Entanglement wedge reconstruction and the
information paradox

Geoffrey Penington1

1Stanford Institute for Theoretical Physics, Stanford University, Stanford CA 94305 USA

Abstract

When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT,
we show that there is a phase transition in the location of the quantum Ryu-Takayanagi
surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon,
at an infalling time approximately the scrambling time β/2π log SBH into the past. We can
immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-
Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the
interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy
of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-
time Hawking modes, despite the absence of a firewall.
By studying the entanglement wedge of highly mixed states, we can understand the state
dependence of the interior reconstructions. A crucial role is played by the existence of tiny,
non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time,
interior operators can only be reconstructed from the Hawking radiation if the initial state
of the black hole is known. As the black hole continues to evaporate, reconstructions become
possible that simultaneously work for a large class of initial states. Using similar techniques,
we generalise Hayden-Preskill to show how the amount of Hawking radiation required to
reconstruct a large diary, thrown into the black hole, depends on both the energy and the
entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-
independent interior reconstruction exists for any code space of microstates with entropy
strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state
dependence to avoid the AMPSS typical-state firewall paradox.

Contents

1
Introduction
2

2
Entanglement Wedge Reconstruction in an Evaporating Black Hole
9
2.1
The Classical ‘Maximin Surface’ . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13

2.2
The Quantum Extremal Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16

2.3
Hayden-Preskill and the Page Curve . . . . . . . . . . . . . . . . . . . . . . . . .
23

2.4
Greybody Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29

3
State Dependence
36
3.1
State Dependence in Entanglement Wedge Reconstruction . . . . . . . . . . . . .
37

3.2
State Dependence in Evaporating Black Holes . . . . . . . . . . . . . . . . . . . .
40

geoffp@stanford.edu

1

arXiv:1905.08255v3  [hep-th]  29 Aug 2020


3.3
Approximation to the Rescue . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44

3.4
Large Diaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45

3.5
Minimal State Dependence
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49

4
Discussion
51
4.1
Summary of Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51

4.2
Entanglement Wedge Reconstruction in Toy Models
. . . . . . . . . . . . . . . .
54

4.3
The Post Evaporation State and the Bulk-to-Boundary Map . . . . . . . . . . . .
55

4.4
The Peak of the Page Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58

4.5
Explicit Interior Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58

4.6
The Information Paradox Beyond AdS/CFT . . . . . . . . . . . . . . . . . . . . .
59

5
Acknowledgements
60

A Cut-offs in Rindler Space
60

B Finite Temperature Infalling Modes
62

C Minimal State Dependence in the SYK Model
66

1
Introduction

By discovering the AdS/CFT correspondence [1,2], Maldacena definitively answered the question
of whether information can escape from a black hole. It can.
While there remains debate about whether information is lost during black hole evaporation
in the real universe [3], in AdS/CFT, the bulk quantum gravity theory in d+1 spacetime dimen-
sions is dual to an ordinary d-dimensional conformal field theory that lives on the asymptotic
boundary of the bulk spacetime. The unitarity of the boundary conformal field theory means
that information must be preserved.
However, on its own, boundary unitarity is not sufficient to consider the information paradox
‘solved’, even in the restricted context of AdS/CFT. We also need to understand what is wrong
with the Hawking calculation [4, 5], which apparently suggests that the radiation should be
completely thermal until the black hole has almost entirely evaporated,1 or at least why the
conclusion of information loss is naïve.
Conventional effective field theory suggests that the bulk evaporation should be semiclassical,
in agreement with Hawking’s calculation, so long as the black hole is large compared to the string
and Planck scales. In this paper, we will assume that this is indeed the case. In particular, we
assume that, from a semiclassical bulk perspective, the Hawking radiation continues to be in
a thermal state (up to greybody factors) that is purified by interior modes, even late in the
evaporation.
However, as we shall show, this does not mean that no information escapes the black hole. By
assuming the quantum version of the Ryu-Takayanagi formula and entanglement wedge recon-
struction, we will show that, at late times, the interior degrees of freedom are not microscopically
independent of the early Hawking radiation. Instead, a large part of the interior is encoded in
the early Hawking radiation, in exactly the same way that the bulk in AdS/CFT is microscopi-
cally encoded in its asymptotic boundary. This is essentially a formal realisation of the notion of
black hole complementarity [6]. We will precisely identify the part of the interior that is encoded

1This is a slight over-simplification. In reality, part of the Hawking radiation will be reflected back into the
black hole, adding ‘greybody factors’ to the radiation that escapes.

2


in the Hawking radiation, and thereby derive all the expected properties of unitary black hole
evaporation.
In particular, we will show that

• Only a non-perturbatively small amount of information escapes the black hole before
the so-called Page time, when the entropy of the Hawking radiation becomes equal to
the Bekenstein-Hawking entropy of the black hole.2 However, the existence of such non-
perturbatively small corrections is crucial in allowing information to later escape.

• A small diary, thrown into the black hole early in the evaporation, can be reconstructed
at the Page time, so long as the state of the black hole is known. A diary thrown into the
black hole after the Page time can be reconstructed after waiting for the scrambling time
β/2π log SBH.3 These twin results are known as the Hayden-Preskill decoding criterion and
were conjectured based on toy models [8]. We also derive generalisations of the Hayden-
Preskill decoding criterion to large diaries and partially unknown black hole states, where
we continue to find exact agreement with toy models.

• The microscopic entanglement entropy of the black hole obeys the so-called Page curve,
which was similarly conjectured based on toy models of black hole evaporation [9]. Before
the Page time, the entanglement entropy is equal to the entropy of the Hawking radiation,
while, after the Page time, it is equal to the Bekenstein-Hawking entropy of the black hole.
Crucially, the firewall paradox, which we discuss below, is avoided because the black hole
interior is partially encoded in the early Hawking radiation.

The firewall paradox [10] suggests that the combination of standard quantum field theory in the
bulk, together with global unitarity, is inconsistent with the existence of a smooth horizon after
the Page time. For a smooth horizon with finite energy density to exist, outgoing modes close
to the horizon must be entangled with the interior outgoing modes, just inside the horizon. As
we evolve forwards in time, these outgoing modes become late-time Hawking radiation.
Furthermore, because the black hole is already close to maximally entangled with the early
Hawking radiation, the late-time Hawking radiation must be entangled with the early Hawking
radiation, to avoid violating unitarity. However, strong sub-additivity means that a single system
cannot be strongly entangled with two different systems at once [11].
We therefore have a
paradox. To resolve the paradox, the authors of [10], known by the acronym AMPS, suggested
that a ‘firewall’, or region of very high energy density, must form at the horizon at some point
at or before the Page time.
The publication of [10] provoked a flood of responses, including [12–17].
Perhaps most
compellingly, in the ER=EPR proposal [18], it was pointed out that the thermofield double
state of a CFT,
�

i
e−βEi/2 |¯i⟩ |i⟩ ,
(1)

is also an example of a black hole that is close to maximally entangled with an external system.
Rather than being entangled with the early Hawking radiation, it is entangled with the second
copy of the CFT. Hence, the AMPS paradox should apply and the thermofield double state

2The Page time is commonly called the ‘halfway point’ in the black hole evaporation, although, because of
the thermodynamic irreversibility of the evaporation and the time dependence of the black hole temperature, it
does not occur halfway through the evaporation either by time or by horizon area/entropy [7].
3In this formula, SBH is the Bekenstein-Hawking entropy of the black hole, and β is the black hole inverse
temperature.

3


should have a firewall at its horizon. However, the thermofield double state is well-known to be
dual to a two-sided Schwarzschild black hole, which has a smooth horizon.
The resolution, in this case, is obvious: the ‘interior modes’ (which are really exterior modes
from the perspective of the second asymptotic boundary) are encoded, from a boundary perspec-
tive, in the second copy of the CFT. Hence, there is no contradiction in the Hawking radiation
being entangled with both the interior modes and the second copy of the CFT; in fact, the first
statement directly implies the second. As one of the results of this paper, we will show that the
firewall paradox for one-sided black holes is resolved in exactly the same way.
To show this, and to show all our other results, we will need to use entanglement wedge
reconstruction. The entanglement wedge reconstruction conjecture was developed in [19–21]
and then established with increasing levels of rigour in [22], [23] and [24]. It has long been
known that bulk operators in AdS/CFT can have multiple, distinct boundary representations,
which are known as ‘reconstructions’. Moreover, there exist reconstructions of any local bulk
operator that act on only part of the boundary. Bulk information is encoded redundantly on
the boundary. This redundancy is best understood in the language of quantum error correction
as the statement that bulk operators, acting on the ‘code space’ of states with a given bulk
geometry, are protected against the erasure of certain boundary subregions [25].
Entanglement wedge reconstruction tells us which part of the bulk is encoded in a given part
of the boundary. A bulk operator can be reconstructed using a given boundary region B, if,
and only if, the bulk operator is contained in a region known as the entanglement wedge b of
the boundary region B.
To define the entanglement wedge of B, we first need to define the Ryu-Takayanagi surface χB
associated to B. This surface was originally defined for static spacetimes as the bulk surface χB
of minimal area A(χB), lying within a static timeslice and homologous to the boundary region
B [26, 27]. However, for general dynamic spacetimes [28], and taking into account quantum
corrections [29, 30], it is the quantum extremal surface, homologous to B, with the smallest
generalised entropy
A(χB)/4GN + Sbulk(χB).

Here, the bulk entropy Sbulk(χB) is the von Neumann entropy of the bulk fields contained in
the entanglement wedge of B, as defined using the candidate surface, and a quantum extremal
surface is defined as a (d − 1)-dimensional surface of extremal generalised entropy.4 The Ryu-
Takayanagi formula, including quantum corrections, states that entanglement entropy of the
boundary region B is equal to the generalised entropy of the Ryu-Takayanagi surface.
The entanglement wedge is now simple to define. It is the bulk region, or, more precisely,
the bulk domain of dependence, bounded by the Ryu-Takayanagi surface χB and the boundary
region B.
An inportant breakthrough was made recently by Almheiri [31], who used entanglement
wedge reconstruction to understand how the ER=EPR proposal continues to resolve the firewall
paradox for a two-sided black hole, even when the black hole is dynamically evolving in time.
He considered a two-dimensional, two-sided black hole, which has an approximate ‘boundary’
description as an entangled state in a pair of SYK models, and imagined extracting Hawking ra-
diation, using absorbing boundary conditions, from one side of the black hole, and then throwing
it into the other side.

4In this paper, we will always use the term quantum extremal surface to refer to any surface that is an
extremum of the generalised entropy. Similarly, classical extremal surface refers to any extremal area surface. We
use Ryu-Takayanagi surface, or quantum Ryu-Takayanagi surface, to refer to the quantum extremal surface of
minimal generalised entropy and classical Ryu-Takayanagi surface to refer to the minimal area classical extremal
surface.

4


When the final state was evolved backwards in time, this time without any interaction
between the two sides, he argued that the Ryu-Takayanagi surface was different from the Ryu-
Takayanagi surface in the initial state. Degrees of freedom had moved from the entanglement
wedge of the ‘evaporating’ side to the entanglement wedge of the ‘growing’ side. Information
was ‘escaping in the Hawking radiation’.
Moreover, this change in Ryu-Takayanagi surface
meant that the interior modes, with which the Hawking radiation on the ‘evaporating’ side was
entangled, were still encoded in the ‘growing’ side. The two-sided black hole therefore continued
to evade the firewall paradox, even as one side of the black hole shrank and the other grew.
The basic conceptual story of this paper will be similar to [31], and indeed to the original
ER=EPR proposal [18]. However, rather than relying on the toy model of the thermofield double
state, which is well understood but does not actually describe an evaporating black hole, we will
work directly with one-sided evaporating black holes.
More specifically, in this paper, we consider an evaporating black hole, formed from col-
lapse, where the Hawking radiation is extracted into an auxiliary reservoir Hrad using absorbing
boundary conditions. By doing so, we will be able to make precise quantitative statements about
where, and when, information is encoded.
Unlike in [31], where only classical Ryu-Takayanagi surfaces were considered, it is crucial,
when studying an evaporating black hole, that we look at quantum RT surfaces. Since we are
extracting the Hawking radiation into an auxiliary reservoir Hrad, the microscopic entanglement
entropy of the black hole is simply the entanglement entropy between the entire boundary Hilbert
space HCFT and the reservoir Hrad. The Ryu-Takayanagi formula states that this entropy is equal
to the generalised entropy of the Ryu-Takayanagi surface χ associated to the entire boundary.
For an evaporating black hole formed from collapse, the only classical extremal surface,
homologous to the entire boundary (i.e. trivial homology), is empty. If entanglement wedge
reconstruction was based on the classical Ryu-Takayanagi surface, the interior of the black hole
would always be encoded in HCFT and no information would ever escape the black hole.5

The empty surface is also a quantum extremal surface, with generalised entropy equal to the
bulk entanglement entropy Srad between the Hawking radiation and the interior of the black
hole. Since we are assuming that the semiclassical Hawking calculation is valid so long as the
black hole is large compared to the string/Planck scales, this bulk entanglement entropy will
continue to grow, in agreement with semiclassical calculations, even after the Page time.
However, even early in the evaporation of the black hole, it will turn out that there also exists
a second, non-empty quantum extremal surface, which lies just inside the event horizon of the
black hole. In Eddington-Finkelstein coordinates, the infalling time of this extremal surface is
exactly the scrambling time, to leading order, before the ‘current time’, when Hawking radiation
was most recently extracted into Hrad.
Initially, this quantum extremal surface will not be the Ryu-Takayanagi surface. Its gener-
alised entropy will be approximately the Bekenstein-Hawking entropy SBH = Ahor/4GN of the
black hole, which is much larger than the generalised entropy Srad of the empty surface. How-
ever, at the Page time, there will be a phase transition and the non-empty quantum extremal
surface will become the Ryu-Takayanagi surface.
From this, one can easily use the Ryu-Takayanagi formula to find the entanglement entropy
S between the CFT and the reservoir. To leading order, it is given by

S = min(Srad, Ahor/4GN).
(2)

5In this case, we would have to believe either in remnants, or in a complete breakdown of the semiclassical
description of the evaporation, as in the firewall proposal. However, remnants are inconsistent with the spectral
density of CFTs and there is no reason within the bulk effective field theory to expect the semiclassical bulk
description to breakdown until the black hole has almost entirely evaporated.

5


The entanglement entropy therefore peaks at the Page time (defined by Srad = Ahor/4GN)
before beginning to decrease. It has long been conjectured that the entanglement entropy of
an evaporating black hole is given by this formula, which is known as the Page curve [9]. In
particular, a version of the Page curve can be derived if we model the CFT boundary dynamics
by a Haar random unitary acting on a large number of qubits.6 Here we derive it directly from
a bulk calculation.7

On its own, an explanation of the Page curve using the Ryu-Takayanagi formula is not
entirely satisfactory. It does not explain why extracting Hawking radiation into the reservoir
should decrease the entanglement entropy.
In particular, it does not resolve the firewall paradox. If the entanglement entropy S is to
decrease over time, the Hawking radiation that is transferred over from the CFT to the reservoir
must itself be entangled with the reservoir. In the semiclassical bulk picture of the evaporation,
however, it is instead entangled with the interior of the black hole.
Fortunately, we also know about entanglement wedge reconstruction. The newly emitted
Hawking radiation is indeed entangled with interior modes, but some of these modes are now in
the entanglement wedge of, and so encoded in, the reservoir Hrad. The same resolution of the
firewall paradox that worked for the thermofield double state also works for evaporating black
holes.
In the thermofield double state, the Hawking radiation is perfectly thermally entangled with
the second CFT. If the late-time Hawking radiation in an evaporating black hole was perfectly
thermally entangled with the reservoir, we would find

dS
dt = −dSrad

dt
<
1

4GN

dAhor

dt
.
(3)

This inequality is strict, even at leading order, because generically black hole evaporation is
a strictly thermodynamic-entropy-increasing process. The entanglement structure of the bulk
modes would therefore be inconsistent with the Page curve.
However, unlike in the thermofield double state, the Ryu-Takayanagi surface of an evaporat-
ing black hole does not lie exactly on the event horizon. Instead it lies an O(GN) radial distance
inside the horizon. Some of the interior outgoing modes are still encoded in the CFT, and so
the new radiation is not perfectly entangled with the reservoir Hrad.
In simple cases, one can explicitly calculate the rate of change in the entanglement entropy
S that results from extracting a small amount of new Hawking radiation. It agrees exactly with
the rate of change we found using the Ryu-Takayanagi formula. This is not a coincidence. In
fact, we shall show that this agreement must always exist; it is a necessary consequence of the
Ryu-Takayanagi surface being an extremum of the generalised entropy.
We are also interested in the question of when information about objects thrown into the
black hole reappears in the Hawking radiation. If a small diary had been thrown into the black
hole more than one scrambling time ago, it would now lie in the entanglement wedge of the
reservoir Hrad. It is therefore in principle possible to recover the state of the diary by looking
only at the Hawking radiation in the reservoir.
At least from a boundary perspective, the
information contained in the diary has escaped the black hole.

6This is slightly ahistorical. The Page curve was conjectured well before AdS/CFT was known. However
the conjecture was still based on the assumption that the black hole evaporation could be modelled by a Haar
random unitary.
7We do, of course, need to assume the Ryu-Takayanagi formula and entanglement wedge reconstruction,
which are both fundamentally holographic ideas. The Page curve cannot be found using the semiclassical bulk
description alone, because it results from the build-up of non-perturbatively small effects. See Section 3.3 for
more details.

