11institutetext: Berkeley Center for Theoretical Physics and Department of
Physics,
University of California, Berkeley, CA 94720, USA22institutetext: Stanford
Institute for Theoretical Physics,
Stanford University, Stanford, CA 94305, USA

# Island Finder and Entropy Bound

Raphael Bousso 2 and Arvin Shahbazi-Moghaddam<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Identifying an entanglement island requires exquisite control over the entropy
of quantum fields, which is available only in toy models. Here we present a
set of sufficient conditions that guarantee the existence of an island and
place an upper bound on the entropy computed by the island rule. This is
enough to derive the main features of the Page curve for an evaporating black
hole in any spacetime dimension. Our argument makes use of Wall’s maximin
formulation and the Quantum Focusing Conjecture. As a corollary, we derive a
novel entropy bound.

##  1 Introduction

###  1.1 Entanglement Wedges and Islands

The quantum-corrected Engelhardt:2014gca , covariant Hubeny:2007xt  Ryu-
Takayanagi Ryu:2006bv  (RT) prescription computes the CFT entropy of a
boundary region in terms of a dual asymptotically AdS bulk spacetime.
Originally an ad hoc proposal, it follows under certain assumptions from a
Euclidean gravitational path integral Lewkowycz:2013nqa ; Faulkner:2013ana ;
Dong:2017xht ; Penington:2019kki ; Almheiri:2019qdq . This derivation implies
that the RT prescription is not tied to the AdS/CFT correspondence but can be
evaluated in any spacetime $M$.

Indeed, RT yields the Page curve Page:1993wv  for the entropy of the bulk
radiation emitted by a black hole Penington:2019npb ; Almheiri:2019psf . The
bulk state and geometry are treated semiclassically. In this approximation,
the radiation is thermal Hawking:1974sw , and its von Neumann entropy $S(R)$
increases monotonically, implying information loss Hawking:1976ra . Using the
same semiclassical solution, the RT proposal computes the radiation entropy
differently, as the generalized entropy111For a partial Cauchy surface
$X\subset M$, $S_{\rm gen}(X)=\text{Area}[\partial X]/4G\hbar+S(X)$, where
$\partial$ denotes the boundary of a set, and $S(X)$ is the renormalized von
Neumann entropy of the density operator of the quantum field theory state
reduced to $X$. See the appendix in Ref. Bousso:2015mna  for a detailed
discussion of this quantity. of the entanglement wedge of the radiation,
$E(R)$:

$S(\mathbf{R})=S_{\rm gen}[E(R)]~{}.$ (1)

The bold-face notation Almheiri:2019yqk  distinguishes the (presumably
correct) entropy computed by RT from the entropy $S(R)$ computed directly from
the semiclassical radiation state (See Bousso:2019ykv ; Bousso:2020kmy  for a
proposal to reconcile the bold and unbold states.)

The original RT prescription defines an entanglement wedge for regions on the
conformal boundary of AdS. In the present context, $R$ is a bulk system, and
the entanglement wedge must be defined as follows Bousso:2020kmy :

  1. 1.

$E(R)=I\cup R$, where $I\subset M$;

  2. 2.

$S_{\rm gen}(I\cup R)$ is stationary under any local variations of the
boundary surface $\partial I$;

  3. 3.

among all such regions globally, $I$ yields the smallest $S_{\rm gen}(I\cup
R)$.

The above definitions apply if $R$ is a nongravitating system external to $M$;
in asymptotically AdS geometries the radiation can be extracted into such a
system Penington:2019npb ; Almheiri:2019psf . We now turn to the case where
$R$ is a weakly gravitating region inside the spacetime. For example, $R$ may
be a distant region occupied by Hawking radiation in an asymptotically flat or
AdS spacetime (see Ref. Bousso:2019ykv  for a detailed setup).

Physically, one expects the RT prescription for a weakly gravitating region to
reduce to that for a nongravitating system, Eq. (1), and we shall assume this
here. $R$ resides in a weakly gravitating region far from any potential island
$I$, so $\partial I\cap\bar{R}=\varnothing$, where an overbar denotes
topological closure. As before, stationarity of $S_{\rm gen}$ is required only
under variations of $\partial I$, not of $\partial R$. (This can be
implemented in a path integral derivation Dong:2020uxp .) Thus, the definition
of $E(R)$ is essentially unchanged.

However, in the presence of gravity, it is simplest to work with the
generalized entropy of the region $\mathbf{R}$ (a cutoff-independent
quantity), so we add its boundary area to both sides of Eq. (1):

$S_{\rm gen}(\mathbf{R})=S_{\rm gen}[E(R)]~{}.$ (2)

It is easy to derive the Page curve, at least approximately, if one ignores
condition 2. We will now summarize this incomplete argument, before discussing
how it can be completed.

Figure 1: Left: evaporating black hole; right: its Page curve. After the Page
time, the semiclassical entropy $S(R)$ of the Hawking radiation in the
asymptotic region $R$ exceeds the Bekenstein-Hawking entropy of the black
hole, $A_{h}/4G\hbar$. The “Hawking partners” in the black hole interior
purify $R$. (Dashed lines indicate entanglement.) Therefore, adjoining
$\hat{I}$ to $R$ decreases the generalized entropy $S_{\rm gen}$. However,
islands must have stationary $S_{\rm gen}$. Solving for this condition exceeds
present analytic control over the entropy. The island finder presented here
sidesteps this obstruction.

The Page time $t_{\rm Page}$ is defined as the time when the black hole and
radiation entropies are equal in the semiclassical analysis:

$S[R(t_{\rm Page})]=\frac{A_{h}(t_{\rm Page})}{4G\hbar}~{}.$ (3)

Let $\hat{I}(t)$ be the black hole interior at time $t$ (see Fig. 1). The
Hawking “partners” in $\hat{I}(t)$ purify the radiation $R(t)$ emitted so far;
hence

$S_{\rm gen}[\hat{I}(t)\cup R]\approx\frac{A_{h}(t)}{4G\hbar}~{}.$ (4)

Before the Page time, this is greater than $S(R)$ by definition, so $\hat{I}$
is not a viable island candidate; one finds that $I(t)=\varnothing$, $E(R)=R$,
and $S(\mathbf{R})=S(R)$. This corresponds to the rising part of the Page
curve, where it agrees with Hawking’s curve. But after the Page time, $S_{\rm
gen}[\hat{I}(t)\cup R]<S(R)$ by Eqs. (3) and (4). Thus, the inclusion of an
island $I(t)\approx\hat{I}(t)\neq\varnothing$ is favored, and we have
$S(\mathbf{R})=S(I\cup R)\approx A_{h}/4G\hbar$. As the black hole evaporates
and its horizon shinks, this yields the decaying part of the Page curve
required by unitarity.

The Page curve result has been extended to asymptotically flat spacetimes
Gautason:2020tmk ; Hartman:2020swn , settings with two layers of holography
Almheiri:2019hni ; Almheiri:2019psy , and eternal black holes Almheiri:2019yqk
. Entanglement islands can also appear in cosmology, where their significance
is less obvious Chen:2020tes ; Hartman:2020khs .

###  1.2 Summary and Outline

Our brief summary of the Page curve result has a major gap. We explained why
condition 3 (global minimization of $S_{\rm gen}$) favors inclusion of the
black hole interior $\hat{I}(t)$ in $E(R)$ after the Page time. However, we
did not show that condition 2 (local extremality) can be satisfied by some
deformation of $\hat{I}(t)$ small enough to preserve condition 3.

One way to fill this gap is to find $I(t)$ exactly, and to verify condition 2.
However, explicit solutions of the quantum extremality condition have been
found only in $1+1$ bulk dimensions Almheiri:2019psf ; Almheiri:2019hni ;
Almheiri:2019yqk , or in toy models of higher-dimensional black holes
Penington:2019npb . The difficulty lies in computing the von Neumann entropy
$S(I\cup R)$. This depends on the detailed state of the dynamics of the
radiation fields, including modes with nonzero angular momentum, and their
interactions. Even free fields scatter nontrivially in a black hole
background, placing an exact calculation out of reach.

In Sec. 2, we develop an alternative way to ensure that condition 2 holds. We
show that the existence of a suitable island $I$ follows from simple
sufficient conditions that are easy to verify:222Ref. Penington:2019npb
presents an elegant existence argument that establishes an island in the
setting of an evaporating black hole. It makes use of properties of the event
horizon and is inequivalent to the more general argument presented here. Let
$I^{\prime}$ be a region that (i) satisfies condition 1, $S_{\rm
gen}(I^{\prime}\cup R)<S(R)$, and suppose that (ii) the generalized entropy of
$I^{\prime}\cup R$ increases under any small outward deformation of $I$, or
decreases under any such deformation. Then there exists an island,
$I\neq\varnothing$; and moreover

$S(\mathbf{R})=S_{\rm gen}(I\cup R)\leq S_{\rm gen}(I^{\prime}\cup R)~{}.$ (5)

We will illustrate in a number of examples that finding a suitable
$I^{\prime}$ is not difficult; in particular, it suffices to understand the
scaling of corrections to simple models of the entropy. Moreover, the upper
bound (5) is powerful enough to establish the main features of the Page curve
for an evaporating black hole.

