-})(1+w^{-}v^{+})}{1+w^{+}w^{-}}\right)\right)$
$\phi_{J}=\frac{\pi\phi_{r}}{\beta
G_{N}}\left(\frac{1-w^{-}w^{+}}{1+w^{-}w^{+}}\right)+\frac{\phi_{0}}{2G_{N}}$

where $v^{\pm}=\pm\exp(\pm 2\pi u/\beta)$ are again the coordinates of the
right boundary observer $R$. $S_{2}$ is obtained from the partition function
$Z=\int P(J_{i})dJ_{i}\left\langle
O_{L}^{\dagger}\sigma^{(2)}_{A}\sigma^{(2)}_{B}O_{R}\right\rangle_{\mathbf{J}}$,
which at late times for a left-right symmetric case Almheiri:2021jwq  gives

$Z\simeq\int
P(J_{i})dJ_{i}\left\langle\sigma_{B}^{(2)}O_{R}\right\rangle_{\mathbf{J}}^{2}.$

Taking the derivatives with respect to $w^{+}$ gives

$-\frac{3c_{2}\left(w^{-}w^{+}+1\right)\left(w^{-}-v^{-}\right)}{\left(v^{-}w^{+}+1\right)\log\left(\frac{\beta}{2\pi\epsilon}\right)}-\frac{c_{1}\left(\left(w^{-}\right)^{2}+1\right)\left(w^{-}w^{+}+1\right)}{w^{-}-w^{+}}-\frac{6\pi
w^{-}\phi_{r}}{\beta G_{N}}=0$ (34)

and $w^{-}$ derivative gives

$-\frac{3c_{2}\left(w^{-}w^{+}+1\right)\left(w^{+}-v^{+}\right)}{\left(v^{+}w^{-}+1\right)\log\left(\frac{\beta}{2\pi\epsilon}\right)}+\frac{c_{1}\left(w^{-}w^{+}+1\right)\left(\left(w^{+}\right)^{2}+1\right)}{w^{-}-w^{+}}-\frac{6\pi
w^{+}\phi_{r}}{\beta G_{N}}=0.$ (35)

In the limit of $w^{-}\to 0,w^{+}\gg 1$, we can find how the island of
Almheiri:2021jwq  shifts because of CFT1 now. This limit makes the (34) into

$\frac{c_{1}}{w^{+}}+\frac{3v^{-}c_{2}}{\log\frac{\beta}{2\pi\epsilon}}-\frac{6\pi
w^{-}\phi_{r}}{\beta G_{N}}=0$

and (35) becomes

$-w^{+}c_{1}-\frac{3(w^{+}-v^{+})c_{2}}{\log\frac{\beta}{2\pi\epsilon}}-\frac{6\pi
w^{+}\phi_{r}}{\beta G_{N}}=0.$

The solution upto order $O(G_{N})$,

$w^{-}=\frac{G_{N}\left(\beta
c_{1}^{2}\log^{2}\left(\frac{\beta}{2\pi\epsilon}\right)+3\beta
c_{2}c_{1}\log\left(\frac{\beta}{2\pi\epsilon}\right)-9\beta
c_{2}^{2}\right)}{18\pi
c_{2}v^{+}\phi_{r}\log\left(\frac{\beta}{2\pi\epsilon}\right)}+\frac{c_{1}\log\left(\frac{\beta}{2\pi\epsilon}\right)}{3c_{2}v^{+}}$
(36) $w^{+}=\frac{\beta
c_{2}v^{+}G_{N}}{2\pi\phi_{r}\log\left(\frac{\beta}{2\pi\epsilon}\right)}.$

If $c_{1}\to 0$, $w^{-}\to-\frac{\beta c_{2}}{2\pi
v^{+}\phi_{r}\log\frac{\beta}{2\pi\epsilon}}$, which gives the island of
Almheiri:2021jwq . However, if $c_{1}$ is large enough, the island moves
inside the black hole $w^{\pm}>0$, which happens for

$c_{1}\log\frac{\beta}{2\pi\epsilon}+\frac{\beta
G_{N}}{6\pi\phi_{r}}\left(2c_{1}^{2}+3c_{2}c_{1}-\frac{9c_{2}^{2}}{\log\frac{\beta}{2\pi\epsilon}}\right)>0.$

Given that $\epsilon$ is very small, $w^{-}>0$ will be typically the case.
Thus we see, the presence of $c_{1}$ has shifted this island inside the black
hole.