6


By modelling the boundary dynamics of the CFT as a fast scrambling unitary, Hayden and
Preskill famously conjectured in [8] that the state of a small diary thrown into a black hole early
in the evaporation could be decoded from the Hawking radiation at the Page time, while the
state of a diary thrown in after the Page time could be decoded after waiting for the scrambling
time. Just as for the Page curve, by assuming entanglement wedge reconstruction, we can derive
the Hayden-Preskill decoding criterion from a bulk description of the evaporation.
So far, we have avoided any discussion of the crucial issue of state dependence. The idea that
there does not exist any single boundary operator that always corresponds to a given interior
bulk operator, and instead different boundary operators must be used for different states, goes
back to Papadodidimas and Raju [32, 33]. As with the ER=EPR proposal, it was partially
inspired as a response to the AMPS firewall paradox. Since then, there has been considerable
work on understanding whether such state dependence exists and, if so, how it works [34–36].
In particular, great progress has been made in the context of the SYK model, a toy model of
quantum gravity, where it was shown that there exists a complete basis (in fact an overcomplete
basis) of pure black hole microstates, whose interior geometries are well understood [37]. Interior
operators can be reconstructed on the boundary for each individual microstate, but there is no
single reconstruction that works for all the microstates. The idea that the state dependence of
interior operators could be interpreted in the language of quantum error correction was suggested
in [38] and developed in detail in [31].
In many ways, however, the simplest case of interior state dependence is the Hayden-Preskill
decoding criterion. As discussed above, a small diary thrown into a known black hole state can
be reconstructed from the Hawking radiation immediately after the Page time. However, to do
this, we have to know the state of the black hole.
If there was a way of extracting information about the diary from the Hawking radiation that
worked for any initial black hole state, then, by linearity, we could also extract information for
highly mixed initial black hole states. But for highly mixed intial states, the Hawking radiation
will look completely thermal until long after the Page time.
So it is clear that the interior
operators describing the state of the diary can only be encoded in the Hawking radiation in a
highly state-dependent way.
It was shown in [38] that state dependence can arise as a consequence of entanglement
wedge reconstruction. By using the formalism of approximate operator algebra quantum error
correction, specifically the results of [39–41], one can show that there only exists a single state-
independent reconstruction on a boundary region B of a given bulk operator and for a given code
space, if the bulk operator is contained in the entanglement wedge of B for all states, both pure
and mixed, with support only in the code space. In contrast, the existence of state-dependent
reconstructions is possible so long as the bulk operator is contained in the entanglement wedge
of B for all pure states.
Suppose, as before, we want to reconstruct a small diary, thrown into the black hole at
an early time, from the Hawking radiation reservoir Hrad. However, rather than knowing the
exact initial state of the black hole, we now only know that the black hole was in some large
code space of possible initial microstates. As a simple example, we can imagine that we started
with a smaller black hole, in a completely unknown state, and then threw in a large amount of
additional energy.
For any pure initial microstate in this code space, the Ryu-Takayanagi surface will jump
to the non-empty quantum extremal surface near the horizon, at the Page time, and so the
entanglement wedge of Hrad will contain the diary. If we knew the initial state of the black
hole, we could reconstruct the diary. On the other hand, for a highly mixed initial state, the
Ryu-Takayanagi surface of the reservoir Hrad will remain empty until much later.
To be able to find a single state-independent reconstruction that works for the entire code

7


space, we need the interior to be in the entanglement wedge of the reservoir, even for such highly
mixed initial states. We therefore need the entropy Scode of the code subspace to satisfy

Scode < Srad − SBH.
(4)

Not only does this agree with a conjecture from [38] based on random unitary toy models, it is
also provides the mechanism by which information is able to escape the black hole. Regardless
of the initial state of the black hole and the state of any diary that was thrown in, the outgoing
Hawking radiation is entangled with interior modes in exactly the same way. However, because
the interior modes are themselves encoded in the reservoir Hrad in a state-dependent way, the
new Hawking radiation still provides information about the state of the black hole to an observer
with access to Hrad.
A similar effect happens before the Page time. At this point, the interior is encoded in the
boundary HCFT rather than the reservoir Hrad. However the encoding is still necessarily state
dependent; if we allow too large a class of initial black hole microstates, the interior will no
longer be contained in the entanglement wedge of HCFT for highly mixed states. To reconstruct
interior operators on the CFT, we need the code subspace of allowed initial microstates to satisfy

Scode < SBH − Srad.
(5)

If the black hole has not evaporated at all, the bulk entanglement entropy Srad is zero. Hence,
(5) suggests that we can reconstruct the interior for code spaces of microstates whose entropy is
almost as large as, but still strictly less than, the Bekenstein-Hawking entropy of the black hole.
We will say that the interior of an unevaporated black hole is encoded in HCFT in a minimally
state-dependent way.
Most of the explicit state-dependent interior reconstructions that have appeared in the liter-
ature, for example [32,33,36,37], are only intended to work for a single black hole microstates,
or a code space with O(1) dimension. However, in an appendix, we show that the Kourkoulou-
Maldacena construction for the SYK model [37] can be trivially extended to work for a set of
microstates with entropy almost as large as the Bekenstein-Hawking entropy. We also show that
minimal state dependence is sufficient to avoid the AMPSS typical-state firewall paradox.
The structure of the paper is as follows. In Section 2, we study entanglement wedge recon-
struction in an evaporating black hole that was formed by collapse. By restricting our attention
to a single initial microstate, we avoid the issue of state dependence. We find the location of
the non-empty quantum extremal surface explicitly, in a simplified evaporation process where
the Hawking radiation is extracted from close to the horizon, in Section 2.2, and then use this
calculation to explain the Hayden-Preskill decoding criterion and the Page curve in Section
2.3. Finally, we show how one can still derive Hayden-Preskill and the Page curve, even when
non-trivial greybody factors are present, in Section 2.4.
In Section 3, we consider large code spaces of initial black hole microstates, and show how the
state dependence of interior reconstructions depends on time. We also generalise the Hayden-
Preskill decoding criterion to large diaries in Section 3.4. In Section 3.5, we argue that the
interior of black holes that have not evaporated at all is encoded in the boundary with only
minimal state dependence.
Finally, Section 4 includes a detailed summary of the results of the paper, as well as dis-
cussion on various topics. In particular, we argue in Section 4.3 that, from a bulk perspective,
information must escape the black hole through a version of the Horowitz-Maldacena final state
proposal [42]. In appendices, we generalise the calculations from Section 2.2 to finite tempera-
ture infalling modes, and show how the Kourkoulou-Maldacena construction can easily be made
minimally state dependent.
After the completion of this manuscript, the author became aware of independent related
work by Almheiri, Engelhardt, Marolf and Maxfield [43], which was published simultaneously.

8


2
Entanglement Wedge Reconstruction in an Evaporating Black
Hole

In this section, we study an evaporating black hole formed from collapse. For simplicity, we
assume throughout that the collapsing matter, and hence the entire spacetime, is rotationally
symmetric. We show that no information about the black hole escapes in the Hawking radiation,
until the Page time, when the bulk entropy Srad of the Hawking radiation becomes equal to the
Bekenstein-Hawking entropy SBH of the black hole. After the Page time, a large part of the
interior of the black hole becomes encoded in the early Hawking radiation. In particular, a
diary thrown into the black hole becomes encoded in the Hawking radiation after waiting for the
scrambling time. The microscopic entanglement entropy of the black hole begins to decrease, in
accordance with the Page curve, because the new Hawking radiation is entangled with interior
modes that are encoded in the early Hawking radiation.
Our main focus will be on black holes with fixed Schwarzschild radius rs in AdS units in the
limit GN → 0. All such black holes are microcanonically stable: if we evolve the system with
reflecting boundary conditions, the black hole will quickly reach equilibrium with the Hawking
radiation and remain constant in size (up to small fluctuations).8

To study the evaporation of microcanonically stable black holes, we instead impose absorbing
boundary conditions.9 The outgoing modes are absorbed by the boundary and so the infalling
modes are always in the vacuum state. The Hawking radiation never returns to the black hole,
which gradually evaporates.
The dynamics of the system are now irreversible. Rather than evolving unitarily, the bound-
ary state |ψ⟩ ∈ HCFT will obey a Markovian master equation [47]. However, as usual with any
quantum channel, we can make the evolution unitary by adding an auxiliary Hilbert space – in
this case, a large Markovian reservoir Hrad that stores the outgoing Hawking radiation once it
reaches the boundary. Such a reservoir is sometimes known as an evaporon [45], although we will
not use this term. The information paradox will be resolved by simply keeping track of which
parts of the bulk are in the entanglement wedge of HCFT and which are in the entanglement
wedge of Hrad.
We assume that the Ryu-Takayanagi surface, associated to a given boundary region, is
defined to be the quantum extremal surface, i.e. surface of extremal generalised entropy

A

4GN
+ Sbulk,
(6)

homologous to the boundary region, with the smallest generalised entropy.
Here, the bulk
entropy Sbulk is the von Neumann entropy of the bulk fields in any spacelike surface bounded
by the Ryu-Takayanagi surface and the boundary region.10 If the overall bulk state is pure,
this bulk von Neumann entropy is simply entanglement entropy between bulk fields inside and
outside the entanglement wedge.
We will also generally assume that this prescription is equivalent to a maximin prescription,

max
Cauchy min
χ

�A(χ)

4GN
+ Sbulk(χ)
�
,
(7)

8This should not be confused with the fact that black holes that are sufficiently small (below the Hawking-Page
transition [44]) in AdS units are thermodynamically unstable, even if their size is fixed in the semiclassical limit
GN → 0.
9For similar use of absorbing boundary conditions to evaporate black holes in AdS/CFT see [31,45,46].
10Bulk causality ensures that this bulk entropy is independent of the choice of spacelike surface. Instead Sbulk
is a function of the bulk domain of dependence of the spacelike surface, which itself depends only on the RT
surface and the boundary region.

9


Figure 1: With reflecting boundary conditions, outgoing modes in a surface ending on the
boundary at time t1 may become ingoing modes on a surface ending at time t2, but the same
degrees of freedom will always be contained in each surface. The bulk entropy cannot depend on
the boundary time. In contrast, with absorbing boundary conditions, outgoing modes at time t1
may escape the bulk in the reservoir Hrad by time t2, and so no longer be contained in a surface
ending at time t2. The bulk entropy, and hence the notion of quantum extremality, depends on
the boundary time.

where one first finds the surface of (globally) minimal generalised entropy within fixed Cauchy
slices, and then selects the Cauchy slice which (globally) maximises this minimal generalised
entropy.11 When this paper first appeared on arXiv, the equivalence of the maximin and extremal
(HRT) prescriptions had only been formally shown for classical surfaces (assuming the null energy
condition) [21], although it was expected to also be true for quantum surfaces [29]. It has since
been shown for quantum surfaces [48] (assuming the quantum focussing conjecture [49]). We
shall therefore only use the maximin prescription to provide intuition about the location of the
quantum RT surface; all our actual results will be found using the extremal surface prescription.
If a classical extremal surface is homologous to any boundary component12 at some time t,
it will also be homologous to the same boundary component at any other time and, trivially,
will still be a classical extremal surface. The classical Ryu-Takayanagi surface therefore cannot
change as a function of the boundary time.13

However, this is only true for a quantum extremal surface when we have reflecting boundary
conditions. The bulk entropy term in the generalised entropy (6) depends not only on local data
at the Ryu-Takayanagi surface, but on the state of the bulk fields in entire bulk region, bounded
by the RT surface and the boundary.
As shown in Figure 1, with reflecting boundary conditions, the same degrees of freedom are
contained in this region, independent of the boundary time. However, with absorbing boundary
conditions, outgoing modes, which are contained in a spacelike surface that ends on the boundary
at time t1, may escape the boundary and not be contained in a spacelike surface ending at a
later time t2. The bulk entropy, and hence, more importantly, the gradient of the bulk entropy,
may depend on the boundary time. A quantum extremal surface for the entire boundary at time
t1 may no longer be a quantum extremal surface for the entire boundary at time t2.
To understand how the information paradox is resolved in AdS/CFT, we will need not only
to find the entanglement wedge of the CFT, but also the entanglement wedge of the reservoir
Hrad. This is not a ‘boundary region’ in the usual sense and we therefore need to be careful
about how to define a Ryu-Takayanagi surface and an entanglement wedge for it.

11As just described, the maximin prescription will not generally pick out a unique surface χ, since we can
generally find maximising Cauchy slices with more than one minimal surface. However, generically, it should pick
out a unique stable surface χ, which continues to be close to a minimal surface under any small perturbation of
the Cauchy slice. This stable surface should be extremal and the quantum RT surface.
12By this we mean a boundary region B, whose boundary ∂B is empty.
13One might worry that the classical Ryu-Takayanagi surface might not be spacelike separated from the bound-
ary at some boundary time. However, this cannot happen, assuming the null energy condition [21].

10


It is a general fact that, if we divide the boundary of a pure holographic state into two
complementary regions, the quantum Ryu-Takayanagi surface will be the same for both regions,
and hence the entanglement wedges of the two regions will also be complementary. The same
will be true here. The Ryu-Takayanagi surface for Hrad will be the same as the RT surface for
HCFT, and the entanglement wedge of Hrad will contain the outgoing modes that were extracted
into the reservoir, along with the bulk domain of dependence of any spacelike surface in the
black hole interior that is bounded only by the RT surface. All the bulk degrees of freedom
will either be in the entanglement wedge of the CFT, or be in the entanglement wedge of the
reservoir Hrad.
However, because our ‘boundary region’ is not actually holographic, let us take a moment to
see explicitly why this is true. The simplest way to do so is to imagine throwing the radiation
in Hrad into the bulk of an auxiliary holographic CFT, with a parametrically smaller Newton’s
constant G′
N (i.e. a parametrically larger central charge) than the original CFT. The small
gravitational coupling G′
N ensures that there won’t be any significant backreaction.14 Since the
entanglement entropy of this auxiliary CFT must be the same as the entanglement entropy of
Hrad, we will define the RT surface of Hrad to be the RT surface of this auxiliary CFT, which
is, by definition the smallest generalised entropy quantum extremal surface homologous to the
entire boundary of this auxiliary bulk geometry (i.e. a closed surface with trivial homology).
Note that, for a one-sided black hole, the entire boundary of the original CFT also has trivial
homology; the two homology constraints are the same.
One might worry that the RT surface that we define in this way could depend on the details
of the auxiliary spacetime, in which case it would not be well-defined as an RT surface for Hrad.
However, it is easy to see that the RT surface cannot contain a non-empty component in the
auxiliary geometry, since in the limit G′
N → 0 the area term will always be dominant and vacuum
AdS contains no classical extremal surface.15 It follows that the only place where a nonemtpy
component of the RT surface can exist is in the original black hole geometry.
The entanglement wedge of the auxiliary CFT is the domain of dependence of any spacelike
surface bounded by the RT surface and the auxiliary boundary. This will therefore include the
entire auxiliary geometry, plus, if the RT surface is non-empty, the domain of dependence of any
spacelike region bounded by the RT surface alone.
The RT surface, and entanglement wedge, of the combination of a boundary region and
a nonholographic system was previously considered in [38]. It was argued there that the RT
surface should be the minimal generalised entropy quantum extremal surface, homologous to the
boundary region, defined with the nonholographic system automatically included as part of the
fields in Sbulk (see for example eqn. 4.14 of [38]). This was justified by similar arguments to those
above (see footnote 15 of [38]). In the special case considered here where the boundary region
is empty and so we only have a nonholographic system, this rule leads to the same conclusions
that we reached above.
We have already argued that the RT surfaces for HCFT and Hrad satisfy the same (trivial)
homology constraint. Moreover, for any given candidate RT surface, the resulting entanglement
wedges for HCFT and Hrad are complementary within a Cauchy slice (of the original black hole

14To avoid talking about more than one holographic CFT, we could instead avoid backreaction by taking a
large number of copies of the original CFT and throwing a small amount of radiation into each one.
15The quantum extremal surface prescription for the RT surface can be derived from the replica trick [30].
In the limit G′
N → 0, the auxiliary spacetimes will be identical in the replicated and unreplicated geometries
(because of the lack of backreaction). It follows that the derivation from [30] can be used to directly calculate
the entropy of Hrad, without having to worry about the auxiliary geometry we introduced here. After this paper
first appeared on arXiv, these replica trick calculations were done explicitly in [50,51], where it was shown that
the relevant topologies for the second half of the Page curve involve spacetime wormholes connecting different
replicas.