In Sec. 3, we consider a different but related problem that yields to a
similar analysis. We consider a spacetime and (internal or external) reference
system $R$ in a pure state. We show that the true entropy $S(\mathbf{R})$
cannot exceed the generalized entropy of appropriate bulk regions. For
example, if $R$ is external, and $M$ is an evaporating black hole spacetime,
an upper bound on $S(\mathbf{R})$ is furnished by the generalized entropy of
the bulk region that can be probed by an asymptotic observer (the black hole
exterior).

##  2 Sufficient Conditions for Islands

In this section we identify sufficient conditions for the existence of an
island. In Sec. 2.1 we begin with the case of an external reference system,
$R\cap M=\varnothing$. In Sec. 2.2 we allow $R$ to intersect with $M$. In Sec.
2.3 we consider various examples in which our conditions easily establish the
presence of islands; in particular, we show that they suffice to derive the
Page curve.

###  2.1 External Reference System

Let $M$ be a semiclassical spacetime which together with a reference system
$R$ is in a pure state. We take $R$ to be external to $M$. For definiteness,
we assume that $R$ is nongravitating; otherwise, simply substitute $S\to
S_{\rm gen}$ when the argument contains $R$.

Figure 2: Island finder. Suppose that $I^{\prime}\cup R$ is quantum normal
(top) or anti-normal (bottom). Then the generalized entropy of $I^{\prime}\cup
R$ decreases along the dashed lines to $\tilde{I}^{\prime}\subset\Sigma$. An
island $I$ with even smaller generalized entropy $S_{\rm gen}(I\cup R)$ must
exist on the maximin Cauchy slice $\Sigma$. If $S_{\rm gen}(I^{\prime}\cup
R)<S(R)$, the island cannot be empty.

Suppose there exists a partial Cauchy surface $I^{\prime}\subset M$ satisfying
the following conditions:

$\displaystyle(i)$ $\displaystyle~{}~{}~{}S_{\text{gen}}(I^{\prime}\cup
R)<S(R)~{};$ (6) $\displaystyle(ii)$
$\displaystyle\begin{cases}k^{\mu}\Theta_{\mu}[I^{\prime}\cup R]\geq
0~{},~{}~{}\ell^{\mu}\Theta_{\mu}[I^{\prime}\cup R]\leq 0~{};\\\
\text{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}or}\\\
k^{\mu}\Theta_{\mu}[I^{\prime}\cup R]\leq
0~{},~{}~{}\ell^{\mu}\Theta_{\mu}[I^{\prime}\cup R]\geq 0~{},\end{cases}$ (7)

where $\Theta$ is the quantum expansion one-form Engelhardt:2014gca ;
Bousso:2015mna  and $k$ and $\ell$ are the outward and inward future-directed
null vectors normal to $\partial I^{\prime}$. Thus, for example,
$k^{\mu}\Theta_{\mu}[I^{\prime}\cup R,y]$ is the rate of change of $S_{\rm
gen}$, per unit transverse area and unit affine parameter length, as
$I^{\prime}\cup R$ is deformed outward along the future-directed null geodesic
orthogonal to $I^{\prime}$ at $y$. We drop the argument $y$ when an equation
holds for all $y$.

The first condition states that adjoining $I^{\prime}$ to $R$ decreases the
entropy, even as the Bekenstein-Hawking entropy of $I^{\prime}$ is included.
The second, Eq. (7), says that $I^{\prime}\cup R$ is quantum normal (its
generalized entropy does not decrease under any small outward deformation of
$I^{\prime}$) or quantum anti-normal (the opposite).

We will now show that these conditions imply the existence of a non-empty
island region $I$. Our proof uses the maximin construction of the HRT surface
Wall:2012uf , which we assume extends to a quantum maximin prescription
Akers:2019lzs : on every Cauchy surface of $M$, one finds the region
$I^{\prime\prime}$ that minimizes $S_{\rm gen}(I^{\prime\prime}\cup R)$. (Note
that $I^{\prime\prime}$ may be the empty set.) One then maximizes the same
quantity over all Cauchy surfaces of $M$. The island $I$ is defined to be the
region $I^{\prime\prime}$ that achieves this maximum. (This is expected to
exist Wall:2012uf ; Akers:2019lzs .)

Note that we define $I$ as an achronal region; hence it is non-unique.
Similarly, the maximin Cauchy slice $\Sigma$ is non-unique. The relevant
unique object is the domain of dependence $D(I)$. (We ignore the degenerate
case where there are two islands with identical generalized entropy but
different $D$.) Any other Cauchy slice of $D(I)$ will be an island if $I$ is,
though not every Cauchy slice of $D(I)$ will be part of a maximin slice
$\Sigma$.

Our goal is to show that $I\neq\varnothing$. In the normal case,
$k^{\mu}\Theta_{\mu}[I^{\prime}\cup R]\geq 0$, we define

$\tilde{I}^{\prime}\equiv D(I^{\prime})\cap\Sigma~{},$ (8)

as the representative of $I^{\prime}$ on $\Sigma$. In the anti-normal case,
$k^{\mu}\Theta_{\mu}[I^{\prime}\cup R]\leq 0$, we define the representative
instead as

$\tilde{I}^{\prime}\equiv J(I^{\prime})\cap\Sigma~{},$ (9)

where $J$ denotes all points that can be reached from $I^{\prime}$ along a
causal curve (the future and past of $I^{\prime}$). In either case, note that
$\tilde{I}^{\prime}$ is obtained from $I^{\prime}$ by deforming along an
orthogonal null congruence with initially negative quantum expansion. We
assume the Quantum Focussing Conjecture (QFC) Bousso:2015mna , that the
quantum expansion cannot increase along a null shape deformation. This implies
that

$S_{\rm gen}(\tilde{I}^{\prime}\cup R)\leq S_{\rm gen}(I^{\prime}\cup R)~{}.$
(10)

Since $\Sigma$ is the maximin Cauchy slice,

$S_{\rm gen}(\tilde{I}^{\prime}\cup R)\geq S_{\rm gen}(I\cup R)~{}.$ (11)

Combined with Eq. (10), this implies333This intermediate result is closely
related to corollary 16b of Wall:2012uf . A simple application of our argument
to asymptotically AdS spacetimes yields the following result: given a partial
Cauchy slice $A$ on the asymptotic boundary of AdS, let $X$ be the RT surface
associated to $A$ with homology slice $H$. Now, consider another surface
$X^{\prime}$, homologous to $A$ with homology slice $H^{\prime}$. If
$H^{\prime}$ is a quantum normal or anti-normal region, then $S_{\rm
gen}(H)\leq S_{\rm gen}(H^{\prime})$.

$S_{\rm gen}(I^{\prime}\cup R)\geq S_{\rm gen}(I\cup R)~{}.$ (12)

Using the assumption in Eq. (6), we get

$S_{\rm gen}(I\cup R)<S_{\rm gen}(R)~{}.$ (13)

Therefore, we conclude that $I\neq\varnothing$.

###  2.2 Distant Reference System

In Section 2.1, the reference system $R$ was external to the spacetime $M$. We
will now allow a system that is wholly or partially inside the spacetime:
$R\cap M\neq\varnothing$.

In order to generalize our island finder to this setting, we shall require
that there exists a partial Cauchy slice $I_{0}$ spacelike to $R$ such that
$I_{0}\cup R$ is a quantum normal region with respect to deformations at
$\partial I_{0}$. That is, we require

$\displaystyle\bar{I_{0}}$ $\displaystyle\subset M-\overline{J(R)}~{},$ (14)
$\displaystyle k^{\mu}\Theta_{\mu}[I_{0}\cup R]$ $\displaystyle\geq 0~{},$
(15) $\displaystyle\ell^{\mu}\Theta_{\mu}[I_{0}\cup R]$ $\displaystyle\leq
0~{}.$ (16)

where as before an overbar denotes closure. As before, $\Theta_{\mu}[I_{0}\cup
R]$ is the quantum expansion one-form, and $k^{\mu}$ and $\ell^{\mu}$ are
future-directed null vectors fields in the normal bundle of $\partial I_{0}$
in the outward and inward directions respectively.

For example, these conditions will be satisfied when $I_{0}$ is the interior
of a sufficiently large, approximately round sphere in an asymptotically flat
spacetime; and $R$ is the exterior of a slightly larger sphere concentric with
the first, or any subsystem thereof (such as the Hawking radiation emitted by
a black hole). Note that we do not require that $R$ be weakly gravitating, but
in many examples of interest this will be the case. Also, we do not require
that Eqs. (15) and (16) hold at $\partial R$.