#### Entropy growth

After substituting (36) in the entropies, the growth only comes from CFT1
part:

$S=\frac{1}{3}c_{1}\log\left(18\beta c_{2}^{2}G_{N}e^{\frac{2\pi
u}{\beta}}\right)\sim\frac{c_{1}}{3}\cdot\frac{2\pi u}{\beta}+\text{constant}$
(37)

Thus CFT1 contribution to matter entropy grows linearly. Without the island,
the journal entropy had been growing only logarithmically.

###  4.4 Paradox resolution and interpretation

Let us summarize the story so far. We’ve noted that the after black hole
entropy saturates, the system $\chi_{1}\cup\chi_{2}$ together own the
complement region. However, as we aruged in subsection 4.1, because
$S(\chi_{1})\sim u$ and $S(\chi_{J})\sim\log u$, the paradox is still not
solved. To resolve it, the entropies of the bath and journal must become
constant or grow at the same rate. The latter is possible here because both
systems have infinite degrees of freedom and therefore can have everlasting
entanglement growth. If we find multiple islands that can give us a
resolution, we will pick the one which gives the minimum entropy growth, in
the spirit of the island formula.

We also argued that $\chi_{1}$ and $\chi_{2}$ would have connected islands,
which with left-right symmetry and no cloning theorem led us to conclude that
gravitational correction will change either the bath entropy to a logarithmic
growth, or journal entropy to a linear growth, such that entropy of both bath
and journal increases at same rate.

In the previous section, we’ve obtained an island for the journal (36) and an
island for the bath (33) - these are the generalizations of islands of
Almheiri:2021jwq  and Almheiri:2019yqk  respectively in this combined setup.
We already see that for journal island (36), the entropy grows linearly, at
the rate (37)

$S\sim\frac{2\pi c_{1}u}{3\beta}.$

Taking the late time limit of (22), the entropy of bath for an empty island,
gives exactly the same rate of growth. Therefore, the island (36) satisfies
all our requirements for a resolution.

The discussion for bath island (33) is hindered because we haven’t been able
to find $G_{\mathbf{J}}$, hence we cannot make definite statements about it.
We however wish to elaborate the following possibility. The island (33) does
make the entropy of the CFT1 a constant, and $S_{2}^{(CI)}$ piece also becomes
constant after some time. Therefore, if $\left\langle\ln
G_{\mathbf{J}}(\eta)\right\rangle$ gives logarithmic growth at the same rate
as that of the journal,

$S\sim\log\frac{u}{m^{2}\alpha^{\prime}\beta}$

then we do have a resolution, and it does so by giving lower entropies for the
bath and the journal, which would make it the preferred canditate. While the
expression for $G(\eta)$ is known in literature for a compact boson
Furukawa:2009mib , it is not clear to us how the variation of boundary
condition can be implemented in it. Another problem in that direction is that
the continuation of compact boson with radius $R$ to $R\to\infty$ is not
smooth, so perhaps one may need other techniques altogether to derive
$G_{\mathbf{J}}(\eta)$.

If one can come up with a method to implement different boundary conditions
even in the compact boson case, some progress may be possible by changing the
probability distribution $P(J)$ to be localized, like a uniform distribution
from $(-a,a)$, such that $a\gg R$. The degrees of freedom of journal are still
infinite, and its entropy is expected to grow logarithmically for empty
island. In that sense, this can allow us to keep most of the structure of the
calculations here, and see what effect $G_{\mathbf{J}}(\eta)$ will make. With
that said, we leave this possibility open for future exploration, and proceed
to describe the consequences of giving the island to journal.

#### Co-ownership in the gap between entanglement wedges

Figure 13 illustrates the gap beteen the points 36 and 27 at some time $u$.
Region $E_{J}$ is the entanglement wedge of the journal, and the green regions
$E_{L},E_{R}$ are the entanglement wedges of the left and right CFTs
respectively. In this setup, the bath has no island, and therefore its
entanglement wedge is empty. But it does not mean that bath’s information is
not of any use.

Figure 13: Interpretation of the journal island: the $w$ plane with islands.
The yellow region $E_{J}$ is the entanglement wedge of journal, green regions
are the entanglement wedges for left and right CFTs; $E_{R}$ has been labelled
above, and the blue regions $S_{l},S_{r}$ are the entanglement wedge of
“journal and bath together”. The figure only labels the right side.