11


geometry plus the auxiliary geometry). Since the overall state of the bulk modes is pure, the
bulk entropy associated to each entanglement wedge will always be the same. It follows that a
quantum extremal surface with respect to HCFT will also be a quantum extremal surface with
respect to Hrad and vice versa. The Ryu-Takayanagi surfaces for HCFT and Hrad will therefore
be the same.
The simplest quantum extremal surface, for both HCFT and Hrad, is, of course, the empty
surface. The generalised entropy of the empty surface will be equal to the bulk entanglement
entropy Srad between the Hawking radiation and the interior of the black hole. Recall that we
assume that the bulk evaporation is semiclassical, so long as the black hole is large compared
to the string and Planck scales. Hence the bulk entanglement entropy Srad will continue to
grow, in accordance with semiclassical calculations, until the black hole has almost completely
evaporated.
Before the Page time, the empty surface will be the Ryu-Takayanagi surface. This can easily
be seen using the maximin prescription. Since the empty surface lies in every Cauchy slice, Srad
upper bounds the generalised entropy of the Ryu-Takayanagi surface. However, it is also easy
to find a Cauchy slice for which the empty surface has minimal generalised entropy. We simply
choose a Cauchy slice that stays a small, but fixed, radial distance inside the event horizon of
the black hole. Within this Cauchy slice, we cannot choose an RT surface χ that excludes the
interior modes, entangled with the outgoing radiation in Hrad, without the area A(χ) of this
surface satisfying

A(χ)
4GN
> Srad.
(8)

The surface of minimal generalised entropy within this Cauchy slice is therefore the empty
surface, with generalised entropy Srad.16

It follows that the RT surface is empty and the interior of the black hole is in the entanglement
wedge of the boundary CFT, and not the entanglement wedge of the Hawking radiation reservoir.
No information has escaped the black hole. The Hawking radiation is thermally entangled with
the interior, which is encoded in the CFT, so the redused density matrix of the reservoir Hrad
will be thermal (with appropriate greybody factors). We can also see from the Ryu-Takayanagi
formula that the entanglement entropy between the CFT and the reservoir will indeed be the
bulk entanglement entropy Srad between the Hawking radiation and the interior.
We have
derived the first half of the Page curve.
As an aside, we emphasize that, because the evaporation is thermodynamically irreversible,
the bulk entanglement entropy Srad is strictly greater than (A0
hor −Ahor)/4GN where A0
hor is the
initial horizon area of the black hole. This means that the Page time, which we recall is defined
by

Srad = Ahor

4GN
,
(9)

occurs when the horizon area Ahor > A0
hor/2, despite commonly being called the halfway point
of the evaporation [7].
What happens after the Page time? The empty surface is still extremal, but it is easy to see
from the maximin prescription that it cannot be the Ryu-Takayanagi surface. In any Cauchy
slice, we can construct a surface homologous to the boundary that is (a) outside the event

16Assuming that the minimal generalised entropy surface is unique, it is sufficient, because of the rotational
symmetry of the system, to only consider rotational symmetric candidate surfaces. However, it is not difficult to
verify that the empty surface does indeed have minimal generalised entropy, within this Cauchy slice, even when
we consider surfaces that are not rotationally symmetric.

12


Figure 2: Schematic drawings of Cauchy slices through the black hole interior, both before
the Page time (left) and after the Page time (right).
The blue lines indicate entanglement
between the interior of the black hole and the reservoir Hrad. Before the Page time, there exist
Cauchy slices where the empty surface is the surface homologous to the boundary with minimal
generalised entropy. It is therefore the Ryu-Takayanagi surface χ. For illustrative purposes, we
draw this surface cutting the entanglement between the interior and the reservoir. After the
Page time, however, no such Cauchy slice exists. Within any Cauchy slice, there will always exist
a surface, near the horizon and homologous to the boundary, with smaller generalised entropy.
The Ryu-Takayanagi surface must become non-empty at the Page time.

horizon, and (b) has only slightly greater area than the current horizon area.17 Since the only
source of greater-than-O(1) bulk entropy is the interior modes that are entangled with radiation
that escaped to Hrad, the generalised entropy of this surface will be less than the generalised
entropy of the empty surface.18
The two cases, before and after the Page time, are shown
schematically in Figure 2.
We therefore conclude that the quantum Ryu-Takayanagi surface must become non-empty
at the Page time. In fact, we will show that there exists a non-empty quantum extremal surface
even before the Page time. At the Page time, there is a phase transition, and the non-empty
quantum extremal surface becomes the Ryu-Takayanagi surface. The main focus of this section
will be on identifying the location of, and the consequences of the location of, this non-empty
quantum extremal surface.

2.1
The Classical ‘Maximin Surface’

We begin with a warm-up. Since the bulk entropy term Sbulk(χ) is subleading compared to the
area term A(χ)/4GN in the formula for the generalised entropy, we shall initially ignore the local
variation in the bulk entropy and instead attempt to simply find a ‘classical maximin surface’

17In fact the area of the ‘classical maximin surface’ that we find in Section 2.1 gives an upper bound on this
area.
18One might worry that late-time Hawking radiation, which has yet to reach the boundary could provide a
source of greater-than-O(1) bulk entropy. However, because the Cauchy slice must be achronal, the infalling time
of any surface in the Cauchy slice cannot be later than the boundary infalling time. Hence, the bulk entropy of
any surface in the exterior will be at most O(1) (once it has been regulated using a cut-off).

13


χc.19

Obviously, for an evaporating black hole formed from collapse, the true classical maximin
surface would be empty. We could fix this issue by temporarily assuming that the original black
hole was two-sided, with one side allowed to evaporate. However, that would be taking this
calculation too seriously. The actual surface that we find will very clearly not make sense as an
actual Ryu-Takayanagi surface of any kind (it won’t be extremal for example); however, it will
turn out to correctly identify the approximate location of the quantum extremal surface, which
we shall find in the later parts of this section. We shall therefore simply ignore the question of
how the black hole formed entirely, and assume that it has been evaporating forever.
Because of the assumed rotational symmetry, the classical maximin surface (and the eventual
quantum extremal surface) should be rotationally symmetric. We therefore only need to consider
rotationally symmetric Cauchy slices.
We also know that the area of an infalling lightcone in an evaporating black hole is monton-
ically decreasing. Hence, given any Cauchy slice, we can increase

min
χ
A(χ)
4GN
(10)

by pushing the Cauchy slice backwards and outwards along infalling lightrays. The maximising
Cauchy slice is therefore simply the past lightcone of the boundary.
In Eddington-Finkelstein coordinates, the metric of a static black hole is

ds2 = −f(r)dv2 + 2dvdr + r2dΩ2,
(11)

where the co-ordinate v labels the initial Schwarzschild time of an infalling lightray and, for an
uncharged AdS-Schwarzschild black hole,

f(r) = 1 + r2

l2 −
16πGNM

(d − 1)Ωd−1rd−2 .
(12)

Since our arguments should also be valid for charged black holes (at least when all the particles
involved in the Hawking radiation are neutral) and BTZ black holes, we will avoid using (12)
directly.
In the semiclassical limit, the evaporation process is very slow. Over any fixed range of
infalling times, the metric will approach the metric of a static black hole of some fixed mass M.
We can therefore approximate the metric of an evaporating black hole by a static black hole
with a mass M, and hence Schwarzschild radius rs (defined by f(rs) = 0), that is slowly varying
with infalling time v; this is known as an ingoing Vaidya metric.20

The radius rl.c.(v) of an outgoing lightcone satisfies

drl.c.
dv
= f(r)

2
≈ 2π

β (r − rs),
(13)

where the approximation is valid in the near-horizon region and the inverse temperature β =
4π/f′(rs). If r′ = rl.c. − rs, we have

dr′
l.c.
dv
= drl.c.

dv
− drs

dv ≈ 2π

β r′
l.c. − drs

dv ,
(14)

19We specify classical maximin surface here, because, unlike the actual quantum extremal surface, this surface
will not be extremal, by either the classical or quantum definition.
20An ingoing Vaidya metric is only a simple approximation of the actual semiclassical metric of an evaporating
black hole; for example, at large radii compared to the Schwarzschild radius, the metric instead resembles an
outgoing Vaidya metric.
For a detailed calculation of the metric of an evaporating black hole in flat space
see [52]. In the limit GN → 0, the only deviation of this metric from the static metric that will be relevant for
our calculations is the infalling-time dependence of the black hole mass.

14


At leading order in the semiclassical limit, we can assume that the inverse temperature β and
the evaporation rate drs/dv are constant.
Integrating (14), we find

rl.c. = rs + C e
2π
β v + β

2π
drs
dv ,
(15)

for some arbitrary constant C. If C > 0, the lightcone will eventually escape the black hole, even
if its radius is initially decreasing. In contrast, if C < 0, the outgoing lightcone will eventually
fall into the singularity.
The causal, or event, horizon of the black hole is defined as the boundary of the causal past of
future asymptotic infinity. In this case, up to subleading corrections from the time dependence
of drs/dv and β, it is the outgoing lightcone (14) with C = 0.21 Its radius rhor is therefore given
by

rhor = rs + β

2π
drs
dv .
(16)

Since drs/dv < 0, this is inside the timelike apparent horizon rs.22

If we instead choose C = rs, we have

rl.c. = rs + rs e
2π
β v + β

2π
drs
dv = rhor + rs e
2π
β v,
(17)

and the lightray escapes the near horizon region at v = O(β).
The radius rl.c., and hence the area Ωd−1rd−1, of the past lightcone reaches a minimum and
begins increasing when it reaches the apparent horizon rs. This occurs at

v = − β

2π log
rs

β |drs/dv| + O(β).
(18)

For small AdS-Schwarzschild black holes,23 the only relevant lengthscale is the Schwarzschild
radius rs.24 Hence, it is easy to see using dimensional analysis and the fact that drs/dv = O(GN)
that

v = − β

2π log SBH + O(β).
(19)

We have therefore found that the classical maximin surface lies on the classical apparent horizon,
one scrambling time into the past.25

21We are assuming here that nothing to radical happens (such as the black hole becoming a white hole) when
the black hole has almost entirely evaporated and the semiclassical description breaks down.
22For our purposes, the apparent horizon is the radius at which the area of an outgoing lightcone is locally
constant.
23For large AdS black holes, there is no clear distinction between radiation that is inside the so-called ‘zone’
near the horizon and radiation that has escaped to the boundary. As a result, a large AdS black hole does not
have a well-defined evaporation rate, even with absorbing boundary conditions; it depends on the details of the
evaporation process.
24 In small, near-extremal Reissner-Nordstr´’om black holes, the inverse temperature β is parametrically large
compared to the Schwarzschild radius rs. The approximation in (13) is therefore only valid when r − rs ≪ r2
s/β.
At larger radii, we have dr/dv = O(1/f(r)) = O(r2
s/(r − rs)2) and so an outgoing lightray escapes the black hole
in an O(β) time. Hence, (18) becomes

v = − β

2π log
r2
s

β2 |drs/dv| + O(β).

25In addition to the O(β) corrections, if the Schwarzschild radius rs is parametrically small in AdS units, we
also need to add the infalling time π lAdS for the outgoing lightcone to get from the black hole to the boundary,
after it has escaped the near horizon region.

15


This is very hopeful: the Hayden-Preskill decoding criterion says that a small diary, thrown
into the black hole after the Page time, should be reconstructable from the Hawking radia-
tion after waiting for the scrambling time. Our calculation suggests that this is because the
entanglement wedge of the Hawking radiation reservoir Hrad now contains the diary.
However, there are two major problems with this classical maximin surface χc as a candi-
date Ryu-Takayanagi surface. Firstly, the surface χc is not a classical extremal surface. It has
extremal area with respect to deformations that stay on the past lightcone, but it certainly does
not have extremal area if we allow deformations that move the surface away from the lightcone.
It therefore cannot be the Ryu-Takayanagi surface according to the HRT extremal surface pre-
scription, even though the HRT and maximin prescriptions are supposed to be equivalent [21].
Secondly, if the surface χc was actually the Ryu-Takayanagi surface, the entanglement wedge
of the boundary CFT would not contain the causal future of the boundary. This would be highly
problematic because the forward time evolution of the CFT is deterministic, and so the future
boundary, although not the past boundary, is in the boundary domain of dependence. The
entanglement wedge of the CFT would not contain its causal wedge.
Both problems have the same cause and will have the same solution. The cause is that the
evaporating black hole spacetime violates the null energy condition. The null energy condition
is needed to prove that the classical maximin surface is the same as the classical extremal
surface [21]. It is also needed to prove that the classical entanglement wedge contains the causal
wedge [21].

2.2
The Quantum Extremal Surface

The problem of quantum effects leading to spacetimes that violate the null energy condition
was the original reason for the conjecture that the Ryu-Takayanagi surface should be a quantum
extremal surface, rather than a classical extremal surface [29]. So, we should definitely be hopeful
that all these problems will go away, once we fully include the effects of the bulk entropy term.
Indeed, heuristically, it is easy to see that the bulk entropy term can push the Ryu-Takayanagi
surface away from the past lightcone. If the RT surface was exactly on the past lightcone, no
outgoing modes would be included in the entanglement wedge of the CFT. Since the entropy
of the outgoing modes is divergent, moving the RT surface a small distance inside the lightcone
should increase the bulk entropy by a formally infinite amount. This strongly suggests that the
quantum RT surface, in the maximin prescription, will be stabilised a small radial distance away
from the past lightcone, creating an actual quantum extremal surface.
Unfortunately, actually calculating the bulk entropy is complicated by the presence of non-
trivial greybody factors. Because the black hole spacetime is curved, outgoing modes close to
the black hole do not necessarily escape to infinity. Instead, there is a non-trivial scattering
process. The curved spacetime wave equation can be rewritten as a flat space wave equation
with a potential barrier. This potential barrier lies an O(rs) distance away from the black hole
horizon and is higher for modes with large angular momentum, causing the Hawking radiation to
be dominated by modes with O(1) angular momentum. The region inside the potential barrier
is known as the zone.
Within the zone, the Hawking radiation is truly black body radiation at the black hole
temperature.26
However, the probability of a Hawking mode escaping depends on its angu-
lar momentum and, importantly, on its Schwarzschild energy. This probability is known as a
greybody factor.

26A more precise statement is that each angular momentum mode looks like two-dimensional black body
radiation. For higher-dimensional black bodies, only angular momentum modes with |J| ≲ Tr are significantly
excited, and only modes with |J| ≪ Tr look like two-dimensional thermal radiation. In contrast, sufficiently
close to the horizon, Rindler modes with |J| ≫ Trs = O(1) will be excited.

16


Outgoing modes in the entanglement wedge of the CFT are entangled both with modes
further in towards the black hole, and with outgoing modes further out – outside of the past
lightcone. The non-trivial greybody factors mean that the outgoing modes outside the past
lightcone are related to later ingoing modes, which are also in the entanglement wedge of the
CFT. This dramatically complicates any explicit calculation of the bulk entropy.
As a simple solution to this problem, we shall therefore temporarily assume that the Hawking
radiation is extracted from deep inside the zone, close to the horizon, before the mixing of ingoing
and outgoing modes occurs. (We will reintroduce the greybody factors in Section 2.4.) By doing
so, we are effectively reducing the problem to a calculation in the two-dimensional effective
theory that governs the near horizon region [53]. For this reason, the results we find in this
section will be essentially identical (once certain constants are fixed appropriately) to those
found independently in an explicitly two-dimensional model in [43].
Extracting the radiation from close to the horizon obviously involves changing the dynamics
of the system compared to the original procedure, where the radiation was extracted near the
boundary. In particular, the boundary Master equation, and its purification using the reservoir
Hrad, will now involve reconstructions of operators deep in the bulk, which are highly non-local
from a boundary perspective.
We emphasize, however, that there is nothing fundamentally unphysical about this. The
outgoing modes will still be extracted at a distance from the horizon (and hence an energy
scale) that is fixed in AdS units as GN → 0; the distance simply needs to be small compared to
the Schwarzschild radius rs. We are therefore still well within the domain of validity of the bulk
effective field theory.
It is important to note that the closer to the horizon we extract the Hawking radiation,
the larger the number of angular momentum modes that are excited.27 Similarly, increasingly
massive fields will be excited very close to the horizon. This can be seen from the explicit form
of the potential barrier in tortoise coordinates [54].
We will assume that we extract some fixed finite number of angular momentum modes for
each field, and that these fields are extracted sufficiently close to the horizon that their mass and
angular momentum can be safely ignored. In effect, we are changing the dynamics of the theory
such that all greybody factors are either zero or one, depending on the angular momentum. We
will also assume that all the light fields are free.
Close to the horizon, the spacetime can be approximated by R1,1 × Sd−1 where the radius
of the sphere Sd−1 is the Schwarschild radius rs. Each angular momentum mode acts as an
independent free field in an effective two-dimensional theory, with a Kaluza-Klein mass m2
KK =
L2/r2
s, which can be ignored at the lengthscales of interest.
Let the number of two-dimensional bosonic modes Nb and fermionic modes Nf. The (1+1)-
dimensional Stefan-Boltzman law [55] states that the rate of energy loss from the black hole is
given by

dM
dv = cevap π

12 β2 ,
(20)

where cevap = Nb + Nf/2.28
The first law of black hole thermodynamics says that βdM =

27Another way of saying this is that modes with larger angular momentum are reflected back into the black
hole at a smaller, but still finite, distance from the horizon.
28Of course, since Hawking radiation is stochastic, this is only the average rate of energy loss. However, so
long as we consider timescales that are large compared to the thermal time β, the average energy change should
be large compared to the fluctuations in the energy change, which can therefore be safely ignored. In our case,
the relevent timescale is the scrambling time, which is indeed very large compared to the thermal time β in the
semiclassical limit. We can also suppress the fluctuations by taking the limit where cevap is large.