Now, suppose there exists a partial Cauchy slice $I^{\prime}\subset D(I_{0})$
satisfying the conditions (6) and (7). That is, $I^{\prime}\cup R$ is quantum
normal or anti-normal, and adjoining $I^{\prime}$ to $R$ reduces the
generalized entropy of $R$. Then there exists a non-empty quantum extremal
region $\hat{I}\subset D(I_{0})$ satisfying $S_{\rm gen}(\hat{I}\cup R)<S_{\rm
gen}(R)$.

Note that this conclusion is weaker than in Sec. 2.1: $\hat{I}$ need not
globally minimize $S_{\rm gen}(I\cup R)$ among all eligible regions, since we
are restricting our search to a subset of $M$. However, the true entanglement
wedge can only have lower entropy, so $S_{\rm gen}(\hat{I}\cup R)$ provides an
upper bound. Note also that when $I^{\prime}\cup R$ is quantum normal, we can
set $I_{0}=I^{\prime}$, so there is no need to identify a larger $I_{0}$
region.

#### Proof, part I

The proof strategy is similar to that in Section 2.1, except that we wish to
restrict our search to the closed set $D(I_{0})$.444A maximin procedure
restricted to entanglement wedges of AdS without reflecting boundary
conditions was considered in Marolf:2019bgj . See also Appendix B of
Brown:2019rox  for a related discussion. We will first assume strict version
of conditions (15) and (16):

$\displaystyle k^{\mu}\Theta_{\mu}[I_{0}\cup R]$ $\displaystyle>0~{},$ (17)
$\displaystyle\ell^{\mu}\Theta_{\mu}[I_{0}\cup R]$ $\displaystyle<0~{},$ (18)

Later, we will demonstrate how to relax this assumption back to conditions
(15) and (16).

Let $\hat{I}$ be the maximin region of $D(I_{0})\cup R$, and let $\Sigma$ be a
Cauchy surface of $D(I_{0})$ on which $\hat{I}$ minimizes the generalized
entropy of $\hat{I}\cup R$ among all subregions of $\Sigma$. Without loss of
generality, we may take $\hat{I}\subset\Sigma$: since any Cauchy surface of
$D(\hat{I})$ is an equally good maximin region, we set $\hat{I}\to\Sigma\cap
D(\hat{I})$. As in Ref. Wall:2012uf , we assume that $\hat{I}$ is stable: any
nearby Cauchy surface $\Sigma^{\prime}$ obtained by infinitesimal deformation
of $\Sigma$ contains a locally minimal region $\hat{I}^{\prime}$
infinitesimally close to $\hat{I}$ with $S_{\rm gen}(\hat{I}^{\prime}\cup
R)\leq S_{\rm gen}(\hat{I}\cup R)$.

It immediately follows from the analysis of Sec. 2.1 that

$\displaystyle S_{\rm gen}(\hat{I}\cup R)<S_{\rm gen}(R)~{}.$ (19)

We will now show that $\partial\hat{I}\cap\partial D(I_{0})=\varnothing$. This
precludes the (unwanted) possibility that $I\cup R$ is maximin but not locally
stationary. It follows that $\hat{I}\cup R$ is a quantum extremal region
Wall:2012uf .

$\partial D(I_{0})$ is the disjoint union of three sets: $\partial I_{0}$, and
two null hypersurfaces $N_{+\ell}$ and $N_{-k}$ that lie in the future and
past of $I_{0}$ respectively. The latter sets are generated by future and
past-directed null geodesics orthogonal to $\partial I_{0}$ in the inward
direction which end at caustics or self-intersections Akers:2017nrr . Let
$\ell^{\mu}$ ($k^{\mu}$) be the normal vector field to $N_{+\ell}$ ($N_{-k}$)
obtained by parallel propagation of $\ell^{\mu}|_{\partial I_{0}}$
($k^{\mu}|_{\partial I_{0}}$).

Let $\Sigma_{M}$ be a Cauchy surface of $M$ that intersects each null
generator of $N_{+\ell}$ and $N_{-k}$ at most at one point. Let
$\Sigma_{N}\equiv\Sigma_{M}\cap D(I_{0})$. By Eqs. (17), (18), and the QFC
Bousso:2015mna ,

$\displaystyle k^{\mu}\Theta_{\mu}[\Sigma_{N}\cup R;p]$
$\displaystyle>0~{}~{}~{}\text{ for all }p\in N_{-k}\cap\partial I_{0}~{};$
(20) $\displaystyle\ell^{\mu}\Theta_{\mu}[\Sigma_{N}\cup R;p]$
$\displaystyle<0~{}~{}~{}\text{ for all }p\in N_{+\ell}\cap\partial I_{0}~{},$
(21)

Suppose for contradiction that there exists a point $q\in\partial
D(I_{0})\cap\partial\hat{I}$. The generator of $\partial D(I_{0})$ that
contains $q$, and hence its tangent vector $\ell^{\mu}$ or $k^{\mu}$, will be
normal to $\partial\hat{I}$ at $q$. (If $q\in\partial I_{0}$, this statement
holds for either generator emanating from $q$.) Since $N_{+\ell}$ ($N_{-k}$)
is nowhere to the past (future) of $\hat{I}$, theorem 1 in Ref. C:2013uza
implies

$\displaystyle k^{\mu}\Theta_{\mu}[\hat{I}\cup R;q]$
$\displaystyle>0~{}~{}~{}\text{ for }q\in N_{-k}\cap\partial I_{0}~{};$ (22)
$\displaystyle\ell^{\mu}\Theta_{\mu}[\hat{I}\cup R;q]$
$\displaystyle<0~{}~{}~{}\text{ for }q\in N_{+\ell}\cap\partial I_{0}~{}.$
(23)

Suppose first that $q\in\partial I_{0}$. In this case the above expansions
imply that a small inward deformation of $\hat{I}$ will decrease the
generalized entropy, in contradiction with the minimality of $S_{\rm
gen}(\hat{I}\cup R)$ among all subregions of the maximin Cauchy surface
$\Sigma$.

We will now demonstrate this rigorously, using the notion of a surface-
orthogonal exponential map Akers:2017nrr ,

$\exp_{K}~{}:~{}NK\to M~{},~{}~{}(p,v)\to c_{p,v}(1)~{},$ (24)

which takes a vector $v$ in the normal bundle $NK$ of a smooth submanifold $K$
at the point $p$ to the point at affine distance 1 along the unique geodesic
through $p$ with tangent vector $v$. We will use $\widetilde{\exp}$ to denote
an exponential map in which the submanifold $\Sigma$ plays the role of $M$ in
the above definition.

Let $\tilde{v}^{\mu}(p)$ be a smooth inward-directed vector field in the
normal bundle to $\partial\hat{I}$, viewed as a submanifold of $\Sigma$. We
define the continuous one-parameter family of inward deformations of $\hat{I}$
as the regions

$\displaystyle\hat{I}(\epsilon)=\text{int}\\{\widetilde{\exp}_{\partial\hat{I}}(p,\epsilon\tilde{v}^{\mu}(p)):p\in\partial\hat{I}\\}~{}.$
(25)

Similarly, in the manifold $M$ we define

$\displaystyle
i(\epsilon)=\hat{I}\cap\text{Int}\\{\exp_{\partial\hat{I}}(p,\epsilon
v^{\mu}(p)):p\in\partial\hat{I}\\}~{},$ (26)

where $v$ is the push-forward of $\tilde{v}$ under the embedding map of
$\Sigma$ into $M$, and $\text{Int}\,X$ denotes the spacetime region spacelike
and interior to $X$. For sufficiently small $\epsilon$, both families are
well-defined. Moreover,

$S_{\rm gen}[\hat{I}(\epsilon)\cup R]=S_{\rm gen}[i(\epsilon)\cup
R]+O(\epsilon^{2})~{}.$ (27)

In $M$, the deformation profile $v^{\mu}\in N\partial\hat{I}$ can be
decomposed as

$\displaystyle v^{\mu}=-ak^{\mu}+b\ell^{\mu}~{},$ (28)

where $a$ and $b$ are positive definite functions. Hence

$\displaystyle\left.\frac{dS_{\rm gen}[\hat{I}(\epsilon)\cup
R]}{d\epsilon}\right|_{\epsilon=0}=\int
dx\sqrt{h}~{}v^{\mu}\Theta_{\mu}[\hat{I}\cup R]~{},$ (29)

where $h$ refers to the intrinsic metric of $\partial\hat{I}$. We now choose
$\tilde{v}\equiv 0$ outside a $\delta$-neighborhood of $q$ in
$\partial\hat{I}$. For small enough $\delta$, $v^{\mu}\Theta_{\mu}[\hat{I}\cup
R]<0$ in the entire $\delta$-neighborhood, by Eqs. (22) and (23) and
continuity. Hence,

$\left.\frac{dS_{\rm gen}[\hat{I}(\epsilon)\cup
R]}{d\epsilon}\right|_{\epsilon=0}=\left.\frac{dS_{\rm gen}[i(\epsilon)\cup
R]}{d\epsilon}\right|_{\epsilon=0}<0~{}.$ (30)

Hence, $\hat{I}$ does not minimize the generalized entropy on $\Sigma$. This
contradicts our assumption that $\hat{I}$ is a maximin region. Therefore, no
such point $q\in\partial\hat{I}\cap\partial I_{0}$ can exist:

$\partial\hat{I}\cap\partial I_{0}=\varnothing~{}.$ (31)

Suppose instead that $q\in\partial D(I_{0})-\partial I_{0}$. In this case,
minimality of $S_{\rm gen}(\hat{I}\cup R)$ on $\Sigma$, together with Eq. (22)
or (23), implies that a small deformation of $\Sigma$ into the interior of
$D(I_{0})$ near $q$ will produce a Cauchy surface of $I_{0}$ whose
minimal-$S_{\rm gen}$ region (together with $R$) has greater generalized
entropy than $\hat{I}\cup R$. But this is impossible if $\hat{I}$ was
constructed by the maximin procedure. Again we will now aim to make this
argument rigorous.