There are gaps $S_{l},S_{r}$ between the entanglement wedges $E_{J}$, $E_{R}$,
and $E_{L}$, which are shown in blue. Just by accessing journal and the left-
right CFTs, one cannot do a bulk reconstruction here. These blue regions
$S_{r},S_{l}$ however are parts of entanglement wedge of
$\chi_{1}\cup\chi_{2}$, which means that they are shared by both bath and
journal in the sense that one needs access to both of them to do the
reconstruction there. As time increases, the equations (36) and (27) show that
this gap keeps on getting bigger, a result similar in spirit to what was seen
in Almheiri:2021jwq  where the effect of the uncertainty of couplings was seen
only at very late times.

#### How gravity connects $\chi_{1}$ and $\chi_{2}$?

The system $\chi_{1}$ is a bath whereas $\chi_{2}$ is an auxiliary system
which cannot directly interact with the bath, but physically we know they are
connected to the same black hole $\chi_{0}$ and thus through gravity we expect
$\chi_{1}$ and $\chi_{2}$ to come to sense each other. Using the island
formula and the constraints in subsection 4.1 we have shown that these systems
do get entangled, but how is this connection manifested through gravity? We do
not know, but are reminded of ER=EPR.

This is a conjecture that in quantum gravity any two entangled particles are
connected by a Einstein-Rosen like bridge Maldacena:2013epr , and if there
were a collection of entangled particles shared by Alice and Bob, their
separate collapse into black holes would lead to a “bridge dynamics” that
ultimately creates an Einstein-Rosen bridge. In other words, the idea is that
in quantum gravity, entanglement between particles is mediated through
gravitational wormholes.

In that language our result would be interpreted as that at Page time, an
Einstein-Rosen bridge forms between $\chi_{1}$ and $\chi_{2}$, whose cross
sectional area grows continuously as a result of increasing entanglement. We
are curious whether such an interpretation can be meaningfully assigned here.

##  5 Comments

Motivated by Almheiri:2021jwq  we have fleshed out a novel kind of information
paradox problem involving three systems in its issues as to what constitutes
the paradox, and using the island formula, entanglement monogamy and
assumption of “connected island”, we have obtained an island which resolves
the paradox, and also detailed the possibility of another island which may be
significant.

We worked with a black hole in thermal equilibrium with a bath here, but the
suggested problem in Almheiri:2021jwq  was for an evaporating black hole
coupled to reservoir and the journal which implemented random boundary
conditions for CFT2 as done here. We’ve noted that case is more difficult but
have also found it more interesting.

In this case, one insight is that the only system with infinite degrees of
freedom would be the journal, which means that its initial logarithmic growth
cannot be substained at late times for there is not enough “fuel”, implying
that journal will have an island which makes sure that this entropy growth is
cured. Another thing to note is, because the CFT2 is not allowed to radiate,
the black hole will never lose all of its energy. Using these insights it
seems to us that at late times, the journal will purify both the radiation and
a remnant, extremal black hole, which would mean that not knowing the unknown
couplings will prevent access also to the remnant and the radiation, but
filling gaps of this expectation with calculations and precision is a
different matter, and we leave it as a future work.

### Acknowledgements

We want to thank Alok Laddha, Loganayanam R, Thomas Hartman, Ayan Mukopadhyay,
Anupam Sarkar, Sujay Ashok, and Yiming Chen for valuable discussions.

##  Appendix A CFT2 entropy for a bulk interval

In this appendix we sketch the derivation of (28), which illustrates the kind
of steps involved in these calculations. Let $z,\bar{z}$ be the coordinates
for the plane, and $w,\bar{w}$ for the disk. The Weyl transformation (17) from
AdS2 to flat disk is

$Z=\left\langle\sigma(w_{1},\overline{w}_{1})\sigma(w_{2},\overline{w}_{2})\right\rangle_{\text{AdS}_{2}}=(\Omega_{1}\Omega_{2})^{\Delta}\left\langle\sigma(w_{1},\overline{w}_{1})\sigma(w_{2},\overline{w}_{2})\right\rangle_{flat\;disk}$

for

$\Omega=\frac{1-|w|^{2}}{2}.$

We make a conformal transformation to a right half plane with coordinates
$z,\bar{z}$, and afterwards do the doubling with appropriate $-$ signs, which
makes

$\left\langle\sigma(w_{1},\overline{w}_{1})\sigma(w_{2},\overline{w}_{2})\right\rangle_{flat\;disk}=\left|\frac{\partial(z,\overline{z})}{\partial(w,\overline{w})}\right|_{1}^{\Delta/2}\left|\frac{\partial(z,\overline{z})}{\partial(w,\overline{w})}\right|_{2}^{\Delta/2}\left\langle\sigma(z_{1})\sigma(z_{2})\sigma(-\overline{z}_{2})\sigma(-\overline{z}_{1})\right\rangle_{\mathbb{C}}$

where

$\left|\frac{\partial(z,\overline{z})}{\partial(w,\overline{w})}\right|^{\Delta/2}=\left|\frac{\partial
z}{\partial w}\right|^{\Delta}.$