17


dAhor/4GN. Hence

dAhor

dv
= cevap π GN

3 β
,
(21)

and

drs
dv =
cevap π GN

3 β (d − 1) rd−2
s
Ωd−1
.
(22)

Substituting (22) into (18) and dropping O(β) terms, we find that the classical maximin surface
for this spacetime occurs at29

vc = − β

2π log SBH

cevap
+ O(β).
(23)

To calculate the quantum extremal surface, we also need to calculate how the bulk entan-
glement entropy depends on the location of the extremal surface.30 Since we are assuming that
all the relevant fields are free and effectively massless at the lengthscales of interest, the ingoing
and outgoing modes are decoupled. The total bulk entropy is therefore simply the sum of the
bulk entropies of the ingoing and outgoing modes.
The infalling modes are in the vacuum state with respect to the infalling time v. Moreover,
if we assume (correctly) that the quantum extremal surface is close to the classical maximin
surface, the entanglement wedge of the boundary CFT will contain infalling modes spread over
approximately the scrambling time, which diverges as GN → 0. Since the entanglement entropy
of the vacuum state grows only logarithmically with system size, we can treat the entanglement
entropy of the infalling modes as approximately independent of the location of the extremal
surface, so long as the cut-off at the extremal surface is fixed in units of infalling time v. By
this, we really mean that the cut-off is equal to ε∂/∂v for some constant ε.
What about the outgoing modes? The only relevant modes are the modes that are extracted
into the reservoir Hrad. At sufficiently short lengthscales, the entropy of these modes will be
given by the Minkowski vacuum formula [57,58]

S = cevap

6
log
�rlc(v) − r
√ε1ε2

�
,
(24)

where rlc is the radius of the outgoing lightcone and ε1 and ε2 are the cut-offs at the quantum
extremal surface and the outgoing lightcone respectively, in units of the radius r.31

29As noted in Footnote 24, for near-extremal black holes, (18) becomes

vc = − β

2π log
r2
s

β2 |drs/dv| + O(β).

Hence, substituting (22), we find that

vc = − β

2π log ∆SBH + O(β),

where ∆SBH = SBH −S0
BH = O(rsSBH/β) with S0
BH the entropy of an extremal black hole with the same charge.
The location we find for the non-empty extremal surface is therefore consistent the similar calculations, for two-
sided black holes in JT gravity, done in [43]. The O(β) corrections are also the same in both calculations [56].
30Since the entanglement entropy of gravitons is not well understood, we shall assume here that no graviton
modes are extracted into the reservoir Hrad.
One would hope that, if we did understand the entanglement
entropy of gravitons, we would find that they would contribute to the location of the quantum extremal surface
in a similar way to other bulk modes.
31Here we are implicitly using the fact that the radius r, like any smooth, non-singular coordinate, is an
approximate affine coordinate along ingoing lightrays at sufficiently small distance scales.

18


Figure 3: The bulk entropy of the region, shown in blue, between the Ryu-Takayanagi surface
and the boundary can be decomposed into the entropy of the ingoing and outgoing modes. The
ingoing modes are in the infalling vacuum, and the bulk region includes ingoing modes spread
over approximately the scrambling time, which diverges in the semiclassical limit. This means
that the gradient, in units of infalling time, of the entropy of the infalling modes tends to zero.
In contrast, as the RT surface approaches the past lightcone of the boundary, there will be
a negative logarithmic divergence in the (renormalised) entropy of the outgoing modes. This
divergence should stabilise the location of the quantum extremal surface a small distance away
from the outgoing lightcone.

The cut-offs ε1 and ε2 are crucial to the calculation and so we take some time to discuss
them in detail. The cut-off ε1 at the quantum extremal surface is unphysical. Since the bulk
entropy would otherwise be formally divergent, we need to cut-off the bulk degrees of freedom at
some fixed proper lengthscale. The lengthscale chosen is arbitrary; in the calculation of physical
quantities, such as entanglement entropies in the boundary CFT, the cut-off dependence should
be cancelled by the scale dependence of the couplings in the effective gravitational theory due
to renormalisation.32

Of course, the cut-off ε1 is not itself a proper lengthscale.
Since the radius r is a null
coordinate, the proper length of the cut-off ε1 is zero.
The proper cut-off εprop is instead
determined by the inner product of the cut-off ε1 with the cut-off at the extremal surface on the
ingoing modes. Specifically

εprop =

�

g
�
ε1
∂
∂r, εin
∂
∂v

�
,
(25)

where g is the metric and εin is the cut-off on the ingoing modes in units of v.
When we claimed that the entropy of the ingoing modes was approximately constant, we
implicitly assumed that the cut-off εin was constant. Since, to leading order, the metric is given

32Of course, even boundary entanglement entropies are not actually well defined because of UV-divergences in
the boundary theory (which correspond to IR-divergences in the bulk theory). However the mutual information
between two boundary regions, for example, is a well defined regulator-independent finite quantity.

19


by

ds2 = 2dvdr,
(26)

everywhere in the near horizon region, this means that the cut-off ε1 should also be constant,
so that the proper cut-off εprop is constant in AdS units.
The physical status of the lightcone cut-off ε2 is very different. The lightcone cut-off ε2 is
related to the cut-off ε0 on the Schwarzschild frequency of the modes that we extract into the
reservoir Hrad. Unlike the cut-off at the extremal surface, this is a physical cut-off that depends
on the dynamics that we use to extract outgoing modes into Hrad.
However the cut-off ε2 is blueshifted as the outgoing modes evolve back in time. If we parallel
transport the cut-off ε2∂/∂r backwards along the past lightcone, we find that

0 = ∇v

�
ε2
∂
∂r

�
,
(27)

= [∂vε2 + Γrvr ε2] ∂

∂r,
(28)

=
�
∂vε2 − 2π

β ε2

� ∂

∂r,
(29)

where we approximated the metric by its leading order (static) approximation (11) and used the
fact that, in the near horizon region, f′(r) ≈ f′(rs) = 4π/β. We therefore find that the cut-off
ε2 is related to the (constant) cut-off ε0 on the Schwarzschild energy of the extracted modes by

ε2 ∝ e
2π
β vε0.
(30)

We emphasize that the infalling-time dependence of this cut-off is purely a product of the
coordinate system that we are using. In Section 2.4, we do a more general calculation, which
includes greybody factors, in Kruskal-Szekeres-like coordinates, where there is no blueshifting,
and so the cut-off at the lightcone would be constant.33 The calculations done here are rederived
as a special case, without any reference to exponentially small cut-offs. For the moment, however,
we shall continue to use Eddington-Finkelstein coordinates, which have a more natural physical
interpretation. For pedagogical purposes, in Appendix A, we also give an example of a simple
Rindler space calculation that illustrates the importance of taking into account the coordinate
dependence of cut-offs.
If we drop terms that are independent of position, it follows from (24) that the bulk entropy
is given by

Sbulk = cevap

6
log (rlc(v) − r) − cevapπv

6β
+ . . .
(31)

Because of the rotational symmetry, it is sufficient to show that the surface is extremal under
perturbations that preserve the symmetry. Varying the infalling time v, while holding the radius

33In many ways, the nicest coordinates to use for the problem are the outgoing Kruskal coordinate U together
with the infalling time v. However, the author is too lazy to rewrite all the calculations in these coordinates.
See [59].

20


Figure 4: The quantum Ryu-Takayanagi surface χq, the classical maximin surface χc, and
the entanglement wedges Erad and ECFT of the reservoir and CFT, in Eddington-Finkelstein
coordinates (left) and in a Penrose diagram (right). In the interests of simplicity, the Penrose
diagram does not include the post-evaporation region, which would be in the top right. The
classical maximin surfaces lies at the intersection of the past lightcone (dashed) with the apparent
horizon rs (dotted), which is outside the event horizon. The quantum RT surface, in contrast,
lies slightly inside the event horizon. Much of the interior is in the entanglement wedge Erad
of the reservoir (green), although part of the interior still lies in the entanglement wedge ECFT
of the CFT (blue). As the black hole continues to evaporate, the RT surface moves forward in
infalling time along a spacelike trajectory, following the red arrow. On timescales that are small
compared to the evaporation time, it remains a fixed radial distance inside the event horizon.

r fixed, we find,

0 = ∂Sbulk

∂v

����
r
+
1

4GN

∂A
∂v

����
r
,
(32)

= ∂Sbulk

∂v
,
(33)

=
drlc/dv
6(rlc − r) − π

6β ,
(34)

rlc − r = β

π
drlc
dv
(35)

= 2(rlc − rs).
(36)

In the last equality, we used (13). Varying the radius r, at fixed infalling time v, we find,

0 = ∂Sbulk

∂r

����
v
+
1

4GN

∂A
∂r

����
v
,
(37)

= −
cevap

6(rlc − r) + (d − 1)Ωd−1rd−2
s

4GN
,
(38)

rlc − r =
2GNcevap

3(d − 1)Ωd−1rd−2
s
,
(39)

= 2β

π
drs
dv = 4(rs − rhor),
(40)

21


Figure 5: Because the absorbing boundary conditions are only deterministic when evolving
forwards in time, the domain of dependence of the boundary is the future boundary.
The
entanglement wedge ECFT of the CFT (light blue) contains the causal wedge CCFT (dark blue).
When time is reversed using standard reflective boundary conditions, the entanglement wedge
still contains the causal wedge because the backreaction on the geometry creates a white hole.

where in the last equality we have used (22). The quantum extremal surface therefore lies at

v = vc + β

2π log 3 = − β

2π log SBH

cevap
+ O(β),
(41)

where vc is the infalling time of the classical maximin surface found in (23), and

r = rs − β

π
∂rs
∂v = rhor − (rs − rhor).
(42)

The extremal surface is twice as far inside the apparent horizon as the event horizon because the
entropy of the Hawking radiation produced by the black hole is twice the Bekenstein-Hawking
entropy lost by the black hole, which can be easily seen by noting that the energy E and entropy
S of black body radiation in two spacetime dimensions are related by

E = 1

2TS.
(43)

The location of the quantum extremal surface is shown in Figure 4, together with the classical
maximin surface and the two entanglement wedges.
It is important to note that the entanglement wedge of HCFT is bounded by the past light-
cone, rather than by the boundary of anti-de Sitter space. This is because the boundary con-
ditions are not deterministic when evolving backwards into the past, without access to Hrad.
The region outside the past lightcone is therefore not in the bulk domain of dependence of a
spacelike surface connecting the RT surface to the boundary.
The causal wedge is the intersection of the causal past and future of the boundary domain of
dependence. Since the boundary time evolution is irreversible, only the future of the boundary
is in its domain of dependence. The causal wedge is therefore the intersection of the exterior of
the black hole with the future of an infalling lightcone from the boundary. The entanglement
wedge contains the causal wedge, as expected.
Of course, we can simply evolve the system back in time using standard reflective boundary
conditions; on the boundary, this corresponds to using the ordinary, uncoupled Hamiltonian for
the CFT. For this spacetime, both the future and past of the boundary are in its domain of

22


Figure 6: If a diary was thrown into the black hole more than the scrambling time into the past
(left), it will now lie in the entanglement wedge of the reservoir Hrad and can in principle be
decoded using only the Hawking radiation. A diary thrown into the black hole more recently
(right) remains in the entanglement wedge of, and encoded in, the CFT.

dependence and so the entire exterior of the black hole will be in the causal wedge. One might
worry that the entanglement wedge will not contain the causal wedge.
Specifically, in our original spacetime, a signal could easily travel backwards in time from
the entanglement wedge of Hrad to the boundary. If this was still possible under an evolution
where the two systems Hrad and HCFT were uncoupled, then we would have a serious problem.
However this fails to take into account the backreaction on the spacetime geometry that
happens when we change the dynamics.34 It is easy to see that the geometry of the spacetime
must change when we change the boundary conditions. Without the Hawking radiation from
Hrad, the black hole cannot grow indefinitely as we evolve the state backwards into the past; it
does not have the energy to do so.
Instead, the discontinuity in the outgoing modes will create a shell of high energy density
at the past lightcone; the energy of this shell will be proportional to the number of modes cevap
that were extracted. As the shell evolves back into the past, it will be blueshifted, creating
large backreaction on the geometry once it is a distance O(cevap GN) from the black hole. This
will create a white hole with Schwarzschild radius O(cevap GN) larger than the original black
hole. The Ryu-Takayanagi surface will now lie slightly inside the bifurcation horizon of the new
spacetime and the entanglement wedge of the CFT will continue to contain the causal wedge.

2.3
Hayden-Preskill and the Page Curve

In this subsection, we show how the Ryu-Takayanagi surface, calculated in Section 2.2, explains
properties of black hole evaporation, such as the Hayden-Preskill decoding criterion and the
Page curve, that have been conjectured based on simple toy models of black hole evaporation.
We start with the Hayden-Preskill decoding criterion [8]. This says that, if an unknown,
small, light diary is thrown into a black hole, whose state is known, at an early stage in its
evaporation, the diary can be decoded from the Hawking radiation almost immediately after the
Page time. If the small diary is instead thrown into the black hole after the Page time, it can
be decoded from the Hawking radiation after waiting for the scrambling time.
This indeed exactly what we see from entanglement wedge reconstruction. After the Page

34See [31] for discussion of essentially the same effect in terms of the dynamics of the boundary particle in
(1 + 1)-dimensional gravity.

23


time, the quantum extremal surface lies near the black hole horizon at an infalling time

v = − β

2π log SBH

cevap
+ O(β).
(44)

Assuming cevap = O(1), this is exactly one scrambling time, plus subleading corrections, before
the current time. A diary thrown into the black hole before this time lies in the entanglement
wedge of, and can be decoded from, the reservoir Hrad containing the Hawking radiation. Any-
thing thrown in after this time lies in the entanglement wedge of the CFT. This is shown in
Figure 6.
Of course, if we actually throw a diary into the black hole, it will have non-zero energy
and will therefore backreact on the geometry. So long as the energy of the diary is O(1), the
backreaction will only change the horizon area by an O(GN) amount. However, the evaporation
of the black hole changes the horizon area by an O(GN) amount over one thermal time β. Hence
it is reasonable to expect that the backreaction will only affect the delay until the diary can be
reconstruction from the Hawking radiation by a subleading O(β) amount. When we study the
reconstruction of large diaries in Section 3.4, we will see that this is indeed the case.
In addition to the well-known scrambling time delay in recovering information thrown into
a black hole, (44) has a small logarithmic correction based on the rate cevap at which Hawking
radiation is extracted from the black hole. While we postpone any formal calculation to future
work, it is easy to see heuristically that this is consistent with the boundary dynamics of the
theory.
In a fast scrambling system, the number of degrees of freedom that a ‘simple’ initial pertur-
bation influences grows exponentially with time. This is sometimes described as an ‘epidemic’
where each ‘infected’ qubit infects an O(1) number of other qubits in each timestep (which in our
case corresponds to an O(β) timescale). After the time given in (44), a O(1/cevap) fraction of the
degrees of freedom will be infected. However, the number of degrees of freedom extracted per
timestep is O(cevap). So it is reasonable to expect that an observer with access to the extracted
Hawking radiation should be able to detect the perturbation, and hence decode the diary, after
the time given in (44).
We emphasize that the state of the diary being encoded in the early Hawking radiation
does not mean that the diary has been magically extracted out of the interior of the black hole
and into the reservoir Hrad. It is ‘still’ in the interior. There exists a single spacetime that
describes the evaporating black hole (which is semiclassical everywhere except for regions of
high curvature). In this spacetime, the diary falls into the black hole and keeps falling until it
approaches the singularity and the semiclassical spacetime breaks down. It cannot ‘no longer’
have this worldline – that’s not how spacetime works. Spacetime does not change over time; it
describes changes over time.
Instead, the encoding of the diary in the Hawking radiation should be understood in terms
of the usual story of holography. An object sitting in the middle of the bulk is not ‘actually’
at the boundary; it is in the middle of the bulk. In the effective field theory that describes the
bulk, it is an independent degree of freedom from all the fields at asymptotic infinity.
Nonetheless, by manipulating the fields at asymptotic infinity in a sufficiently complicated
way, we can make the bulk effective field theory breakdown and thereby manipulate the object in
the middle. Microscopically, the fields at asymptotic infinity contain all the degrees of freedom
of the theory.
We should therefore not be too surprised that, at a microscopic level, the diary in the interior
is not an independent degree of freedom from the radiation in the reservoir, and hence, with
sufficiently complicated manipulations of the reservoir, one can, in principle, manipulate the

24


Figure 7: The next Hawking modes to escape the black hole and be extracted into Hrad will
be entangled with interior modes that are mostly in the entanglement wedge of the reservoir
Hrad. This causes the entanglement between the black hole and the reservoir to decrease as the
new radiation is extracted, in accordance with the Page curve. (Left: Eddington-Finkelstein
coordinates; right: a Penrose diagram.)

diary.35

We have understood the Hayden-Preskill decoding criterion by analysing the entanglement
wedge, HCFT or Hrad, that particular ingoing modes are in. The Page curve, and a resolution
of the firewall paradox, will follow from analysing the entanglement wedge of outgoing modes.
We first note that, since we know the location of the Ryu-Takayanagi surface, it is easy to
find the entanglement entropy between the Hawking radiation reservoir Hrad and the conformal
field theory HCFT (and hence the black hole) using the Ryu-Takayanagi formula. After the Page
time, the entanglement entropy is given to leading order by the Bekenstein-Hawking entropy
Ahor/4GN of the black hole, plus a subleading correction from the bulk entropy term. Since
we have already calculated the entanglement entropy before the Page time, at the start of this
section, we have therefore successfully derived the entire Page curve.
However, on its own, this is somewhat unsatisfying. It does not explain why moving outgoing
Hawking modes, which we naively thought were unentangled with the earlier radiation, from
the CFT to the reservoir Hrad should somehow decrease the entanglement between the two. It
does not explain how the AMPS firewall paradox [10] is avoided.
Fortunately, in addition to the Ryu-Takayanagi formula, we also know about entanglement
wedge reconstruction.
Again, we start with heuristic arguments and then progress to more
precise statements.
Consider a Hawking mode that escapes the black hole slightly into the
future. As shown in Figure 7, we can heuristically think of this mode as being entangled with
a partner mode behind the horizon. The partner modes will be in the entanglement wedge of
Hrad. Hence moving this Hawking quanta from HCFT to Hrad will decrease the entanglement
between the two. The same ER=EPR resolution [18] to the firewall paradox that worked for
the two-sided black hole also works for a one-sided evaporating black hole.
Of course, this is only an approximate heuristic picture. For free fields, Rindler modes, with
a given Rindler frequency, outside and inside the horizon are indeed perfectly entangled with
one another. However such modes are completely delocalised within the exterior and interior

35This is not to say that there aren’t serious conceptual questions that remain to be understood about the
relationship between the microscopic boundary theory and the effective bulk theory; it is just that they are
fundamentally they same conceptual problems that always exist in holography, even without any black holes.