Figure 3: Maximin restricted to the domain of dependence (“wedge”) $D(I_{0})$
returns a region $\hat{I}$ on a maximin slice $\Sigma$. If $I_{0}\cup R$ is
quantum normal, then $\partial\hat{I}$ cannot intersect $\partial D(I_{0})$
(dashed). Left: if $\hat{I}\cap\partial I_{0}\neq\varnothing$, then $S_{\rm
gen}(\hat{I}(\epsilon)\cup R)<S_{\rm gen}(\hat{I}\cup R)$, contradicting the
min of maximin. Right: if $\hat{I}\cap\partial D(I_{0})-\partial I_{0}$ then
$S_{\rm gen}(\hat{I}(\epsilon)\cup R)>S_{\rm gen}(\hat{I}\cup R)$ on the
deformed slice $\Sigma(\epsilon)$ violates the max of maximin.

For definiteness, we assume that $q\in N_{+\ell}$. (If instead $q\in N_{-k}$,
the time reverse of our argument applies.) In any open neighborhood $O(q)$,
$\Sigma$ (and hence $\hat{I}$) must enter the interior,
$O(q)\cap\Sigma\cap\text{int}[D(I_{0})]\neq\varnothing$, or else Eq. (23)
would violate the minimality of $S_{\rm gen}(\hat{I}\cup R)$ on $\Sigma$.
Hence the inward-directed vector field $t^{\mu}$ orthogonal to
$\partial\hat{I}$ and tangent to $\hat{I}$ is spacelike in an open
neighborhood of $q$ on $\partial\hat{I}$. Moreover, $\Sigma$ must contain the
null generator segment of $N_{+\ell}$ connecting $\partial I_{0}$ to $q$, or
$\Sigma$ would fail to be achronal.

Let the achronal hypersurfaces $\Sigma(\epsilon)$ be a smooth one parameter
deformation of $\Sigma$ such that $\Sigma(\epsilon)$ agrees with $\Sigma$
outside a $\delta$-neighborhood of $q$ (and everywhere for $\epsilon=0$). We
also require that $\Sigma(\epsilon_{2})$ is nowhere to the future of
$\Sigma(\epsilon_{1})$ if $\epsilon_{1}<\epsilon_{2}$. The stability
assumption guarantees the existence of a smooth one-parameter family of
regions $\hat{I}(\epsilon)$, each of which locally minimizes $S_{\rm
gen}[\hat{I}(\epsilon)\cup R]$ on the corresponding $\Sigma(\epsilon)$. By the
max step of maximin,

$\frac{dS_{\rm gen}[\hat{I}(\epsilon)\cup R]}{d\epsilon}\leq 0~{}.$ (32)

For small enough $\epsilon$, there exists an infinitesimal vector field
$w^{\mu}$ in the normal bundle of $\partial\hat{I}$ in a neighborhood of $q$
such that

$\partial\hat{I}(\epsilon)=\\{\exp_{\partial\hat{I}}(p,\epsilon
w^{\mu}(p)+O(\epsilon^{2})):p\in\partial\hat{I}\\}~{}.$ (33)

We have

$\frac{dS_{\rm gen}[\hat{I}(\epsilon)\cup R]}{d\epsilon}=\int
dx\sqrt{h}~{}w^{\mu}\Theta_{\mu}{[\hat{I}\cup R]}~{}.$ (34)

Since $t^{\mu}$ is spacelike, it is linearly independent of $\ell^{\mu}$, so
there exists a unique decomposition

$w^{\mu}=ct^{\mu}+d\,(-\ell^{\mu})~{}.$ (35)

with $c$ and $d$ nonnegative functions on $\partial\hat{I}$ that vanish
outside an open neighborhood of $q$. Hence

$\frac{S_{\rm gen}[\hat{I}\cup R]}{d\epsilon}=\int
dx\sqrt{h}~{}[ct^{\mu}\Theta_{\mu}+d\,(-\ell^{\mu})\Theta_{\mu}]~{}.$ (36)

The first term is positive-definite by the minimality of $S_{\rm
gen}[\hat{I}\cup R]$ on $\Sigma$; the second is positive by Eq. (23). Hence

$\frac{S_{\rm gen}[\hat{I}\cup R]}{d\epsilon}>0~{},$ (37)

which contradicts Eq. (32). Therefore,

$\partial\hat{I}\cap N_{+\ell}=\varnothing~{},$ (38)

and a time-reversed argument implies

$\partial\hat{I}\cap N_{-k}=\varnothing~{}.$ (39)

Together with Eq. (31) this establishes that

$\hat{I}\subset\text{int}[D(I_{0})]~{};$ (40)

hence $\hat{I}$ is locally quantum extremal.

#### Proof, part II

We will now discuss what happens if we relax the conditions (17) and (18) to
their non-strict versions (15) and (16). We will argue that while in this case
$\hat{I}$ might not be contained in $\text{int}[D(I_{0})]$, an island
candidate still exists, i.e., there exists $\hat{I}\subset D(I_{0})$ such that
$\hat{I}\cup R$ is quantum extremal and $S_{\rm gen}(\hat{I}\cup R)<S_{\rm
gen}(R)$.

Let us start with the case where there exists at least a point $p\in\partial
I_{0}$ where $k^{\mu}\Theta_{\mu}[I_{0}\cup R;p]>0$ and a point $q$ ($p=q$ is
allowed) where $\ell^{\mu}\Theta_{\mu}[I_{0}\cup R;p]<0$. Then,
$\partial\hat{I}$ cannot be a cross section of $N_{-k}$ ($N_{+\ell}$) because
then by the QFC there would be a point $r$ in the cross section, along the
same generator as $p$ ($q$), where $k^{\mu}\Theta_{\mu}[\hat{I}\cup R;r]>0$
($\ell^{\mu}\Theta_{\mu}[\hat{I}\cup R;r]<0$). As discussed above, this
contradicts maximin.

If $\partial\hat{I}$ only partially coincides with $\partial D(I_{0})$ then
there must exist a point $r$ in the boundary of the intersection set such that
in any sufficiently small neighborhood of it $\partial\hat{I}$ and $N_{-k}$
($N_{+\ell}$) do not coincide. In the $\hbar\to 0$ limit, Lemma B of C:2013uza
implies that there exists a point $s$ in a neighborhood of $r$ such that
$k^{\mu}\theta_{\mu}[\hat{I};s]>0$ ($\ell^{\mu}\theta_{\mu}[\hat{I};s]<0$).
Here, we will assume that the Lemma B of C:2013uza  continues to hold when we
replace the classical expansion with the quantum expansion.555Our assumption
is motivated by the semiclassical generalization of many similar conditions on
the classical expansion Bousso:2015mna . Note that if the von Neumann entropy
term in the quantum expansion is $O(G\hbar)$ while the classical expansion is
$O(1)$, Lemma B trivially generalizes to the version with quantum expansions.
We will then conclude that $k^{\mu}\Theta_{\mu}[\partial\hat{I}\cup R;s]>0$
($\ell^{\mu}\Theta_{\mu}[\partial\hat{I}\cup R;s]<0$). However,
$s\in\text{int}[D(I_{0})]$, so a non-zero quantum expansion at $s$ is not
allowed by maximin.

Next, we consider the case where $\ell^{\mu}\Theta_{\mu}[I_{0}\cup R]=0$ (The
case with $k$ and $\ell$ exchanged is similar by time-reflection symmetry). By
the previous arguments, $\partial\hat{I}$ cannot intersect $N_{-k}$. Also, by
the quantum version of Lemma B, $\partial\hat{I}$ cannot intersect $N_{+\ell}$
only in part. We will therefore focus on discussing the case where
$\partial\hat{I}$ is a cross section of $N_{+\ell}$.