Given the requirements, the map is

$z=\left(\frac{1-w}{1+w}\right),\quad|\partial_{w}z|^{\Delta}=\left(\frac{2}{|1+w|^{2}}\right)^{\Delta}.$

Using Cardy-Calabrese formula for 2 disjoint intervals Calabrese:2009dicft
the four point function

$\left\langle\sigma(z_{1})\sigma(z_{2})\sigma(-\overline{z}_{2})\sigma(-\overline{z}_{1})\right\rangle_{\mathbb{C}}=\frac{G_{n}(\eta)}{[(z_{1}-z_{2})(\overline{z}_{1}-\overline{z}_{2})\eta]^{\Delta}}$

where

$\eta=\frac{(z_{1}+\bar{z}_{1})(z_{2}+\bar{z}_{2})}{(z_{1}+\bar{z}_{2})(z_{2}+\bar{z}_{1})}.$

Cardy and Calabrese didn’t consider the CFT to have different boundary
conditions on each sheet, and therefore their conformal invariant cross ratio
does not have a $\mathbf{J}$ dependence. We expect however that conformally
invariant piece of the correlation function would stil remain the same, as it
is independent of any boundary conditions, and thereby expect $\mathbf{J}$
dependence to be present only in a multiplicative factor. Thus for our case,

$\left\langle\sigma(w_{1},\overline{w}_{1})\sigma(w_{2},\overline{w}_{2})\right\rangle_{flat\;disk}=\left(\frac{2}{|1+w_{1}|^{2}}\right)^{\Delta}\left(\frac{2}{|1+w_{2}|^{2}}\right)^{\Delta}\frac{G_{n}(\eta,\mathbf{J})}{[(z_{1}-z_{2})(\overline{z}_{1}-\overline{z}_{2})\eta]^{\Delta}}$
$=\left(\frac{2}{|1+w_{1}|^{2}}\right)^{\Delta}\left(\frac{2}{|1+w_{2}|^{2}}\right)^{\Delta}\frac{G_{n}(\eta,\mathbf{J})}{[(z_{1}-z_{2})(\overline{z}_{1}-\overline{z}_{2})\eta]^{\Delta}}$

$=\left({|1+w_{1}|^{2}\cdot|1+w_{2}|^{2}}\right)^{-\Delta}\cdot\frac{4^{\Delta}\cdot
G_{n}(\eta,\mathbf{J})}{[(z_{1}-z_{2})(\overline{z}_{1}-\overline{z}_{2})\eta]^{\Delta}}$
$=\left(\frac{(w_{1}\overline{w}_{2}-1)(\overline{w}_{1}w_{2}-1)}{(w_{1}\overline{w}_{1}-1)(w_{1}-w_{2})(\overline{w}_{1}-\overline{w}_{2})(w_{2}\overline{w}_{2}-1)}\right)^{\Delta}G_{n}(\eta,\mathbf{J}).$

So

$Z=\left(\frac{1-|w_{1}|^{2}}{2}\right)^{\Delta}\left(\frac{1-|w_{2}|^{2}}{2}\right)^{\Delta}\left\langle\sigma(w_{1},\overline{w}_{1})\sigma(w_{2},\overline{w}_{2})\right\rangle_{flat\;disk}$
$=\left[\frac{(w_{1}\overline{w}_{2}-1)(\overline{w}_{1}w_{2}-1)}{(w_{1}-w_{2})(\overline{w}_{1}-\overline{w}_{2})}\right]^{\Delta}\frac{G_{n}(\eta,\mathbf{J})}{4^{\Delta}}$
$S_{2}=\frac{c_{2}}{6}\ln\left(\frac{(w_{1}-w_{2})(\overline{w}_{1}-\overline{w}_{2})}{(1-\overline{w}_{1}w_{2})(1-\overline{w}_{2}w_{1})}\right)+\ln\mathcal{G}_{n}(\eta,\mathbf{J})+\text{const.}$

Finally, putting $\bar{w}_{2}=w^{+},w_{2}=-w^{-}$, and by using the left-right
symmetry $\bar{w}_{1}=w_{1}^{+}=w^{-},w_{1}=-w_{1}^{-}=-w^{+}$, gives

$S_{2}^{(CI)}=\frac{c_{2}}{6}\ln\left(\frac{(-w^{+}+w^{-})(w^{-}-w^{+})}{(1+(w^{-})^{2})(1+(w^{+})^{2})}\right)$

which is the claimed result (28).