25


Figure 8: Hawking modes that will escape the black hole only an O(β) time into the future
are entangled with interior modes that are almost entirely in the entanglement wedge of the
CFT. This is very different from a simple random unitary toy model, but is consistent with toy
models where the thermodynamic entropy (i.e. number of qubits) increases over time. (Left:
Eddington-Finkelstein coordinates; right: Penrose diagram.)

respectively. Localised modes outside the horizon will not be perfectly entangled with their
reflection inside the horizon, unless the modes have support in only a very narrow range of
Rindler frequencies, and hence are delocalised across a large region in Rindler units.
In this case, since part of the interior is in the entanglement wedge of the CFT, we cannot
find modes, with support in only a narrow range of Rindler frequencies, whose reflection inside
the horizon will be entirely in the entanglement wedge of Hrad. We should therefore expect that
the thermal outgoing radiation will be mostly entangled with the reservoir Hrad, but also be
somewhat entangled with the CFT. Moving the Hawking quanta from HCFT to Hrad will decrease
the entanglement entropy, but by less than the entropy of the Hawking quanta themselves. This
agrees with the Page curve, since the total thermodynamic entropy of the CFT and reservoir is
increasing over time and hence

1

4GN

dAhor

dv
> −dSrad

dv ,
(45)

even at leading order. We will do a formal calculation that finds perfect agreement between the
bulk entanglement structure and the Page curve below.
Interestingly, and somewhat counterintuitively, Hawking quanta that escape at a time only
O(β) into the future will be almost perfectly entangled with interior modes that lie almost
entirely in the entanglement wedge of HCFT, as shown in Figure 8. (In this case, by making
the outgoing Hawking mode escape sufficiently far, although still an O(β) time, into the future,
we can indeed consider outgoing wavepackets with support in only a narrow range of Rindler
frequencies, but with ‘mirror’ interior modes that are entirely within the entanglement wedge of
HCFT.) They will therefore be almost completely unentangled with the reservoir Hrad.
This is in sharp contrast with the most naïve random unitary toy models of black hole
evaporation. If our model consists of a single random unitary acting on the initial black hole
state and then qubits being released one by one as Hawking radiation, we find that any single
qubit of Hawking radiation is almost perfectly entangled with any set of more than half the
qubits. Hence, if we collect Hawking radiation until after the Page time, throw away qubits of
Hawking radiation for a while, and then finally collect one more qubit of Hawking radiation, we

26


Figure 9: A simple toy model of black hole evaporation that takes into account the increase in
thermodynamic entropy as the black holes evaporates. At each timestep, two qubits escape the
black hole as Hawking radiation, but then a random isometry is applied to the black hole that
increases the number of qubits by one. The total number of qubits (corresponding to the total
thermodynamic entropy) therefore increases by one qubit at each timestep. Even if we collect
the radiation qubits until long after the Page time, qubits that are radiated only a few timesteps
into the future will be completely unentangled with the radiation that we have collected.

still find that the additional qubit of Hawking radiation is almost perfectly entangled with the
early Hawking radiation that we collected.
However such a model does not take into account the fact that the total combined thermody-
namic entropy of the black hole and Hawking radiation, which corresponds in the toy model to
the total number of qubits, is strictly increasing over time as the black hole evaporates. A more
sophisticated toy model, which does take into account this thermodynamic entropy increase,
involves a series of nested random isometries, as shown in Figure 9 [38].
At each step, qubits are extracted into the Hawking radiation, but then a random isometry
is applied to the black hole so that the number of black hole qubits decreases by less than the
number of Hawking radiation qubits increases. It can easily be seen using Page’s theorem [60]
that an additional qubit of Hawking radiation will be almost totally uncorrelated with the early
Hawking radiation, so long as a large, but O(1), number of qubits are thrown away in between.36

If the infalling modes are in a thermal state at a temperature equal to the temperature of the
black hole, there will be no increase in thermodynamic entropy. We study finite-temperature
infalling modes in Appendix B, where we indeed find that, if the temperature of the infalling
modes is equal to the black hole temperature, the Hawking radiation will be completely unen-
tangled with any CFT degrees of freedom, even when it escapes far into the future. We also
find, in several separate cases such as thermal infalling modes and pure infalling modes with
constant energy density, that information stops escaping the black hole and the Hawking radi-
ation becomes completely thermal at exactly the moment when this becomes consistent with
unitarity.
Having described heuristically how entanglement wedge reconstruction allows us to avoid the
firewall paradox, let us now do a more formal calculation of the change in entanglement entropy
from extracting bulk Hawking modes. In fact, we shall prove that this change in entanglement

36We call the early radiation Hilbert space HE, and the black hole Hilbert space, when we stop collecting
radiation, HBH. There is then a random isometry V : HBH → HT ⊗ HQ ⊗ HZ, where HT contains the thrown
away radiation, HQ is the final collected qubit and HZ contains the remaining black hole qubits. When we stop
collecting the Hawking radiation, the state |ψ⟩ ∈ HBH ⊗ HE will be close to maximally entangled. Since we are
after the Page time, |HE| ≫ |HBH| and so |ψ⟩ only has support in a subspace ˜
HE ⊆ HE with | ˜
HE| = |HBH|.
However, since | ˜
HE ⊗ HQ| ≪ |HT ⊗ HZ|, the reduced density matrix of the state V |ψ⟩ on HE ⊗ HQ will be
very close to maximally mixed. There will be essentially no entanglement, or even correlation, between the early
radiation and the final collected qubit.

27


entropy will necessarily always agree with the Ryu-Takayanagi formula. However, we first start
by calculating it explicitly.
We want to calculate the change in entanglement entropy from outgoing modes, over some
small time range δv, being transferred from HCFT to Hrad. We need the time δv to be small,
because otherwise the Ryu-Takayanagi surface, and hence the entanglement wedges, of HCFT and
Hrad will depend on whether the transferred modes are included in HCFT or Hrad. By making
δv very small, we ensure that all bulk modes (other than the transferred ones) are encoded in
one of the two Hilbert spaces, even if the transferred modes themselves are not counted as part
of either Hilbert space. One can then calculate the change in entanglement over longer time
periods by integrating these infinitesimal changes.
Outgoing modes that are between the Ryu-Takayanagi surface and the past lightcone of the
boundary are encoded in the CFT, while all other outgoing modes are encoded in the reservoir
Hrad.37 As discussed at the beginning of this section, since the overall state of the outgoing
modes is pure, we will find the same change in entanglement entropy if we look at the change
in the entropy of the outgoing modes in HCFT, or the entropy of the outgoing modes in Hrad.
However, since the modes in the CFT consist of a single interval, the change in their entropy is
more natural to calculate.
From (13), it is easy to see that extracting the Hawking radiation for an additional time δv
will move the radius rlc of the outgoing lightcone by

δrlc(v) = −2πδv

β
(rlc(v) − rhor(v)).
(46)

Assuming we keep the cut-off on the extracted outgoing modes constant, extracting the addi-
tional Hawking radiation will change the cut-off ε2 in units of r by

δε2 = −2πδv

β
ε2.
(47)

To derive this equation, we note that the cut-off ε2 only depends on the difference between the
infalling time at which the radiation is extracted and the infalling time of the quantum extremal
surface.
Hence, extracting radiation for an additional time δv has the same effect on ε2 as
moving the extremal surface backward in infalling time by δv; (47) is therefore an immediate
consequence of (30).
Using (24), we find that

δS = cevap

6
δrlc

rlc − r − cevap

12
δε2
ε2
(48)

= −cevap π δv (rlc(v) − rhor(v))

3 (rlc − r)
+ cevap π δv

6
(49)

= −cevapπδv

12β
=
1

4GN

dAhor

dv
δv,
(50)

where in the second equality we used (46) and (47), in the third equality we used (36) and (40)
and in the last equality we used (21). The change in entanglement entropy exactly agrees with
the change in entanglement entropy that one finds using the Ryu-Takayanagi formula.

37We are ignoring the fact that entanglement wedge reconstruction is only approximate here. Since the separa-
tion between the outgoing lightcone and the extremal surface grows linearly in the limit of large cevap, the effect
of the reconstruction errors should become small in this limit. Furthermore, it is reasonable to hope that the
effect of errors in the reconstruction of bulk modes on each side will cancel and so we will still find the correct
answer even at small cevap. As we shall see, it seems that this is indeed the case.

28


At first glance, these two methods of calculating the change in entanglement entropy appear
very different, despite the perfect quantitative agreement between them. In the Ryu-Takayanagi
formula, the change comes from the change in the horizon area of the black hole, with its some-
what mysterious association with entropy, while, in (48), the change occurs because outgoing
bulk modes are entanglemed with other outgoing bulk modes in the entanglement wedge of Hrad.
However, it is not a coincidence that they give the same answer.
The Ryu-Takayanagi
formula is really calculating the change in the generalised entropy A/4GN + Sbulk of the Ryu-
Takayanagi surface, not just the change in the area; it is just that, in this case, the bulk entropy
happens to stay approximately constant (because of the approximate time translation invariance
of the evaporation process). The bulk entropy calculation can also be thought of as a change
in generalised entropy; just one in which the RT surface for which the generalised entropy is
evaluated, and therefore the area term, stay fixed.
In the Ryu-Takayanagi formula calculation, the change in entropy is given by the difference
between the new generalised entropy for the new Ryu-Takayanagi surface and the old gener-
alised entropy for the old Ryu-Takayanagi surface. In the bulk entanglement calculation, the
change in entropy is the difference between the new generalised entropy and the old generalised
entropy, when both are evaluated using the old Ryu-Takayanagi surface. However, by definition,
the generalised entropy is constant at leading order if we perturb the Ryu-Takayanagi surface.
Since we need δv to be infinitessimally small to do the bulk entanglement calculation, the two
calculations will always give the same answer. Because the Ryu-Takayanagi surface is a quantum
extremal surface, there can never be a firewall paradox. The bulk entanglement structure will
always be consistent with the Ryu-Takayanagi formula.

2.4
Greybody Factors

As we discussed in Section 2.2, there is nothing genuinely unphysical about evaporating a black
hole in AdS/CFT by extracting black-body Hawking radiation from well inside the zone. How-
ever, if we eventually want to understand four dimensional black holes in flat space that evap-
orate naturally (without an external super-observer extracting Hawking radiation from near
the horizon), it is important to understand what happens when there are non-trivial greybody
factors.
Although we will not be able to explicitly calculate the location of the quantum extremal
surface when greybody factors are present, it turns out that we will still be able to derive both
the Hayden-Preskill decoding criterion and the Page curve.38

We first recall our argument, from the very beginning of this section, that the maximin
prescription implies that the Ryu-Takayanagi surface must become non-empty at the Page time,
even when the greybody factors are non-trivial. After this time, assuming that the quantum
maximin prescription is valid, there must exist a non-empty quantum extremal surface.39 We
shall show that this is indeed the case

38Since the entanglement entropy of gravitons is not understood, we shall still assume that no graviton modes
are extracted using the absorbing boundary conditions and hence that their entanglement entropy can be ignored.
Of course, in flat space, gravitons will always contribute to the Hawking radiation, so understanding their
entanglement entropy precisely is an important task for future work. However, since the relevant graviton modes
simply become ordinary light scalar fields in a (1 + 1)-dimensional reduction of the evaporation, as do all the
other bosonic modes that contribute to the evaporation, it seems reasonable to expect that their contributions
to the bulk entropy will be qualitatively the same as any other mode.
39In fact, since we continue to assume rotational symmetry, this argument would not actually require the full
power of the assumption of quantum maximin. Instead, we can restrict our maximisation and minimisation to
rotational symmetric Cauchy slices and surfaces χ respectively, thereby avoiding most of the subtleties that would
be involved in defining the quantum maximin prescription and showing its equivalence to the quantum extremal
surface prescription.

29


What can we say about the location of this quantum extremal surface? We know that the
entanglement wedge must contain the causal wedge [29], so the extremal surface must lie in
the interior of the black hole.40
However we also know that there does not exist a classical
extremal surface anywhere in this spacetime.
This means that, at the non-empty quantum
extremal surface, the gradient of the bulk entropy term must be O(1/GN) (at least in Eddington-
Finkelstein coordinates where the gradient of the area is O(1) everywhere).
Since the bulk entropy itself is O(1), the only way that this can happen is if the extremal
surface is very close in Eddington-Finkelstein coordinates to a point where the bulk entropy
diverges. The extremal surface must therefore approach the past lightcone of the boundary,
which is always outside the event horizon and only approaches the event horizon at infalling
times that are far into the past.41 In Eddington-Finkelstein coordinates, the quantum extremal
surface must therefore both approach the event horizon, with respect to the radius r, and diverge
into the infinite past, with respect to the infalling time v, in the limit GN → 0. Again, we shall
explicitly verify that this is the case.
It is helpful at this point to switch from Eddington-Finkelstein coordinates to lightlike,
Kruskal-Szekeres-like coordinates. At radii close to the event horizon, and over infalling timescales
that are small compared to the evaporation time, the metric of the evaporating black hole in
Eddington-Finkelstein coordinates is given by

ds2 = −4π

β (r − rs(v))dv2 + 2dv dr + r2dΩ2
d−1,
(51)

where we can assume that the inverse temperature β and the evaporation rate drs/dv are
constant at leading order. Substituting the Kruskal-like coordinates,

V = β

2π exp(2πv/β),
(52)

and

U = (rhor(v) − r) exp(−2πv/β),
(53)

where rhor = rs + β(drs/dv)/2π as in (16), we find

ds2 = −2dUdV + r2(U, V )dΩ2
d−1.
(54)

Note that the definition (53) is only intended to be valid in the near horizon region where
outgoing lightrays escape exponentially in Eddington-Finkelstein coordinates. More generally,
the coordinate U should be defined in the exterior region by

U ∝ − exp(−2πu/β),
(55)

where u is the boundary time at which an outgoing lightray reaches the boundary, and then
the metric should be analytically extended to the interior where U > 0. This will ensure that
the coordinates U, V are exactly lightlike everywhere. However, we are only interested in the
near horizon region where the definition (53) and the metric (54) are valid.42 Note that V > 0

40We shall prove explicitly, later in this section, that the quantum extremal surface that we find is indeed inside
the event horizon of the black hole.
41Technically, the bulk entropy will also diverge near the future lightcone. However there cannot be a quantum
extremal surface near the future lightcone, because, at the future lightcone, the bulk entropy will only diverge as
function of the infalling time v, while dA = (d − 1) rd−2
s
Ωd−1dr.
42Even within the near horizon region, our conventions for U and V differ from the more standard conventions
for Kruskal coordinates in AdS space by constant factors [54]. However, within the near horizon region, our
convention will be somewhat more convenient.

30


everywhere; the infinite past with respect to infalling time v corresponds to the limit V → 0+.
Also note that the past lightcone of the current boundary time (i.e. v = 0) is at Ul.c. = −O(rs).43

Our basic strategy will be to show that ∂Sbulk/∂U should approach a well-defined O(1) limit
as V → 0, whereas
1

4GN

∂A
∂U = O
� V

GN

�
.

We will also find that, in the same limit, ∂Sbulk/∂V = O(1/V ), while

1

4GN

∂A
∂V = O
� 1

GN

�
+ O
� 1

V

�
.

We will therefore be able to argue that the quantum extremal surface must be at some fixed U
that is independent of GN and at V = O(GN), which corresponds to an infalling time exactly
one scrambling time (plus subleading corrections) into the past.
As with the Eddington-Finkelstein coordinates r, v, the metric (54) is approximately constant
in the near horizon region in terms of the coordinates U, V . We can therefore consistently cut-
off ingoing modes at the quantum extremal surface with a constant cut-off in units of V , and
outgoing modes with a constant cut-off in units of U. Since

∂V
∂v = exp(2πv/β),
(56)

and

∂U
∂r = exp(−2πv/β),
(57)

have non-trivial infalling-time dependence, the gradient of the entropy of either the ingoing or
outgoing modes alone will depend on the set of cut-offs that we use. In particular, in Section 2.2,
when the cut-offs were constant in Eddington-Finkelstein coordinates, there was an increase in
bulk entropy, when moving the RT surface forwards in infalling time along an outgoing lightcone,
that came from outgoing bulk modes . With constant cut-offs in Kruskal-like coordinates, the
same increase in bulk entropy exists, but it comes from the infalling modes. The gradient of the
total bulk entropy is the same in both cases.
An advantage of using constant cut-offs in units of U and V is that outgoing modes are
not blueshifted, with respect to U, as we evolve them backwards in time. The outgoing modes
that are contained in the entanglement wedge of HCFT will be determined only by U, and we
don’t have to worry about blueshifting the cut-off at the outgoing lightcone. If the cut-off at
the quantum extremal surface is also constant in units of U, the entropy of the outgoing modes
in the entanglement wedge of the CFT will be independent of V . To calculate ∂Sbulk/∂V we
therefore only need to worry about the ingoing modes near the quantum extremal surface.
In Section 2.2, the infalling modes were in the vacuum state with respect to the infalling time
v. We therefore argued that, so long as the cut-off was constant in units of v, the gradient of
the entropy of the infalling modes would tend to zero as the quantum extremal surface diverged
into the infinite past in the semiclassical limit.
When there are non-trivial greybody factors, with part of the Hawking radiation being
reflected back into the black hole, the infalling modes will instead be in a mixed state that is
invariant with respect to translations in infalling time. The mixed-state infalling modes will be

43As discussed in Section 2.1, for near-extremal black holes, we actually have Ul.c = −O(r2
s/β). Also, recall
that we are assuming for convenience that the past lightcone escapes the near-horizon region at v = 0. Hence
for parametrically small AdS-schwarzschild black holes, the current boundary time is really v = πlAdS + O(β).
These subtleties will be unimportant for our purposes; the key point is that Ul.c. is independent of GN.