Associated with any cross section $L$ of $N_{+\ell}$, we can define a partial
Cauchy slice $\Sigma_{L}$ of $D(I_{0})$ which intersects $N_{+\ell}$ at $L$.
Let $L_{1}$ be the latest cross section of $N_{+\ell}$ such that any other
cross section $L$ in the past of $L_{1}$ satisfies
$\ell^{\mu}\Theta_{\mu}[\Sigma_{L}\cup R]=0$.666If such $L_{1}$ does not
exists, then $N_{+\ell}$ is a semi-infinite stationary null hypersurface
which, at least classically, by the no-hair theorem needs to be a semi-
infinite portion of the horizon of a Kerr-Newman black hole. And since
$k^{\mu}\Theta_{\mu}[I_{0}\cup R]\geq 0$, $\partial I_{0}$ needs to lie fully
in the past of the bifurcation surface which provides us with a classical
extremal surface on $N_{+\ell}$. We then expect to find a quantum extremal
region $\hat{I}\cup R$ such that $\partial\hat{I}$ is either on or near this
classical extremal surface. By the QFC, any cross section $L$ which is partly
in the future of $L_{1}$ needs to contain at least a point $r$ at which
$\ell^{\mu}\Theta_{\mu}[\Sigma_{L}\cup R;r]<0$. Hence, $\partial\hat{I}$ must
be a cross section in the past of $L_{1}$. Furthermore, a maximin Cauchy slice
$\Sigma$ corresponding to $\hat{I}$ cannot intersect the future of $L_{1}$ as
it would violate the minimality of $S_{\rm gen}(\hat{I}\cup R)$. $\Sigma$ then
has to leave $N_{+\ell}$ in the past of $L_{1}$ or on $L_{1}$. Let $L_{2}$ be
the cross section at which $\Sigma$ leaves $N_{+\ell}$. Then,
$k^{\mu}\Theta_{\mu}[\Sigma_{L_{2}}\cup R]\leq 0$ or else the min condition is
violated. Since $k^{\mu}\Theta_{\mu}[I_{0}\cup R]\geq 0$, we expect that there
exists a cross section $L_{3}$ between $\partial I_{0}$ and $L_{2}$ such that
$k^{\mu}\Theta_{\mu}[\Sigma_{L_{3}}\cup R]=0$. In the classical limit in
particular, we expect that the results of Andersson:2007gy  showing a similar
existence on spacelike Cauchy surfaces can be applied here by taking a limit
of spatial Cauchy surfaces that approach $N_{+\ell}$. Together with
stationarity along $N_{+\ell}$, this would imply that $\Sigma_{L_{3}}\cup R$
is quantum extremal.

###  2.3 Examples

We will now present some examples where the above sufficient conditions (i)
and (ii) provide an efficient diagnostic for the existence of islands. We will
require no detailed calculations of matter entropy and its derivatives, nor
will we be forced to assume special symmetries, low dimensions of spacetime,
or adopt other toy models. Our sufficient conditions establish the existence
of an island and its key properties, at the cost of not exactly locating the
island.

In a final example, we show that neither of the two sufficient conditions can
be eliminated.

#### Evaporating black hole after the Page time

For concreteness, we pick asymptotically flat boundary conditions, though our
conclusion will not depend on this choice. We furthermore assume that the
black hole mostly radiates massless particles, as will be the case if its
initial mass is sufficiently large.

We consider an evaporating black hole formed in a pure state. At late times,
the spacetime will be approximately spherically symmetric. Let
$r=(A/4\pi)^{1/2}$ be the area radius of spheres. Near the horizon, the metric
is well-approximated by

$ds^{2}=-\left(1-\frac{r_{s}(v)}{r}\right)dv^{2}+2dv\,dr+r^{2}d\Omega^{2}~{},$
(41)

Due to evaporation, $r_{s}(v)$ decreases slowly with retarded time $v$:
$dr_{s}/dv\sim-O(G\hbar/r_{s}^{2})$. For $r\gg r_{s}$, the metric is well-
approximated instead by the outgoing Vaidya metric Abdolrahimi:2016emo , but
this will not be important in our analysis.

Figure 4: Evaporating black hole after the Page time. Hawking radiation has
accumulated in $R$. As shown on the left, the boundary of the past of $R$,
denoted by $r_{0}(v)$, intersects the stretched horizon (shown in purple) at
the sphere $A_{s}$, which together with the $A_{h}$ sphere on the event
horizons reside at retarded time $v_{s}$. We consider candidate regions
$I^{\prime}$ with boundary $\partial I^{\prime}$ on a causal horizon spacelike
to $R$ (grey regions). The generalized second law implies that $S_{\rm
gen}(I^{\prime}\cup R)$ increases under future outward deformations. For
future inward deformations, quantum normalcy follows from the (trivial)
classical normalcy in the dark grey subregion, which is chosen to keep quantum
corrections to the expansion small. The $I^{\prime}$ that minimizes $S_{\rm
gen}(I^{\prime}\cup R)$ subject to these restrictions is shown in pink. Its
boundary is located a $\Delta v=r_{s}\log c_{1}$ to the future of $A_{h}$, as
shown on the right. We show that it provides an extremely tight upper bound on
the true entropy $S(\mathbf{R})=S_{\rm gen}(I\cup R)\leq S_{\rm
gen}(I^{\prime}\cup R)=A_{h}/4G\hbar+O(1)$.

Let $u$ be retarded time on $\mathscr{I}^{+}$, and let $R$ be the portion of
$\mathscr{I}^{+}$ given by $u\leq u_{0}$; see Fig. 4. $R$ is a reservoir that
contains the Hawking radiation emitted until the time $u_{0}$. The boundary of
the past of $R$ is given by $u=u_{0}$. We will be interested in the behavior
of the metric only near the retarded time when this surface intersects the
stretched horizon $r_{s}$, so it will be sufficient to set $r_{s}$ to that
value and neglect its $v$-dependence from here on. Let $A_{s}=4\pi r_{s}^{2}$
be the area of the stretched horizon where it meets the past of $R$. Let
$A_{h}=4\pi r_{h}^{2}$ be the area of the event horizon where it intersects
the future of $A_{s}$; the areas satisfy

$A_{s}=A_{h}+O(G\hbar)~{}.$ (42)

We choose $u_{0}$ late enough so that $S(R)>A_{h}/4G\hbar+\log c_{1}$, where
$\log c_{1}$ will be small in a sense made precise below. That is, $R$ extends
to after the Page time, with a little room to spare. We seek an $I^{\prime}$
that satisfies our sufficient conditions while placing a tight bound on the
entropy $S(\mathbf{R})$.

Let $I^{\prime}(r,v)$ be a Cauchy slice of the interior of the sphere $(r,v)$.
Since $S_{\rm gen}(I^{\prime}\cup R)$ is well-defined only for achronal
$I^{\prime}\cup R$, we require $I^{\prime}(r,v)\subset M-J^{-}(R)$, or
$u>u_{0}$. In the ingoing coordinates of Eq. (41), $u=u_{0}$ corresponds to a
function $r_{0}(v)$, defined implicitly by $v=u_{0}+2r_{*}(r_{0})$, where
$r_{*}(r)=r+r_{h}\log(\frac{r}{r_{h}}-1)$. Near the horizon, this satisfies

$\Delta r_{0}(v)\equiv
r_{0}(v)-r_{h}=r_{h}\exp\left(\frac{v-u_{0}}{2r_{h}}-1\right)~{}.$ (43)

We thus require $r<r_{0}(v)$ for the boundary of $I^{\prime}$.

Quantum normalcy of $I^{\prime}\cup R$ requires that the generalized entropy
grows along both of the null directions away from $I^{\prime}$. Any future
outward light-cone outside the horizon is a null surface of constant $u$ that
reaches $\mathscr{I}^{+}$. Hence it is a causal horizon. The generalized
second law of thermodynamics Bekenstein:1972tm ; Wall:2011hj  applies to all
causal horizons. It implies that the future outward quantum expansion at
$I^{\prime}$ is positive if $\partial I^{\prime}$ is outside the horizon. (In
more general settings that are not exactly spherically symmetric, we can
accomplish the same goal by choosing $\partial I^{\prime}$ to be a cut of a
causal horizon.) We thus require $r>r_{h}$.

The past outward classical expansion is trivially positive:
$\partial_{r}A=8\pi r$. Quantum normalcy follows if the quantum correction,
$4G\hbar\,\partial_{r}S(I^{\prime}\cup R)$, is negligible, i.e., if
$\partial_{r}S(I^{\prime}\cup R)\ll r/G\hbar$. Let $\Sigma$ be a global Cauchy
slice containing $I^{\prime}\cup R$ and define
$I^{\prime}_{c}=\Sigma-(I^{\prime}\cup R)$. By purity of the global quantum
state, $S(I^{\prime}\cup R)=S(I^{\prime}_{c})$.