## References

  * (1) S.W. Hawking, _Breakdown of predictability in gravitational collapse_ , _Phys. Rev. D_ 14 (1976) 2460.
  * (2) G. Penington, _Entanglement Wedge Reconstruction and the Information Paradox_ , _JHEP_ 09 (2020) 002 [1905.08255].
  * (3) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, _Replica Wormholes and the Entropy of Hawking Radiation_ , _JHEP_ 05 (2020) 013 [1911.12333].
  * (4) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, _The entropy of Hawking radiation_ , _Rev. Mod. Phys._ 93 (2021) 035002 [2006.06872].
  * (5) A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, _The page curve of hawking radiation from semiclassical geometry_ , 1908.10996v2.
  * (6) A. Almheiri, R. Mahajan and J. Maldacena, _Islands outside the horizon_ , 1910.11077.
  * (7) A. Almheiri and H.W. Lin, _The Entanglement Wedge of Unknown Couplings_ , 2111.06298.
  * (8) T. Hartman, E. Shaghoulian and A. Strominger, _Islands in asymptotically flat 2d gravity_ , _Journal of High Energy Physics_ 2020 (2020) .
  * (9) K. Goto, T. Hartman and A. Tajdini, _Replica wormholes for an evaporating 2d black hole_ , 2011.09043v1.
  * (10) A. Almheiri, R. Mahajan and J. Santos, _Entanglement islands in higher dimensions_ , _SciPost Physics_ 9 (2020) .
  * (11) A. Recknagel and V. Schomerus, _Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes_ , Cambridge Monographs on Mathematical Physics, Cambridge University Press (2013), 10.1017/CBO9780511806476.
  * (12) J. Sully, M.V. Raamsdonk and D. Wakeham, _Bcft entanglement entropy at large central charge and the black hole interior_ , 2004.13088v1.
  * (13) P. Di Francesco, P. Mathieu and D. Senechal, _Conformal Field Theory_ , Graduate Texts in Contemporary Physics, Springer-Verlag, New York (1997), 10.1007/978-1-4612-2256-9.
  * (14) A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, _The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole_ , _JHEP_ 12 (2019) 063 [1905.08762].
  * (15) G. Sárosi, _Ads 2 holography and the syk model_, 1711.08482v6.
  * (16) D.A. Trunin, _Pedagogical introduction to syk model and 2d dilaton gravity_ , 2002.12187v4.
  * (17) J. Engelsöy, T.G. Mertens and H. Verlinde, _An investigation of ads 2 backreaction and holography_, 1606.03438v3.
  * (18) A. Almheiri and J. Polchinski, _Models of ads 2 backreaction and holography_, 1402.6334v3.
  * (19) T.J. Hollowood and S.P. Kumar, _Islands and page curves for evaporating black holes in jt gravity_ , _Journal of High Energy Physics_ 2020 (2020) 94.
  * (20) P. Calabrese and J. Cardy, _Entanglement entropy and quantum field theory_ , hep-th/0405152v3.
  * (21) P. Calabrese, J. Cardy and E. Tonni, _Entanglement entropy of two disjoint intervals in conformal field theory_ , 0905.2069v2.
  * (22) J.L. Cardy, O.A. Castro-Alvaredo and B. Doyon, _Form factors of branch-point twist fields in quantum integrable models and entanglement entropy_ , 0706.3384v2.
  * (23) T. Takayanagi, _Holographic dual of a boundary conformal field theory_ , _Physical Review Letters_ 107 (2011) .
  * (24) H.Z. Chen, Z. Fisher, J. Hernandez, R.C. Myers and S.-M. Ruan, _Evaporating black holes coupled to a thermal bath_ , 2007.11658v2.
  * (25) P. Hayden, R. Jozsa, D. Petz and A. Winter, _Structure of states which satisfy strong subadditivity of quantum entropy with equality_ , _Communications in Mathematical Physics_ 246 (2004) 359.
  * (26) S. Furukawa, V. Pasquier and J. Shiraishi, _Mutual information and boson radius in a $c=1$ critical system in one dimension_, _Physical Review Letters_ 102 (2009) .
  * (27) J. Maldacena and L. Susskind, _Cool horizons for entangled black holes_ , _Fortschritte der Physik_ 61 (2013) 781.