31


Figure 10: In the left figure, infalling modes near the quantum extremal surface are in a infalling-
time-translation invariant mixed state. These mixed state modes (black) are purified by modes
deep in the interior, together with modes that escaped into the reservoir (both red). Both of
these are in the entanglement wedge of the reservoir Hrad. In the right figure, outgoing modes
near the quantum extremal surface (black) are entangled with outgoing modes in the interior, but
also with outgoing modes that recently escaped into the reservoir and late-time infalling modes
that were reflected back into the black hole (all in red). The first two are in the entanglement
wedge of Hrad, while the last is in the entanglement wedge of the CFT. The bulk entropy of
the modes in the entanglement wedge of the CFT depends on U no longer depends on U in the
same simple way that it did when no greybody factors were present in Section 2.2.

purified by outgoing modes that escaped the black hole into the reservoir, as well as modes deep
in the interior of the black hole. As shown in Figure 10, these modes are all in the entanglement
wedge of Hrad.
In the semiclassical limit, when the extremal surface diverges into the infinite past, there will
be no entanglement between ingoing modes near the quantum extremal surface and outgoing
modes in the entanglement wedge of the CFT. We therefore find

2πV

β
∂Sbulk

∂V

����
U
= ∂Sbulk

∂v

����
U
= cevapπ

6β
− dSin

dv ,
(58)

where dSin/dv ≥ 0 is the constant entropy per unit infalling time of the infalling modes assuming
a constant cut-off in units of the infalling time v. dSin/dv can in principle be calculated from a
numerical approximation of the reflection coefficients for modes escaping the near-horizon zone.
(See, for example, similar calculations in [61].) The additional term cevapπ/6β shows up because
the cut-off at the extremal surface should be constant in units of V , rather than constant in
units of v, if we want to ignore the outgoing modes. Since

∂
∂V

����
U
= e−2πv/β ∂

∂v

����
U
,
(59)

and the logarithmic divergence (cevap/12) log(1/ε) with respect to the cut-off length ε is univer-
sal,44 we can immediately obtain (58).
Formally, there are infinitely many angular momentum modes and so cevap is infinite. How-
ever, modes with large angular momentum are almost entirely reflected back into the black hole.
The ingoing modes are in a thermal state at the same inverse temperature β as the black hole.

44Recall that ε is the cut-off on only the ingoing modes, at only one end of an interval.

32


They therefore satisfy [57,58]

dSin
dv
= cevapπ

6β
.
(60)

It follows that only the finite number of low angular momentum modes, which actually partially
escape the black hole, contribute to (58). So long as we include the same modes in calculating
dSin/dv that we use in calculating cevap, (58) should be independent of the choice of any sufficient
large angular momentum cut-off on the modes we consider.
From (52) and (53), we have

r = rhor(v) − 2π

β UV,
(61)

and hence

∂r
∂V

����
U
=
β

2πV
drs
dv − 2πU

β
.
(62)

It follows that

1

4GN

∂A
∂V

����
U
= (d − 1)rd−2
s
Ωd−1

4GN

∂r
∂V

����
U
,
(63)

=
β2

2πV
dM
dv − π(d − 1) rd−2
s
Ωd−1 U

2βGN
.
(64)

In the second equality we used (62) and the first law of black hole thermodynamics βdM =
dAhor/4GN. The quantum extremal surface must therefore satisfy

0 =
1

4GN

∂A
∂V

����
U
+ ∂Sbulk

∂V

����
U
,
(65)

rhor − r = 2π

β UV =
2GN β

π (d − 1) rd−2
s
Ωd−1

�
β dM

dv + cevapπ

6β
− dSin

dv

�
,
(66)

where we used (58) and (64).
It can be shown, as follows, that the right hand side of (66) must be non-negative, and so
the extremal surface is inside the event horizon of the black hole. Suppose that the infalling
modes were in a thermal state at inverse temperature β′. We would then have

dM
dv = dM

dv = cevapπ

12
( 1
β′2 − 1

β2 ),
(67)

and [57,58]

dSin
dv
= cevapπ

6β′ .
(68)

The right hand side of (66) would then be given by

GN β2 cevap

6 (d − 1) rd−2
s
Ωd−1

� 1

β − 1

β′

�2
≥ 0,
(69)

which is non-negative at any inverse temperature β′.45
Since thermal states have maximal
entropy for any fixed energy flux, the right hand side of (66) must therefore be non-negative for

45For detailed calculations of quantum extremal surfaces for finite temperature infalling modes, see Appendix
B.

33


any state of the infalling modes, thermal or otherwise. The quantum extremal surface is always
inside the event horizon.
So far we have only demanded that the surface be extremal if we vary V at constant U. The
quantum extremal surface should also be extremal when we vary U at constant V . From (61),
we have

1

4GN

∂A
∂U

����
V
= −(d − 1) rd−2
s
Ωd−1 V

4GN
.
(70)

What about the variation of the bulk entropy term? By varying U, we change the outgoing modes
that are included in the entanglement wedge of the CFT. These outgoing modes are entangled
with the other outgoing modes, both inside the quantum extremal surface and outside the past
lightcone of the boundary.
In Section 2.2, all the outgoing modes that were not in the entanglement wedge of the CFT
were in the entanglement wedge of the reservoir. We therefore found

∂Sbulk

∂U

����
V
=
cevap

6(U − Ul.c),
(71)

where Ul.c. < 0 labels the past lightcone of the boundary.46

When greybody factors are present, this will no longer be the case, as shown in Figure 10.
Outgoing modes outside the past lightcone will be partially reflected back into the black hole,
and end up as ingoing modes, which are in the entanglement wedge of the CFT. The functional
form of ∂Sbulk/∂U will therefore be much more complicated.
However, in the semiclassical limit where the extremal surface diverges into the infinite past,
there will still be no entanglement between outgoing modes in the entanglement wedge of the
CFT and ingoing modes near the extremal surface. The gradient ∂Sbulk/∂U will therefore be
independent of V , so long as V is sufficiently small. Indeed, this follows directly from the fact that
∂Sbulk/∂V is independent of U, by the symmetry of mixed partial derivatives. We conclude that
∂Sbulk/∂U has a well-defined limit as V → 0, which is some (presumably complicated) function
of U.
If the surface is extremal, we must have

0 =
1

4GN

∂A
∂U

����
V
+ ∂Sbulk

∂U

����
V
,
(72)

∂Sbulk

∂U

����
V
= (d − 1) rd−2
s
Ωd−1 V

4GN
,
(73)

U ∂Sbulk

∂U

����
V
= β

2π

�
β dM

dv + cevapπ

6β
− dSin

dv

�
,
(74)

where in the first equality we used (70) and in the last equality we used (66). The right hand
side of (74) is constant over timescales that are small compared to the evaporation time, while
the left hand side is a function of U.
If a non-empty quantum extremal surface is to exist in the near horizon region for any
sufficiently small GN, as we expect from the maximin prescription, there must exist a solution
U0 to (74). Importantly, since (74) does not depend on GN, this solution must be independent
of GN.

46Of course, our actual calculations in Section 2.2 were in Eddington-Finkelstein coordinates, but were
equivalent to (71).
With constant cut-offs in Kruskal-like coordinates, the outgoing bulk entropy Sout =
(cevap/6) log((U − Ul.c)/√ε1ε2) where the cut-offs ε1 and ε2 are both constant in units of U.

34


In the simple example from Section 2.2 where there are no greybody factors, dM/dv is given
in (20), dSin/dv = 0 and ∂Sbulk/∂U is given in (71). Hence

U0 = −Ul.c./3,
(75)

in agreement with our calculations in Eddington-Finkelstein coordinates.
What about when greybody factors are present? In this case, it turns out that we can still
calculate ∂Sbulk/∂U in the limit U → ∞, and this is sufficient to argue that a solution U0 must ex-
ist. For this argument it is simplest to use the Rindler-like coordinates, uL = −(β/2π) log(U/rs)
for U > 0 and uR = −(β/2π) log(|U|/rs) for U < 0.47 Up to a GN-independent shift, uR is the
time at which an outgoing lightray would reach the boundary, while uL is its ‘mirror’ coordinate
inside the event horizon.
The outgoing modes are near the horizon are in the thermofield double state with respect
to these Rindler coordinates. A key point will be that the thermofield double state only has
significant correlation between the left and right region when uR = uL ±O(β). If the RT surface
is at (U RT , V RT ), the interior outgoing modes are in the entanglement wedge for uL > uRT
L
=
−(β/2π) log(U RT /rs).
What about the exterior outgoing modes?
For uR/β ≫ 0, they are completely in the
entanglement wedge, since they haven’t had time to escape. For uR/β ≪ 0, the situation is
more complicated. In this case, the outgoing near-horizon modes are encoded in a combination
of the Hawking radiation that has escaped into Hrad, and the reflected ingoing modes at an
infalling time v = uR + O(β). For uR ≫ vRT = (β/2π) log(V RT /rs), the reflected modes are
in the entanglement wedge, but the escaped Hawking radiation is not. For uR ≪ vRT , neither
is in the entanglement wedge. Finally, since we are assuming that (66) holds, we note that
uRT
L
≫ vRT .
In summary, for uL, uR ≪ vRT , none of the interior or exterior outgoing modes are in the
entanglement wedge. For uRT
L
≫ uL, uR ≫ vRT , the interior modes are not in the entanglement
wedge and only the reflected part of the exterior modes is in the entanglement wedge.
For
0 ≫ uL, uR ≫ uRT
L , the interior modes and the reflected part of the exterior modes are in the
entanglement wedge. Finally for uL, uR ≫ 0, all the modes are in the entanglement wedge.
Since the thermofield double state has a local entanglement structure, as discussed above,
then we can ignore any effects that come from the finite size of each of these regions, in the
semiclassical limit GN → 0. Instead there are only three contributions to the gradient of the
bulk entropy as a function of uRT
L . The first is that increasing uRT
L
increases the range of uL, uR
for which only the reflected part of the interior modes is in the entanglement wedge.
This
gives a contribution equal to the entropy density dSin/dv of the reflected modes. The second
is that increasing uRT
L
decreases the range of uL, uR for which both the interior modes and the
reflected part of the exterior modes are in the wedge. Since the tripartite state of interior modes,
reflected exterior modes and escaped exterior modes is pure, this gives a contribution equal to
−dSrad/dv. Finally, we need the cut-off to be constant in units of U, we means the cut-off
length must exponentially grow as a function of uL. This gives a contribution to the gradient
of −πcevap/6β. We therefore find that

U ∂Sbulk

∂U
= − β

2π
∂Sbulk

∂uL
= β

2π

�dSrad

dv
− dSin

dv + πcevap

6β

�
.
(76)

To build some intuition for this formula, note that, in the limit U → ∞ and radius r given in
(66),

∂Sbulk

∂v

����
r
= −2πU

β
∂Sbulk

∂U

����
V
+ 2πV

β
∂Sbulk

∂V

����
U
= −dSrad

dv .
(77)

47The factors of rs are included only to make the logarithms dimensionless.

35


This is in exact agreement with our heuristic picture from way back in Figure 2 of Bell pairs,
entangled between the interior and the Hawking radiation, that are evenly distributed along the
wormhole.
So long as the black hole is evaporating and hence dM/dv < 0, it follows from (76) that the
left hand side of (74) should be larger at large U than the right hand side, which is independent
of U.48 Meanwhile, at the horizon, the left hand side is zero, while the right hand side is positive,
as shown in (69). Hence, assuming that the derivative ∂Sbulk/∂U is a continuous function of
the location of the rotationally symmetric surface, then by the intermediate value theorem there
must indeed exist a solution U0 to (74), as we expected from the maximin arguments.49

Since we know a solution U0 must exist, we can substitute it into (66) and find that

V =
GN β2

π2 U0 (d − 1) rd−2
s
Ωd−1

�
β dM

dv + cevapπ

6β
− dSin

dv

�
.
(78)

Hence

v = β

2π log 2πV

β
= β

2π log SBH + O(β).
(79)

In the last equality, we assumed for simplicity that all other scales are held fixed in the semiclas-
sical limit GN → 0. We can therefore derive the Hayden-Preskill decoding criterion even when
the greybody factors are non-trivial.
We can also find the Page curve (up to subleading corrections) using the Ryu-Takayanagi for-
mula. Furthermore, our argument from Section 2.3, showing that the outgoing modes automac-
tically have exactly the right entanglement to reproduce the Page curve, without any firewall
paradox, only used entanglement wedge reconstruction and the fact that the Ryu-Takayanagi
surface is an extremum of the generalised entropy. All our main results can therefore be derived,
even in the presence of greybody factors.

3
State Dependence

In deriving the results of Section 2, we were careful to focus on a single spacetime where a black
hole was formed by collapse in some particular initial state of the matter fields. Although the
details of how the black hole was formed did not affect any of our calculations, we always implic-
itly assumed that those details were known by the observer reconstructing interior operators.
We never considered the task of reconstructing a diary from the Hawking radiation with the
initial state of the black hole partially, or completely, unknown.
We therefore avoided the issue of whether, and to what extent, reconstructions of bulk
interior operators necessarily depend on the initial state of the black hole. Such questions will

48If dM/dv > 0, then this will stop being true, and the quantum extremal surface will stop existing, at exactly
the moment when the rate of change of the Bekenstein-Hawking entropy 1/4GNdAhor/dv = βdM/dv becomes
greater than the rate of increase in the entropy of the Hawking radiation. This is exactly the moment when
boundary unitarity becomes consistent with no information ever escaping the black hole. See Appendix B for a
more detailed discussion of this.
49One might wonder whether there could exist multiple solutions and hence multiple non-empty quantum
extremal surfaces. We first note that, even if there did exist multiple solutions, the solution that minimised
the generalised entropy would be independent of GN, for sufficiently small GN, and so we could simple ignore
the other solutions. However, in practice, it seems that the left hand side of (74) should be a monotonically
increasing function of U and so only one solution will exist. If no exterior modes were in the entanglement wedge
of the CFT, we would have ∂Sbulk/∂U ∝ 1/U and the left hand side of (74) would be constant. The existence of
exterior outgoing modes in the entanglement wedge of the CFT should only slow the rate of decay of ∂Sbulk/∂U
as a function of increasing U.

36


be the focus of this section. In particular, we will find that this state dependence is crucial
in resolving the information paradox. If the interior partners of the late-time Hawking modes
were encoded in the early radiation in a state-independent way, there would be no way for the
final state of the Hawking radiation to depend on the initial state of the matter falling into the
black hole, even if the final state was pure rather than thermal. The state dependence allows
the entanglement of early and late radiation to depend on the state, despite the entanglement
of late radiation and interior modes being fixed. This allows information to escape.
We begin the section by briefly reviewing results from [38] that show how state dependence
can arise in entanglement wedge reconstruction.

3.1
State Dependence in Entanglement Wedge Reconstruction

Entanglement wedge reconstruction is best understood in the language of holographic quantum
error correction [25]. Bulk operators in AdS/CFT are only well defined within the “code sub-
space” Hcode ⊆ HCFT of boundary states with the correct smooth bulk geometry. The claim of
entanglement wedge reconstruction can then be phrased as follows: the noisy quantum channel,
mapping states ρ in the code space to their restriction ρB to some boundary region B, forms
an approximate operator algebra quantum error correcting code for the bulk operators in the
entanglement wedge b of B. This means that there exists a decoding channel D such that, for
all states ρ in the code subspace,

D(ρB) ≈ ρb,
(80)

where ρb is the restriction of the bulk state ρ to the algebra of operators associated with the
entanglement wedge b of B.50 We can then use the adjoint channel D† to map bulk operators φb,
acting within the entanglement wedge, to boundary ‘reconstructions’ φB = D†(φb), acting only
on region B, whose action is (approximately) the same as the bulk operator φb on states in the
codespace. It is important to note that the decoding channel D is in general highly non-unique;
two very different boundary reconstructions of the same bulk operator may be both be valid, so
long as their action on states in the code subspace is (approximately) the same.
Because the spacetime geometry is dynamical, the entanglement wedge of a given boundary
region depends on the bulk geometry of the particular CFT state that we are interested in. Even
if we only consider a code space of bulk states with a fixed spacetime geometry, the quantum
extremal surface may be state-dependent because of the bulk entropy term. When we say that
operators in the entanglement wedge can be reconstructed in the boundary region, we should be
careful to specify which states need to have their entanglement wedge contain the bulk operator.
The initial derivation of entanglement wedge reconstruction in [23] suggested that one could
always reconstruct a bulk operator so long as it was contained within the entanglement wedge
of the boundary region for all pure states in the code space of states for which the reconstruc-
tion was meant to work. However, this derivation ignored the fact that entanglement wedge
reconstruction is only approximate at finite GN. (More specifically it ignored the fact that the
equality between bulk and boundary relative entropies [22] is only approximate at finite GN.)
It should therefore only be trusted for code spaces whose dimension is relatively small (and, in
particular, independent of GN).
A more rigorous derivation of entanglement wedge reconstruction, using the tools of approx-
imate quantum error correction, makes it clear that the bulk operator needs to be contained
in the entanglement wedge even for mixed states with support only in the code space [24].51

50Here, restriction can be thought of as a partial trace, although it is really the projection of the state onto
the von Neumann subalgebra associated with the bulk region b.
51For an alternative derivation that is more directly equivalent to the derivation in [23], but which reaches the
same conclusion as [24], see [38].