Dimensional analysis dictates that the leading divergence in the renormalized
entropy scales as

$\partial_{\rho}S(I^{\prime}_{c})\sim O(1/\rho)~{},$ (44)

where $\rho=r_{0}-r$ and we may assume $\rho\ll r_{0}$. To see this, suppose
first that the only available scales are $r$ and $\rho$. Terms stronger than
Eq. (44) would be of the form $\partial_{r}S(I^{\prime}_{c})\sim
r^{n}/\rho^{n+1}$ with $n>0$ and positive coefficient. Such a term would imply
that at fixed $\rho$, $dS/dr<0$, which is not physically sensible. In the
presence of an additional mass scale $m\sim\hbar/\lambda$, an enhancement of
Eq. (44) would have to take the form
$\partial_{r}S(I^{\prime}_{c})\sim\lambda^{n}/\rho^{n+1}$, $n>0$. Formally,
this is an enhancement for $\rho\ll\lambda$, but physically, a mass scale
cannot have any physical effect in this UV regime.

Hence, quantum normalcy is assured if we require

$r_{0}-r>c_{1}\frac{G\hbar}{r}~{},~{}~{}c_{1}\gg 1~{}.$ (45)

To summarize, we may consider any $I^{\prime}$ whose boundary is in the range

$r_{h}\leq r\leq r_{0}(v)+c_{1}\frac{G\hbar}{r}~{},~{}~{}c_{1}\gg 1~{}.$ (46)

We now minimize $S_{\rm gen}(I^{\prime}\cup R)$ over this range. By purity,
$S_{\rm gen}(I^{\prime}\cup R)=4\pi r^{2}+4G\hbar S(I^{\prime}_{c})$. Along
any ingoing light-cone, the classical area decreases rapidly and $\Delta r$
only increases as we go to smaller $r$, so we are driven to the smallest $r$
in the search space, the event horizon. Scanning in the other null direction,
along the event horizon, $S_{\rm gen}(I^{\prime}_{c})$ will decrease towards
the past, by the GSL.

Hence we obtain the tightest upper bound on $S(\mathbf{R})$ by choosing
$I^{\prime}$ to be the interior of the event horizon, as early as is possible
while maintaining Eq. (45). With the boundary of $I^{\prime}$ on the event
horizon, $r-r_{0}=\Delta r_{0}\propto\exp(v/2r_{h})$ by Eq. (43). Moreover, at
$A_{h}$, we have $r_{0}=r_{s}$ and hence $\Delta r_{0}\sim O(G\hbar/r_{h})$.
To grow this by the factor of $c_{1}$ demanded in Eq. (45), we must choose
$v=v_{s}+r_{h}\log c_{1}$, where $v_{s}$ is the $v$-coordinate of $A_{s}$ and
$A_{h}$.

To summarize, the optimal choice of $I^{\prime}$ is

$(r,v)=(r_{h},v_{s}+r_{h}\log c_{1})~{},~{}~{}c_{1}\gg 1~{}.$ (47)

The true entropy $S(\mathbf{R})$ is upper bounded by

$S_{\rm gen}(I^{\prime}\cup R)=\frac{\pi
r^{2}}{G\hbar}+S(I^{\prime}_{c})=\frac{A_{h}}{4G\hbar}+O(\log c_{1})~{}.$ (48)

Note that the $O(G\hbar)$ area difference between the event horizon and the
stretched horizon is negligible. The $O(\log c_{1})$ term captures both the
(negative) correction to the horizon area due to evaporation since $A_{h}$,
and the (positive, and larger) correction due to the von Neumann entropy of
$S(I^{\prime}_{c})$.

We stress that this upper bound is quite tight. The correct $S(\mathbf{R})$ is
given by Eq. (48) with $O(\log c_{1})$ replaced by $O(1)$. Recall that $c_{1}$
should be large enough to overcome any $O(1)$ coefficients that might enhance
the von Neumann entropy in an exact calculation. But it is itself $O(1)$ in
that sense, and $\log c_{1}$ is even smaller. In particular, we can always
choose $\log c_{1}\ll\log\log(A_{h}/G\hbar)$ in the semiclassical limit.

We also emphasize that exact spherical symmetry is not crucial; our argument
only relies on the scaling behavior of the relevant terms.

#### Recollapsing flat universe

Our next

Figure 5: Spatially flat radiation-dominated universe with negative
cosmological constant, purified by a reference universe (thermal Minkowski
space, right). If we choose a large enough reference region $R$ at $t_{\rm
Mink}=0$, then the region $I^{\prime}$ at the turnaround time $t=0$ satisfies
our sufficient conditions. Therefore an island $I$ must exist.

example was studied in detail in Ref. Hartman:2020khs .777Ref. Hartman:2020khs
considered a different but related question to ours: given a region $I$ in the
cosmology, can one find a region $R$ in the reference spacetime such that $I$
is an island with respect to $R$. By contrast, we specify a reference region
$R$ and use our sufficient conditions to establish the existence of an island
for it. Consider a radiation-dominated, spatially flat Friedmann-Robertson-
Walker (FRW) universe $M$ with cosmological constant $\Lambda$, purified by a
thermal state on a Minkowski background $M_{R}$ without gravity. The metric of
$M$ and $M_{R}$ is

$\displaystyle ds^{2}$
$\displaystyle=-dt^{2}+a(t)^{2}(dr^{2}+r^{2}\,d\Omega^{2})~{},$ (49)
$\displaystyle ds_{R}^{2}$
$\displaystyle=-dt_{R}^{2}+dr_{R}^{2}+r_{R}^{2}\,d\Omega_{R}^{2}~{}.$ (50)

Without loss of generality, one can set the scale factor $a(0)=1$ at the
turnaround time $t=0$, when $da/dt=0$ and hence $-\Lambda/8\pi G=\rho_{\rm
rad}$. A thermofield double (TFD) state is constructed at $t=0$, $t_{R}=0$.

A simple implication of the TFD state suffices for the purposes of our
analysis. Consider two spatial regions, one in $M$ at $t=0$ and the other in
$M_{R}$ at $t_{R}=0$. The von Neumann entropy of their union vanishes
approximately, if they have the same spatial coordinates. If the regions are
unequal, then the von Neumann entropy of their union will be given by the sum
of the thermal entropy of the nonoverlap portions:

$S=s_{\rm rad}(\hat{V}+\hat{V}_{R})~{},$ (51)

where $\hat{V}$ and $\hat{V}_{R}$ are the volumes of the nonoverlap portions
in $M$ and in $M_{R}$, and the entropy density is $s_{\rm rad}\sim\rho_{\rm
rad}^{3/4}$. These statements receive corrections on scales below the thermal
length scale, $\lambda\sim\rho_{\rm rad}^{-1/4}$.

Now choose $R\subset M_{R}$ to be a ball of radius $r_{R}$ at $t_{R}=0$, and
$I^{\prime}\subset M$ a ball of radius $r<r_{R}$ at $t=0$. By time symmetry
around $t=t_{R}=0$, the region $I^{\prime}\cup R$ will be quantum normal or
anti-normal. We have

$S_{\rm gen}(I^{\prime}\cup R)-S(R)=\frac{\pi r^{2}}{G\hbar}-\frac{4\pi s_{\rm
rad}}{3}r^{3}~{}.$ (52)

To satisfy our second condition, this must be negative, so we require

$r>r_{\rm crit}\equiv\frac{3}{4\pi Gs_{\rm rad}}$ (53)

This condition on $I^{\prime}$ can be satisfied, and hence an island $I\subset
M$ must exist, for a sufficiently large reference region, $r_{R}>r_{\rm
crit}$.

Going beyond spherical symmetry, we can choose $R$ to be any convex reference
region of arbitrary shape in $M_{R}$, and let $I^{\prime}$ be the identical
coordinate region in $M$. Then $I^{\prime}\cup R$ will be normal or anti-
normal by convexity and time symmetry. Moreover,

$S_{\rm gen}(I^{\prime}\cup R)-S(R)=\frac{A[\partial
I^{\prime}]}{4G\hbar}-\left(V_{R}+O(A[\partial R]\lambda)\right)s_{\rm rad}$
(54)

will be negative for any sufficiently large region of fixed shape. Any such
references region must have an island $I$.

Note that $I$ will not be the identical coordinate region to $R$, because of
the $O(A[\partial I^{\prime}]\lambda)$ corrections to the von Neumann entropy.
Moreover, in the nonspherical case, minimization of the area term will favor a
more round shape for $I$ than for $R$.

#### Bag of gold

Figure 6: A time-symmetric Cauchy slice of a bag-of-gold geometry. The bag has
large entropy (grey) compared to the area of its throat, and it purifies the
reference system $R$. Then it is easy to find a (classically and quantum)
anti-normal region $I^{\prime}$ (pink) such that $S_{\rm gen}(I^{\prime}\cup
R)<S(R)$. Hence, there must exist an island $I$. $I$ is expected to be
approximately the interior of the classically minimal surface labelled
$r_{0}$.

Next, we discuss a time-symmetric slice of a “bag-of-gold” geometry
Marolf:2008tx , shown in Fig. 6. Its defining feature is the existence of an
arbitrarily large volume of space behind a throat (a minimal area surface) of
fixed area. To construct it, we glue the interior of a sphere of radius
$r_{1}$ of a closed FRW universe, at the time of recollapse, $da/dt=0$, to the
exterior of a sphere of the same size behind the bifurcation surface in a
maximally extended Schwarzschild spacetime Horowitz:1983vn . The corresponding
spatial metrics are:

$\displaystyle ds_{\rm in}^{2}$ $\displaystyle=a(d\chi^{2}+\sin^{2}\chi
d\Omega^{2})~{},~{}~{}$ $\displaystyle 0\leq\chi\leq\chi_{1}~{};$ (55)
$\displaystyle ds_{\rm out}^{2}$
$\displaystyle=\left(1-\frac{r_{0}}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}~{},~{}~{}$
$\displaystyle r_{0}\leq r\leq r_{1}~{},$ (56)

where in the second line we omitted the portion of the Schwarzschild metric
outside of the bifurcation surface as it will not be needed for the analysis
below.

Let $r_{0}$ be the radius of the throat, and let $\chi$ be the angle at which
the metrics are glued. The gravitational constraints imply $\chi_{1}>\pi/2$
and

$\displaystyle a\sin\chi_{1}$ $\displaystyle=r_{1}~{},$ (57) $\displaystyle
a\sin^{3}\chi_{1}$ $\displaystyle=r_{0}~{}.$ (58)

The Friedmann equation implies that the energy density in the bag is

$\rho=\frac{1}{8\pi Ga^{2}}~{}.$ (59)

Now suppose that the bag contains thermal photon radiation purified by a
external reference system $R$. The entropy density in the bag is
$s\sim\rho^{3/4}$, and hence

$S(R)\sim(G\hbar)^{-3/4}a^{3/2}~{}.$ (60)

Let $I^{\prime}$ be the interior of some sphere $r^{\prime}$ between the edge
of the bag, $r_{1}$, and the throat, $r_{0}$; hence

$S_{\rm gen}(I^{\prime}\cup R)\sim\frac{r^{\prime 2}}{G\hbar}~{}.$ (61)

By time reversal symmetry, $I^{\prime}\cup R$ is quantum normal or anti-
normal. Moreover, we can achieve $S(R)>S_{\rm gen}(I^{\prime}\cup R)$, by an
arbitrarily large margin, by taking $a$ large while holding $r^{\prime}$ and
$r_{0}$ fixed. (This will only increase $r_{1}$.) Hence, our conditions are
satisfied, and a nontrivial island $I\subset M$ must exist.

Importantly, this construction is insensitive to the spherical symmetry that
we assumed for simplicity. It is also insensitive to the addition of
perturbative matter near the throat. Such modifications can affect the precise
position of the island, which may be very hard to determine. But so long as
they are small enough, our sufficient conditions will hold, and they guarantee
the existence of an island.

#### Collapsing star (an example without islands)

Figure 7: Collapse of a spherical star that is maximally entangled with a
distant reservoir $R$. $I^{\prime}$ (pink) is the interior of a sphere
surrounding the star at a time close to the singularity. Hence $\partial
I^{\prime}$ has a small area, and condition (i) can be satisfied by an
arbitrarily large margin. But $I^{\prime}$ is quantum trapped and so fails to
satisfy condition (ii). Indeed, $R$ does not possess any island in this
spacetime.

To illustrate the importance of condition (ii), let us discuss a case for
which condition (i), Eq. (6), is satisfied, but condition (ii), Eq. (7), is
violated. Consider an Oppenheimer–Snyder spacetime: a black hole formed by the
collapse of a “star,” modeled as a spherical, homogeneous ball of dust.

Suppose that the star is in a maximally mixed state with entropy $S_{\rm
star}$ and let $R$ be an early portion of $\mathscr{I}^{+}$ which contains
only a purification of the star (and no Hawking radiation), giving
$S(R)=S_{\rm star}$. We choose $I^{\prime}$ to be the interior of a sphere
just outside of the star and very close to the singularity (see Fig. 7). Then
$S_{\rm gen}(I^{\prime}\cup R)\approx A(\partial I^{\prime})/4G\hbar$. Picking
$S_{\rm star}$ large with $A(\partial I^{\prime})$ held fixed, we can arrange
for

$\displaystyle 1\ll S_{\rm gen}(I^{\prime}\cup R)\ll S(R)~{}.$ (62)

The first inequality ensures semiclassical control at $\partial I^{\prime}$.
The second states that condition (i) is satisfied (by an arbitrarily large
margin).

However, $\partial I^{\prime}$ is a classically trapped surface, i.e.
$\theta_{k}<0,\theta_{\ell}<0$. And since $\partial I^{\prime}$ is not close
to $\partial J^{-}(R)$, we expect quantum corrections to be small:
$\Theta_{\ell}=\theta_{\ell}+O(G\hbar)$. Condition (ii) is therefore violated.

Indeed, there are no islands associated to $R$ in this spacetime. To see this,
note that there are no classically extremal spheres. As in the previous
example, near $\partial J^{-}(R)$, quantum corrections to $\theta_{\ell}$ can
become large; but $\partial J^{-}(R)$ stays far from the horizon and so has
large classical (and quantum) expansion everywhere.

This example shows that condition (ii) is essential. So is condition (i), of
course. For example, suppose we chose $R$ to be a later portion of
$\mathscr{I}^{+}$. As before, $R$ contains only the purification of the star,
but no Hawking radiation. Since $\partial J^{-}(R)$ gets close to the horizon,
where $\Theta_{k}$ can vanish, there will be a quantum extremal region
$\tilde{I}$ with $\Theta_{\ell}$=0. However, this region fails to be an island
because $S_{\rm gen}(\tilde{I}\cup R)>S(R)$.

##  3 New Entropy Bound

In this section, we will show that in a globally pure state, the entropy of a
reference system $R$ cannot exceed the generalized entropy of suitable
asymptotic regions. We consider an external reference system in Sec. 3.1, and
we generalize to $R\subset M$ in Sec. 3.2. We discuss examples in Sec. 3.3.

###  3.1 External Reference System

Given an external reference system $R$, an island $I\subset M$ in a
semiclassical spacetime $M$ is defined as a region that is quantum extremal
and homologous to $R$ (i.e., $\partial I\subset M$), such that $S_{\rm
gen}(I\cup R)$ is minimal among all such regions. In section 2.1 we identified
sufficient conditions for $I\neq\varnothing$.

We will now employ similar techniques to derive an entropy bound on the exact
entropy of the reference system, $S(\mathbf{R})$, assuming that this is
computed by the “island formula”:

$\displaystyle S(\mathbf{R})=S_{\rm gen}(I\cup R)~{}.$ (63)

Following Ref. Almheiri:2019hni , we denote $\mathbf{R}$ in boldface when
referring to the exact (nonperturbatively computed) state of the region. We
write $R$ when referring to the semiclassically computed state. For
simplicity, we will assume that the global quantum state is pure,

$S(R\cup M)=0~{},$ (64)

though generalizations can easily be considered.

Let $I^{\prime}_{c}\subset{M}$ be any partial Cauchy slice of $M$ that is
quantum normal or anti-normal:

$\begin{cases}k^{\mu}\Theta_{\mu}[I^{\prime}_{c}]\geq
0~{},~{}~{}\ell^{\mu}\Theta_{\mu}[I^{\prime}_{c}]\leq 0~{};\\\ \text{or}\\\
k^{\mu}\Theta_{\mu}[I^{\prime}_{c}]\leq
0~{},~{}~{}\ell^{\mu}\Theta_{\mu}[I^{\prime}_{c}]\geq 0~{},\end{cases}$ (65)

where $k$ and $\ell$ are the future-directed null vector fields orthogonal to
$\partial I^{\prime}_{c}$. We also require that $I^{\prime}_{c}$ is
“asymptotic,” though only in the weak sense that in the conformally
compactified spacetime,

$\partial\Sigma\subset\partial I^{\prime}_{c}~{},$ (66)

where $\Sigma$ is a Cauchy slice of $M$. That is, $I^{\prime}_{c}$ must
contain the asymptotic region of $M$, but it may extend deep into the interior
of $M$. A simple example of a region $I^{\prime}_{c}$ that satisfies Eqs. (65)
and (66) is the exterior of a sufficiently large approximately round sphere.

Let $I^{\prime}$ be the complement of $I^{\prime}_{c}$ on some global Cauchy
slice of $M$. By Eqs. (64) and (65), $I^{\prime}\cup R$ will be anti-normal or
normal. By Eq. (66), $I^{\prime}$ is homologous to $R$. Our notation reflects
the fact that $I^{\prime}$ shares these properties with the region denoted
$I^{\prime}$ in Sec. 2.1.