37


Rather than directly considering mixed states, it is often more natural, and is mathematically
equivalent, to assume that the mixed states are purified by an arbitrary ‘reference system’ HR,
whose dimension is equal to the code space dimension.52 In this picture, the bulk operator must
be contained in the entanglement wedge of B for all pure states, including entangled states, in
Hcode ⊗ HR.
If, instead, the bulk operator is only contained in the entanglement wedge of B for all pure
states in the code space (with no reference system), entanglement wedge reconstruction will still
be possible, but only if we allow the reconstruction to be state-dependent [38].
It is a general fact about quantum error correction that, if exact state-dependent recon-
struction of operators is possible all states in a finite-dimensional code space, then exact state-
independent reconstruction is also possible for that code space [38, 62].53 This is why it was
sufficient to only consider pure states in [23], where the reconstruction errors were ignored.
Even if the state-dependent reconstruction is only approximate, state-independent recon-
struction will necessarily also be possible (with a somewhat larger error), so long as the code
space is not too large. In holography, this corresponds to the fact that entanglement with a
reference system cannot affect the location of the Ryu-Takayanagi surface in the semiclassical
limit, so long as the dimension of the code space is O(1), because any entanglement with the
reference system will give a subleading correction to the generalised entropy.
In contrast, if the dimension of the code space is very large, for example when one considers
a large number of black hole microstates, approximate state-dependent reconstruction may be
possible, even when state-independent reconstruction is not.
A simple example of this was studied in [38]. It shows up when one considers code spaces
consisting of a large number of black hole microstates, plus bulk degrees of freedom outside the
horizon, as illustrated in Figure 11. We emphasize that, unlike in the rest of this paper, we will
not be interested in the details of the interior of these black hole microstates.
Suppose that we try to reconstruct bulk operators in a simply connected region B consisting
of slightly more than half of the boundary.
For any pure black hole microstate, the Ryu-
Takayanagi surface, with area A1, lies between the black hole and the complementary region ¯B.
The entanglement wedge of region B contains the black hole.
However, for a two-side black hole, such as the thermofield double state, the homology
constraint means that the Ryu-Takayanagi surface lies between region B and the black hole,
so long as the area A2 of this surface is less than the area A1 plus the horizon area A0. The
entanglement wedge will no longer contain the bulk region b′ that lies between the two extremal
surfaces.
Replacing the second CFT by an arbitrary reference system cannot affect whether bulk
operators in region b′ can be reconstructed in region B. Region b′ therefore cannot be encoded
in region B for any purification of the thermal density matrix. So long as we correctly use the
quantum extremal surface prescription to define the Ryu-Takayanagi surface, we do indeed find
that this is the case. Instead of classical area and a homology constraint, we now have a large
amount of bulk entanglement between the black hole and the reference system. However, the
quantum Ryu-Takayanagi surface, and its generalised entropy, are unchanged.
The black hole does not need to be maximally entangled with the reference system in order
to exclude region b′ from the entanglement wedge. We only need the entanglement entropy S

52The reference system HR should not be confused with the Hawking radiation reservoir Hrad that we use
when studying evaporating black holes. The second is an actual physical system, whereas the first is purely a
mathematical ‘accounting trick’.
53In the Schrödinger picture, which is usually used to describe quantum error correction, this corresponds to
the fact that exact universal subspace quantum error correction implies full quantum error correction [62].

38


(a)
(b)

Figure 11: The exterior geometry of a black hole, with horizon area A0, in AdS/CFT. The
boundary is divided into two regions B and ¯B; there exists a locally minimal surface separating
B from ¯B on either side of the black hole, with areas A2 and A1. These divide the bulk into
three regions b, b′ and ¯b, where region b′ lies between the two minimal surfaces and contains the
black hole. If A2 > A1, region b′ is contained in the entanglement wedge of region B for all pure
microstates, shown in Figure 11a. However, if the black hole is entangled with a reference system
with Sbulk/4GN > A2 − A1, as shown in Figure 11b, the Ryu-Takayanagi surface will jump to
A2 and the entanglement wedge of B will no longer contain region b′. As a result, a state-
independent reconstruction of region b′ on region B only exists for code spaces of microstates
with dimension |Hcode| ≤ e(A2−A1)/4GN .

to satisfy

S > A2 − A1

4GN
.
(81)

At face value, we now have something of a paradox. The bulk region b′ is encoded in region
B for any pure microstate, but for sufficiently entangled state it can be reconstructed on the
combination of region ¯B and the reference system. By linearity, a reconstruction that works
for any pure state will also work for entangled states.54 But the no cloning theorem says that
quantum information can’t be simultaneously encoded in region B, and in the combination of
region ¯B and the reference system.
The resolution, of course, is the fact that entanglement wedge reconstruction can only be
made state independent, if the bulk operator is contained in the entanglement wedge even for
mixed states with support only in the code space. In this case, the reconstruction will have
to be state dependent, precisely when the entropy S of the code space satisfies (81). There is
therefore no single reconstruction that could be used for the entangled state.
Again, we emphasize that this resolution is only consistent because of the approximate
nature of entanglement wedge reconstruction. Otherwise it would be impossible for a state-
dependent reconstruction to exist for every state in the code space, without state-independent
reconstruction also being possible. There would be no way to evade the paradox described above.
If we make the code space Hcode of microstates as large as possible, so that

log |Hcode| ≈ SBH = A0

4GN
,
(82)

54For approximate reconstructions, this fact is somewhat non-trivial to prove. However, it is indeed true, up
to a dimension-independent increase in the error size [63].

39


at leading order, a single reconstruction will only exist for any subspace of the code space with
dimension less than |Hcode|α, where

α = A2 − A1

A0
.
(83)

This is an example of something known as an ‘α-bit code’ [62]. Indeed, many of the results
about state-dependence in evaporating black holes that we will derive in this section can be
rephrased in the language of [62] as statements about the existence (or non-existence) of α-bit
codes for various values of α.
However, since the terminology of ‘α-bits’ was developed for
asymptotic quantum resource equalities and is, perhaps, more misleading than clarifying in the
present context, we will not use it any further in this paper.
Using the results of [62], it is possible to put strict lower bounds on the size of the non-
perturbative error that must exist to avoid a cloning paradox. Specifically we find that the error
in the reconstruction of region b′ on region B for a code space Hcode must be at least e−O(∆S),
where

∆S = A2/4GN − A1/4GN − log |Hcode|,
(84)

is the minimum difference, for any state in the code space, between the generalised entropies of
the two extremal surfaces [38].

3.2
State Dependence in Evaporating Black Holes

Exactly the same effects that were described in [38] also happen in evaporating black holes. The
boundary HCFT and the reservoir Hrad play the roles of the two boundary regions HB and H ¯B.
Suppose, as in Section 2.3, that we throw a small diary into a black hole. This time, however,
rather than knowing the initial state of the black hole exactly, we only know that the initial
state was in some large code space of possible states.
For example, we can imagine starting with a small black hole, with Bekenstein-Hawking
entropy Scode, in a completely unknown state. If we then throw a large amount of additional
energy (this time in a known state) into this black hole, we end up with a larger black hole,
whose state is partially unknown. It lies in a code space of possible states, which has entropy
log |Hcode| = Scode.
When can we reconstruct the diary from the Hawking radiation? As discussed in Section 2,
for any pure initial black hole state, there is a phase transition in the Ryu-Takayanagi surface
at the Page time. After this point, the diary will be in the entanglement wedge of the Hawking
reservoir Hrad (assuming it was thrown into the black hole at least one scrambling time into the
past).
Unfortunately, our lack of knowledge about the state of the black hole prevents us from
taking advantage of this fact. Instead, we can only successfully decode the state of the diary
once a state-independent reconstruction becomes possible that works for the entire code space
of possible initial states.
Such a reconstruction will only be possible once the diary is contained in the entanglement
wedge of the reservoir Hrad even for highly mixed states in the code space of initial microstates
(or equivalently code space states that are highly entangled with a reference system HR). These
states will have a large bulk entropy in the interior, which will increase the generalised entropy
of the non-empty quantum extremal surface for Hrad, as shown in Figure 12. Note that, because
we are no longer considering bipartite pure states, the Ryu-Takayanagi surfaces for Hrad and
HCFT will no longer necessarily be the same. The Ryu-Takayanagi surface of Hrad will therefore

40


Figure 12: After the Page time, the entanglement wedge of the reservoir contains most of the
interior for any pure initial black hole microstate. However, if we only know that the initial
state was in some large class of possible microstates, we cannot take advantage of this fact to
do a Hayden-Preskill recovery, unless the interior is also contained in the entanglement wedge
for states in the code space of possible microstates that are highly entangled with a reference
system. This entanglement entropy increases the generalised entropy of the non-empty quantum
extremal surface (dotted lines) and can make the Ryu-Takayanagi surface χrad become empty
(solid line), preventing us for doing a Hayden-Preskill reconstruction until the black hole has
evaporated further.

remain empty for highly mixed states until

Srad − SBH > Scode.
(85)

Immediately after the Page time, we need to know the exact initial state of the black hole in
order to reconstruct the diary. However, as the black hole continues to evaporate, the amount
of state-dependence required in the reconstruction decreases; a single reconstruction can work
for an increasingly large class of microstates. Happily, (85) agrees exactly with the amount of
state-dependence required for the Hayden-Preskill decoding criterion in simple random unitary
toy models [38].
This state dependence does not just make the entanglement wedge version of Hayden-Preskill
compatible with toy models. It also provides the mechanism by which information about the
initial state of the black hole is able to escape out into the Hawking radiation. The Hawking
radiation that escapes the boundary is always entangled with interior modes in the same way,
regardless of the initial state of the black hole. However, the way that the interior modes are
encoded in Hrad depends on the initial state of the black hole. The Hawking radiation will be
purified by a different subsystem of Hrad, depending on the initial state of the black hole.55 As

55We emphasize that, for any single state, the subsystem that purifies some Hawking quanta is not uniquely

41


a result, the reservoir Hrad, plus the additional Hawking radiation, contains more information
about the initial state of the black hole than the reservoir alone.
The bulk evaporation is
consistent with information escaping, even though the state of the Hawking radiation and the
interior mode does not care about the initial microstate of the black hole.
Interestingly, even if the initial microstate is completely unknown, and so the code space
entropy Scode is equal to the initial Bekenstein-Hawking entropy of the black hole, (85) will be
satisfied long before the black hole has completely evaporated, because of the thermodynamic
entropy increase from the evaporation. The information in the diary, as well as all the information
about the initial state of the black hole, will be revealed, in a completely state-independent way,
even while the black hole is still an O(1) fraction of its original size. This is closely related to
the fact that, even for a completely thermal initial black hole state, the von Neumann entropy
of the reservoir will peak and begin decreasing, even while the black hole is still an O(1) fraction
of its initial size, as discussed by Page in [7]. For black holes in our universe, both events occur
when the black hole has approximately 90% evaporated [7].
The same effect happens with reconstructions of the interior on the boundary CFT, before
the Page time. In this case, a large amount of bulk entropy in the interior will increase the
generalised entropy of the empty surface for HCFT, and so can make the Ryu-Takayanagi surface
for HCFT become non-empty. As shown in Figure 13, the Ryu-Takayanagi surface of HCFT will
only be empty for highly mixed/entangled states if

Scode < SBH − Srad.
(86)

The interior can therefore only be reconstructed on HCFT in a state-independent way for code
spaces with entropy Scode satisfying (86). Initially, we don’t need to know very much about the
state of the black hole to reconstruct the entire interior on the boundary CFT. However, as the
black hole evaporates, the reconstruction becomes more and more state-dependent. Eventually,
just before the Page time, one needs to know the exact initial state of the black hole. Again,
(86) agrees with toy models.
The part of the interior that lies between the non-empty quantum extremal surface and the
boundary is always encoded in the CFT in a completely state independent way, both before
and after the Page time. No amount of bulk entanglement with a reference system can stop the
entanglement wedge of the CFT from including this region.
The non-empty extremal surface lies at a radius O(GN) inside the event horizon of the black
hole. An outgoing lightcone starting from this extremal surface will therefore hit the singularity
after an infalling time equal to the scrambling time plus an O(β) correction after the infalling
time of the extremal surface. This infalling time is within O(β) of the ‘current’ time, when the
last radiation was extracted into Hrad. The worldline of an observer, who jumps into the black
hole at an O(β) time into the future, will remain entirely within the entanglement wedge of the
CFT. The entire experience of the observer in the interior, until the curvature becomes large
close to the singularity, will be encoded in the CFT in a completely state-independent way.
This is a somewhat comforting fact. It means that the accessible part of the black hole
interior is always encoded in the state of the black hole itself, plus O(1) number of recent quanta
of outgoing Hawking radiation.56 The initial state of the black hole does not matter, nor do

defined because the decoding channel used to reconstruct the interior mode on Hrad is not unique. The point
here is that there is no subsystem that purifies the Hawking quanta for all the possible initial states.
56This is somewhat similar to recent work by Yoshida [64], where it was shown, in a qubit toy model of a black
hole, that swapping an O(1) number of degrees of freedom and then applying a scrambling unitary was sufficient
to make new Hawking radiation be unentangled with the early Hawking radiation, even long after the Page time.
Yoshida therefore argued that swapping in the new degrees of freedom had made the interior be encoded in the
black hole degrees of freedom, plus the purification of these new degrees of freedom, with no dependence on the

42


Figure 13: Before the Page time, the interior is in the entanglement wedge of the CFT for any
pure state. However, if the initial state is allowed to be highly entangled with a reference system,
the Ryu-Takayanagi surface χCFT for the CFT may jump to the non-empty quantum extremal
surface. (In contrast, the RT surface χrad of the reservoir will remain empty.) This means that
a reconstruction of the interior that acts only on the CFT, and not on the reservoir, will have
to be at least somewhat state dependent, if the code space is too large.

manipulations, even arbitrarily complicated ones, of Hawking radiation that escaped from the
black hole a long time in the past.
In particular, so long as rs ≪ lAdS, an observer who jumps into the black hole from the
boundary will never leave the entanglement wedge of the CFT at the time that they left the
boundary. This seems to still be true, even for large AdS black holes, at least when the Hawking
radiation is extracted from inside the zone, as in Section 2.2. Potentially, this is important for
precomputation versions of the firewall paradox [65], where an observer attempts to extract a
mode from Hrad, which is expected to take exponential time [17], before jumping into the black
hole.
However, as discussed in Appendix B, by making the infalling modes be at a temperature
very close to the black hole temperature, we can make the extremal surface be arbitrarily close to
the event horizon. By tuning the state of the infalling modes, we can therefore always ensure that
the observer is able to escape the entanglement wedge of the CFT and encounter the infalling
mode that they extracted into Hrad. The fundamental answer to the precomputation version
of the firewall paradox seems to simply be that we can indeed manipulate the interior using
Hrad, so long as we are able to do very complicated, non-semiclassical manipulations. After all,
it is well known that it is possible to manipulate the interior of a two-sided black hole in the
thermofield double state, just by acting on the left CFT.

initial black hole state. In our case, it is simply the continuous increase with time of the combined thermodynamic
entropy of the black hole and Hawking radiation that makes part of the interior be encoded in the CFT. See
Appendix B for an example of the extremal surface that lies exactly on the event horizon because there is no net
increase in thermodynamic entropy.