However, here we do not assume the inequality $S_{\rm gen}(I^{\prime}\cup
R)<S(R)$, and hence we will not be guaranteed the existence of an island
$I\neq\varnothing$. This does not affect the maximin analysis performed in
Sec. 2.1: (anti-)normalcy of $I^{\prime}\cup R$ implies that the true island
$I$ satisfies

$S_{\rm gen}(I\cup R)\leq S_{\rm gen}(I^{\prime}\cup R)~{},$ (67)

regardless of whether $I$ is the empty set or not. Using Eqs. (63) and (64),
we thus find the entropy bound

$S(\mathbf{R})\leq S_{\rm gen}(I^{\prime}_{c})~{}.$ (68)

###  3.2 Distant Reference System

The bound (68) generalizes to the case where $R\subset M$, subject to
appropriate modifications. (It is easy to generalize further to the case where
$R$ is partly internal to $M$ and partly an external system.) We shall assume
that gravity is negligible in $R$, so that the notion of an exact state of
$\mathbf{R}$ can be made precise. The island rule can then be adapted to
compute the generalized entropy of $R$:

$\displaystyle S_{\rm gen}(\mathbf{R})=S_{\rm gen}(I\cup R)~{},$ (69)

where $I$ is an island (possibly the empty set), as described above. The
relevant homology rule is $I\subset\mbox{int}[M-\overline{J(R)}]$, where $J$
denotes the union of the causal past and future.

We again assume global purity, $S(M)=0$. To obtain a bound on $S_{\rm
gen}(\mathbf{R})$, we consider a spatial region $I^{\prime}_{c}$ that
satisfies the following conditions (see Fig. 8):

  * •

For $I^{\prime}$ to be of the correct homology type, without directly
referring to $I^{\prime}$,888Our goal is to formulate a bound in terms of
quantities that are accessible to an asymptotic observer. we require that
$I^{\prime}_{c}$ is adjacent to $R$ in $M$; and in the conformally
compactified spacetime $\tilde{M}$, $I^{\prime}_{c}$ contains any conformal
boundary portions not covered by $R$:

$\partial I^{\prime}_{c}\supset\partial(\tilde{\Sigma}-\bar{R})~{},$ (70)

where $\tilde{\Sigma}\supset R$ is a Cauchy slice of $\tilde{M}$, and
$\bar{R}$ denotes the closure of $R$ in $\tilde{M}$.

  * •

$I^{\prime}_{c}$ is quantum normal or anti-normal under shape deformations of
its inner boundary in $M$, i.e., at $(\partial I^{\prime}_{c}-\partial R)\cap
M$ .

  * •

$I^{\prime}_{c}$ contains a region $I_{0,c}$ that is quantum anti-normal at
$\partial I_{0,c}-\partial R$. (Normal is not allowed in this criterion.)

Global purity implies that $I^{\prime}\cup R$ will be quantum normal or anti-
normal at $\partial I^{\prime}$. It also guarantees quantum normalcy of
$I_{0}\cup R$, where $I_{0}\equiv\Sigma-I_{0,c}-R$ and $\Sigma$ is a Cauchy
surface that contains $I_{0,c}$ and $R$. By Sec. 2.2, the maximin procedure
restricted to the wedge $D(I_{0})$ will return a region
$\hat{I}\subset\mbox{int}[D(I_{0})]$ that satisfies the homology rule and has
stationary $S_{\rm gen}(\hat{I}\cup R)$ under shape deformations at
$\partial\hat{I}$.

Figure 8: The entropy $S(\mathbf{R})$ of an external or distant reference
system $R$ must be less than the generalized entropy of any region
$I^{\prime}_{c}$ that is normal or anti-normal (blue).

Note that $\hat{I}$ may be empty; and $S_{\rm gen}(\hat{I}\cup R)$ need not be
globally minimal, since the true island $I$ may not be contained in
$D(I_{0})$. However, we have

$S_{\rm gen}(I\cup R)\leq S_{\rm gen}(\hat{I}\cup R)~{},$ (71)

The quantum (anti-)normalcy of $I^{\prime}$ implies

$S_{\rm gen}(\hat{I}\cup R)\leq S_{\rm gen}(I^{\prime}\cup R)~{}.$ (72)

Using Eqs. (69) and global purity, we thus find the entropy bound

$S_{\rm gen}(\mathbf{R})\leq S_{\rm gen}(I^{\prime}_{c})~{}.$ (73)

Recall that $\partial I^{\prime}_{c}\supset\partial R$, so in any situation
where the generalized entropy can be separated into a regularized entropy and
Bekenstein-Hawking term, the area terms associated with $\partial R$ will
cancel, and Eq. (73) reduces to Eq. (68).

###  3.3 Examples and Discussion

The bound (73), and its external $R$ version (68), are powerful and versatile.
They require knowledge only of $R$ and $I^{\prime}_{c}$, but not of the rest
of the spacetime $M$. The only nontrivial condition, Eq. (65), can be easily
verified. Often the quantum expansion is dominated by the classical expansion,
so that it is easy to check whether $I^{\prime}_{c}$ is quantum (anti-)normal;
yet the bound remains nontrivial.

Figure 9: Not a Schwarzschild black hole. This could be a highly dynamical
spacetime far from a stationary black hole solution. $I^{\prime}_{c}$ is is
quantum anti-normal, so $S(\mathbf{R})\leq S_{\rm gen}(I^{\prime}_{c})$. In
the left example, $R$ is on the conformal boundary. Quantum anti-normalcy
follows from the GSL (future null congruence), and from the classical area
theorem (past null congruence) if quantum corrections are small. In the right
example, $R$ is inside the spacetime. Quantum anti-normalcy follows by the
area theorem and smallness of corrections if $\partial I^{\prime}_{c}$ stays
far enough from the horizon.

For example, by the generalized second law, causal wedges of a boundary region
must be quantum anti-normal.999Note that the causal wedge is not in general a
domain of dependence. The region that is always quantum normal is the maximal
Cauchy evolution of the causal wedge. This follows both for asymptotically
anti-de Sitter and flat spacetimes. When $R$ is disjoint from the conformal
completion of $M$, this ensures the quantum anti-normalcy of $I^{\prime}_{c}$.

The requirement that $R$ and $I^{\prime}_{c}$ be spacelike separated prevents
the application of the GSL when $R$ is part of the conformal completion of
$M$. For instance, suppose that $R$ is a subset of $\mathscr{I}^{+}$. The GSL
can still be applied to any future causal horizon associated to
$\mathscr{I}^{+}$ regions including $R$, guaranteeing $k^{\mu}\Theta_{\mu}\geq
0$. However, all past horizons will intersect the past of $R$, blocking the
application of the GSL to $I^{\prime}_{c}$ along them. To establish that
$\ell^{\mu}\Theta_{\mu}\leq 0$, we can use the classical area law on past
horizons, so long as quantum corrections to $\ell^{\mu}\theta_{\mu}$ is
negligible (see Fig. 9, left).

More generally, when $R$ is a subset of $M$, the classical area law ensures
condition (65) if the quantum corrections to both expansions are suppressed
(see right Fig. 9). This causal wedge method for finding $I^{\prime}_{c}$
suggests a nice physical interpretation of $I^{\prime}_{c}$ as a region that
can be explored geometrically by asymptotic observers.

The bound thus tells us that $S(\mathbf{R})$ cannot be greater than the
generalized entropy of any causal wedge region (subject to quantum effects on
the expansion remaining negligible). If $I^{\prime}_{c}$ is a whole Cauchy
surface $\Sigma$ of $M$, this reduces to the trivial statement that
$S(\mathbf{R})\leq S(\Sigma)$. (In this case, by purity, equality must hold.)
But if $I^{\prime}_{c}$ has a boundary in $M$, the bound is nontrivial.
Indeed, a quantum anti-normal causal wedge can reach very close
($O[(G\hbar)^{1/2}]$ distance) to a black hole horizon. For a black hole after
the Page time, this means that bound becomes nearly saturated. The bound then
implies the nontrivial statement of unitarity.

The bound simplifies if $I^{\prime}_{c}$ contains little matter entropy, so
that

$S_{\rm gen}(I^{\prime}_{c})\approx\frac{A}{4G\hbar}~{},$ (74)

In this case,

$S(\mathbf{R})\lesssim\frac{A}{4G\hbar}~{}.$ (75)

The entropy deliverable to an asymptotic observer by a spacetime causally
explored to an inner boundary of area $A$ cannot be greater than $A/4G\hbar$.

## Acknowledgements

ASM would like to thank Raghu Mahajan for discussions. This work was supported
in part by the Berkeley Center for Theoretical Physics; by the Department of
Energy, Office of Science, Office of High Energy Physics under QuantISED Award
DE-SC0019380 and under contract DE-AC02-05CH11231; and by the National Science
Foundation under Award Numbers 1820912 (RB) and 2014215 (ASM).

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