43


3.3
Approximation to the Rescue

Just like the α-bit codes found in [38] and summarised in Section 3.1, the results that we found
in Section 3.2 only make sense because entanglement wedge reconstruction is approximate. This
fact should be somewhat apparent from our discussion in Section 3.1 and 3.2. However, in the
interests of clarity, we now give a simple, explicit example of a paradox that would otherwise
occur.
Suppose that we allow a black hole to evaporate until slightly before the Page time, storing
the Hawking radiation in a reservoir H1. We then allow it to continue to evaporate until slightly
after the Page time, storing the Hawking radiation in a different reservoir H2. Let the entropy
of the Hawking radiation in H1 be (1 − δ)SBH and the entropy of the Hawking radiation in H2
be 2δSBH. where δ > 0 is small and SBH is the Bekenstein-Hawking entropy of the black hole
after all the evaporation has taken place. For reasons that will become clear, we shall refer to
the combined state of the evaporating black hole and the Hawking radiation as the ‘entangled
state’.
The black hole has evaporated beyond the Page time. The Ryu-Takayanagi surface of Hrad
is therefore non-empty and lies just inside the horizon of the black hole. Most of the interior of
the black hole is encoded in H1 ⊗ H2.
Now suppose that we do a complete measurement of H2 in some arbitrary basis. Regardless
of the outcome of such a measurement, and regardless of the basis that we measure in, the
Ryu-Takayanagi surface for the CFT will now be empty and so the interior will be encoded in
HCFT. We shall refer to the states that result from such a measurement as ‘pure states’, in
contrast with the original ‘entangled state’, because they are pure states in HCFT ⊗ H1.
We now have exactly the same apparent paradox that was found in [38] and discussed in
Section 3.1. For any pure state, the interior is encoded in the CFT. However, in the entangled
state, it is encoded in H1 ⊗ H2 and so, by the no-cloning theorem, it cannot be encoded in the
CFT. The boundary Hilbert space HCFT plays the role of the boundary subregion Hilbert space
HB that we had access to in Section 3.1; the early Hawking radiation reservoir H1 plays the role
of the complementary boundary subregion Hilbert space H ¯B and the later Hawking radiation
reservoir H2 plays the role of the reference system HR.
As before, the resolution of this paradox is two-fold. Firstly, the reconstruction of the interior
on HCFT necessarily depends on the state of the interior modes that were previously entangled
with H2. Hence, there is no single reconstruction that will work for all possible states of those
interior modes. This would be necessary to reconstruct the interior of the entangled state in
HCFT.
However, this resolution only works because the reconstruction is approximate. Since the
reduced density matrix of the entangled state on H2 consists of a large number of approximately
independently and identically distributed thermal modes, the smooth max entropy of H2 in the
entangled state is approximately equal to its von Neumann entropy 2δSBH, up to subleading
corrections of order O(
√

S) [66]. By the definition of the smooth max entropy, this means that
we can construct a code space Hcode ⊆ HCFT ⊗ H1 satisfying

log |Hcode| = 2δSBH + O(
�

SBH),
(87)

such that the entangled state can be approximated, with very high fidelity, by a state in Hcode ⊗
H2.
Any interior operator O should have a state-independent global reconstruction Ocode on
HCFT ⊗ H1 that works for the entire code space Hcode. If entanglement wedge reconstruction
were exact, then, for any state |ψ⟩, the state-dependent reconstruction Oψ
CFT that acts only on

44


the CFT would satisfy

Oψ
CFT |ψ⟩ = Ocode |ψ⟩ .
(88)

However, as shown in [62,67], this would imply that there must also exist a state-independent
reconstruction Ocode
CFT that works for the entire code space and acts only on the CFT. The cloning
paradox can only be resolved if there is an error term in (88) with size at least exp(−O(δ SBH))
[62].
This error is tiny; it is non-perturbatively small in GN for any fixed δ > 0. It is expected
that non-perturbative exp(−O(S)) corrections to the bulk physics of black holes in AdS/CFT
must exist in order for the decay of correlators to be consistent with boundary unitarity [68].
However, there has been debate about whether such tiny errors can explain the large-scale “O(1)”
paradoxes that show up in evaporating black holes. The answer is that they can and do. If the
code space of allowed microstates has exponentially large dimension, exponentially small errors
can be amplified in very entangled states and become O(1) in size.
Once we have measured H2, regardless of the measurement outcome we obtain, a diary in
the interior of the black hole will be encoded in HCFT and only has a non-perturbatively small
effect on the state of H1. However, the entanglement with H2 amplifies these tiny differences,
so that orthogonal diary states are almost exactly orthogonal on H1 ⊗H2. No magic is required;
just the same mechanism that occurs in random unitary toy models [62].

3.4
Large Diaries

So far we have assumed that any diaries thrown into the black hole are small, both in energy
and entropy. We have therefore been able to ignore both their backreaction on the geometry
and their contribution to the bulk entropy. In this section, we remove those assumptions.
If a heavy diary is thrown into a black hole before the Page time, it can still be reconstructed
immediately after the Page time (so long as the entropy of the diary is small). The only change
is that the Page time will be delayed by the increase in the horizon area of the black hole caused
by the diary. By almost identical arguments to those in Section 3.2, if the diary also has a large
entropy Sdiary, we have to wait until

Srad − SBH ≥ Sdiary,
(89)

so that the diary is contained in the entanglement wedge of Hrad even for highly mixed diary
states.
A more interesting situation occurs when a large diary is thrown into the black hole after the
Page time. Let us first consider the case where the entropy of the diary is small, but the energy
is large. The diary now causes a large backreaction on the geometry that significantly increases
the horizon area. We assume, for simplicity, that the diary is encoded only in s wave modes
and so the rotational symmetry of the spacetime is preserved. After we throw the diary into the
black hole, the Bekenstein-Hawking entropy of the black hole will again be significantly larger
than its entanglement entropy; heuristically, we will have made the black hole young again.
What happens to the quantum extremal surface in such a spacetime? Before the diary is
thrown into the black hole, the quantum extremal surface will continuously move forwards in
infalling time, at a radius just inside the event horizon, as radiation escapes the black hole into
Hrad. However, radiation that would have escaped shortly after the diary was thrown into the
black hole will instead fall into the larger black hole created by the backreaction of the diary.
The actual radiation that escapes instead comes from close to the new event horizon, which,
being teleological, already began moving out from the apparent horizon rs along an outgoing
lightcone in anticipation of the diary falling in. Even at very late times, the Hawking radiation

45


comes from outside the past lightcone of the boundary, which in turn will always lie outside the
event horizon, although it will approach the horizon exponentially at any fixed time as we evolve
the boundary forwards in time.
Using (36) and (39), it is therefore easy to see that the quantum extremal surface stops
tracking along the horizon after the diary is thrown into the black hole, and instead asymptotes
to a radius

r = rs −
GNcevap

3(d − 1)Ωd−1rd−2
s
,
(90)

at the infalling time v when

rhor(v) = rs +
GNcevap

3(d − 1)Ωd−1rd−2
s
.
(91)

For simplicity, we are assuming here, as in Sections 2.2 and 2.3, that the Hawking radiation is
extracted from close to the horizon and so there are no greybody factors.
If the change in horizon area δAdiary caused by the diary is small compared to the original
horizon area Ahor, we find that

v = vdiary − β

2π log δAdiary

cevapGN
+ O(β),
(92)

where vdiary is the infalling time at which the diary is thrown into the black hole. In deriving (92),
we have used the fact that the event horizon is an outgoing lightcone, obeying (15), and that, at
vdiary, the radius of the event horizon should be approximately equal to the new Schwarzschild
radius of the black hole.
The entanglement wedge of Hrad will not contain the diary. The large amount of energy
thrown into the black hole has stopped any information from escaping. The location of the
extremal surface and the entanglement wedges in shown in Figure 14.
However, this quantum extremal surface will not remain the Ryu-Takayanagi surface forever.
As new radiation escapes from the black hole into Hrad, the bulk entropy, calculated using this
quantum extremal surface, will increase. Heuristically, the new Hawking radiation is entangled
with interior modes that lie close to the new, larger black hole horizon and hence lie in the entan-
glement wedge of HCFT. More formally, as in (47), the cut-off ε2 in (24) shrinks exponentially
as

ε2 ∝ exp(−2πvrad/β)
(93)

where vrad is the ‘current’ infalling time (i.e. the point in time where we last extracted Hawking
radiation into Hrad). In contrast, the radial distance

rl.c. − r ≈ rhor − r,
(94)

between the past lightcone and the RT surface is approximately constant. As a result, we find
using (24) that

∂Sbulk
∂vrad
= cevapπ

6β
.
(95)

This is the increase in entropy that one finds with thermal outgoing modes that are purified by
degrees of freedom in HCFT [57, 58]. By adding energy, we have stopped information escaping
the black hole and made the Hawking radiation be purely entangled with the CFT. The black
hole did indeed become young again.

46


Figure 14: When a large diary is thrown into a black hole, the radius rhor of the event horizon
(solid line) begins increasing in anticipation of the diary falling in, while the radius rs of the
apparent horizon (dotted line) continues to slowly decrease until the diary is actually thrown into
the black hole. As the black hole continues to evaporate, the past lightcone (dashed line) of the
current boundary remains outside the event horizon. The Ryu-Takayanagi surface χ1 asymptotes
to a point approximately the scrambling time before the diary was thrown in. There is a second
quantum extremal surface χ2 at an infalling time after the diary is thrown in. Initially, this
extremal surface is not the RT surface because its area is larger than the area of χ1. However,
eventually, at the new Page time, there will be a phase transition, with χ2 becoming the new
RT surface, and the diary can finally be reconstructed from the Hawking radiation, so long as
its entropy is small. (Left: Eddington-Finkelstein coordinates, right: a Penrose diagram.)

The surface χ1 at (90), (92) remains a quantum extremal surface even at boundary times long
after the diary was thrown into the black hole. However, it is not the only non-empty quantum
extremal surface at such late times. There will also be a second non-empty quantum extremal
surface χ2 that lies, as usual, approximately the scrambling time before the current boundary
time. Because of the increase in horizon area created by the diary, the generalised entropy of
the extremal surface χ2 will initially be significantly larger than the generalised entropy of the
surface χ1.
However the generalised entropy of χ1 steadily increases over time because of the increase in
bulk entropy discussed above. Meanwhile, the generalised entropy of χ2 decreases over time as
the black hole evaporates. Eventually, after the new Page time of the black hole, the generalised
entropy of χ2 will become smaller than the generalised entropy of χ1. The later extremal surface
χ2 will be the Ryu-Takayanagi surface, and the diary will finally be in the entanglement wedge
of Hrad and can be decoded.
If the diary had a large amount of entropy, as well as a large amount of energy, we would
have to wait even longer in order to recover the diary. Even after the new Page time, when the
Ryu-Takayanagi surface has a phase transition for pure diary states, the Ryu-Takayanagi surface
of Hrad will still not contain the diary for highly mixed diary states. To actually recover the diary
from the Hawking radiation, we need to wait until the diary is contained in the entanglement
wedge of Hrad, even for such highly mixed states, as shown in Figure 15. This requires

Snew
rad − δAdiary + δAevap

4GN
> log |Hdiary|
(96)

where Snew
rad is the bulk entropy of the new radiation emitted after the diary was thrown into

47


Figure 15: If a heavy diary, left, is thrown into the black hole, it can only be reconstructed
from Hrad once the generalised entropy (solid lines) of the surface χ1 for Hrad is greater than
the generalised entropy (dotted lines) of the surface χ2.
The generalised entropy of χ1 has
contributions from both the area term and the entropy of Hawking radiation emitted after the
diary was thrown into the black hole. In contrast, so long as the diary has small entropy, the
surface χ2 only has a contribution from its area, plus an O(1) bulk entropy correction. However,
if the diary also has a large entropy (right), the diary also needs to be in the entanglement
wedge, even when it is in a highly mixed state, or is entangled with a reference system. This
increases the generalised entropy of χ2.

the black hole, δAdiary > 0 is the change in horizon area from throwing the diary into the black
hole, δAevap < 0 is the change in horizon area from the black hole evaporation after the diary is
thrown into the black hole and Hdiary is the Hilbert space of the diary.
Note that the generalised second law implies that

δAdiary/4GN ≥ log |Hdiary|
(97)

and

Snew
rad ≥ −δAevap.
(98)

Hence, whenever (96) is satisfied, we will also have

Snew
rad > log |Hdiary|.
(99)

It follows that there is always sufficient entropy in the new Hawking radiation to encode the
diary. Entanglement wedge reconstruction is consistent with quantum capacity bounds, so long
as we consider mixed (or entangled) states in the code space.
It can be verified (96) is consistent with random unitary toy models [38], although, as dis-
cussed in Section 2.3, most of the focus in random unitary models has been on evaporation
that is either perfectly or approximately thermodynamically reversible, where (97) and (98) are
equalities.
As in Section 3.2, the period when the interior reconstruction depends on the state of the
diary does not merely make the amount of information encoded in the Hawking radiation be

48


compatible with quantum capacity bounds; it also provides the mechanism by which information
about the state of the diary escapes the black hole. The new Hawking radiation is entangled
with the same interior modes, regardless of the state of the diary. However, because the encoding
of those interior modes in Hrad depends on the state of the diary, an observer with access to the
reservoir Hrad can learn information about the diary from the new Hawking radiation.
So far, both in this section and in Section 2.3, we have not worried too much about the
errors that exist in entanglement wedge reconstruction, even though we showed in Section 3.3
that their existence was crucial to the consistency of our results. Reconstruction errors are also
present in the Hayden-Preskill protocol in random unitary toy models of black holes. Since all
our other results have been consistent with random unitary toy models, we might hope that
the error in the Hayden-Preskill entanglement wedge reconstruction will also be consistent with
random unitary toy models.
Unfortunately, the actual size of the errors in entanglement wedge reconstruction remains
unknown. However, the lower bound on their size that was derived in [38] and discussed briefly
in Section 3.1 suggests that, for reconstruction with error ε to be possible, the difference ∆S
between the generalised entropy of an extremal surface for which the bulk operator would not
be in the entanglement wedge and the generalised entropy of the Ryu-Takayanagi surface, where
the bulk operator is in the entanglement wedge, must satisfy

∆S ≥ O(log 1

ε).
(100)

Note that (100) needs to be satisfied for all states, both pure and mixed, in the code space. The
exact coefficient in (100) depends on how the error is measured, and we will not worry about it
here.
The generalised entropy of the extremal surface χ1, where the diary is not in the entanglement
wedge of Hrad, increases by an O(1) amount in O(β) time. Similarly, the generalised entropy of
the extremal surface χ2, where the diary is in the entanglement wedge of Hrad, decreases by an
O(1) amount in O(β) time.
If we make the strong assumption that the lower bound on the error ε derived in [38] is
approximately saturated, we find that to reconstruct the diary with error ε, we need to wait for
an additional time
O(β log(1

ε),

even after the condition (96) is satisfied. Up to the (unstated) linear coefficient, this agrees, yet
again, with random unitary toy models [8,62].

3.5
Minimal State Dependence

So far, we have avoided talking about state dependence for interior operators in black holes that
have not evaporated at all, where there is no auxiliary reservoir Hrad. Yet this is the situation
in which state dependence is most commonly discussed [32,33,37].
There is a very good reason for our reticence. Every proof of entanglement wedge reconstruc-
tion assumes that there is a global isometry from the bulk code space to the larger boundary
Hilbert space. It is this isometry, combined with a partial trace over some of the boundary
degrees of freedom, that creates a noisy quantum channel and, potentially, a quantum error cor-
recting code. Yet, so long as such an isometry exists, there must always exist state-independent
global boundary reconstructions.
If the CFT is the only Hilbert space and pure bulk states correspond to pure boundary
states, interior operators cannot be encoded in the CFT in a state-dependent way, within the
framework of quantum error correction.

49


Nonetheless, suppose we take as an axiom the idea that boundary reconstructions are state
independent if, and only if, the bulk operator is contained in entanglement wedge even for states
in a ‘code space’ of the bulk effective field theory that are entangled with a reference system.57

If a code space of interior states Hcode satisfies

log |Hcode| < SBH,
(101)

the entanglement wedge of the boundary should always include the interior, even if the state
is entangled with a reference system.58. A single reconstruction of interior operators should
therefore exist that works for the entire code space.
In contrast, if

log |Hcode| ≥ SBH,
(102)

then we can make states in Hcode ⊗ HR where the RT surface is non-empty, and part of the
interior is no longer contained in the entanglement wedge of the boundary.
To distinguish this idea from most of the literature on interior state dependence, which has
focussed on constructing interior operators for a single microstate, or at most a small code space
of microstates with O(1) dimension, we shall refer to it as ‘minimal state dependence’. We
emphasize, however, that this paper is far from the first to suggest it, see, for example, the
discussion near the end of [32].
As discussed, given that we are taking results derived using quantum error correction and
applying them outside of that framework, we should be somewhat cautious about this idea. In
particular, rather than making too many claims about code spaces for which (102) is true, it
seems better to focus on the fact that, so long as a code space satisfies (101), we should be
relatively confident that everything is well behaved and that all bulk operators have global,
state-independent boundary reconstructions that work for the entire code space.
An important question is how small

∆S = SBH − log |Hcode|
(103)

can be without causing any problems. However, since we don’t have the tools to answer such a
question with any confidence, we shall be maximally cautious and assume that ∆S needs to be
non-zero at leading order. In other words, we shall require

log |Hcode| ≤ (1 − δ)SBH,
(104)

for some δ > 0, which should be fixed in the semiclassical limit.
We can now show how minimal state dependence neatly avoids the so-called AMPSS argu-
ment [65] that generic black hole microstates must have firewalls.
The AMPSS argument goes as follows. Consider the subspace of CFT states within some
narrow, but O(1) width, energy band M ≤ E ≤ M + δM. If M is sufficiently large and δM is
sufficiently small then all such states have a bulk description as a black hole. We then assume
that there exists some state-independent operator b†
ω that acts as a raising operator for an
interior Hawking mode, as well as an inverse operator (1 + b†
ωbω)−1 bω.
Acting with the operator b†
ω decreases the Schwarzschild energy by ω and so maps our
subspace of CFT states into the energy band M − ω ≤ E ≤ M + δM − ω. But the number of

57We emphasize that this may no longer be a code space of a quantum error correcting code in the traditional
sense.
58The dimension of the code space here includes both allowed interior degrees of freedom, as well as any degrees
of freedom describing an end-of-the-world brane that are also included in the code space, if our geometry ends in
such a brane. See, for example, [37,69] for discussion of interior geometries ending on end-of-the-world branes.

50


states in this energy band is smaller than the number in our original energy band by a factor
of approximately e−βω. So the map cannot be invertible, contradicting our original assumption.
The authors of AMPSS conclude that b†
ω, and hence the interior, cannot exist.
It is, of course, well known that the argument above breaks down if the operator b†
ω is state
dependent. Our point here is simply to emphasize that this is still true even if the operator b†
ω
is only minimally state dependent.
Up to logarithmic corrections, the entropy of the maximally mixed state in the energy band
M ≤ E ≤ M + δM is equal to the Bekenstein-Hawking entropy. Assuming minimal state-
dependence, we won’t be able to find a single state-independent operator b†
ω. Our code space is
too large. If instead we restrict to a random code subspace, within the energy band and with
entropy at most (1 − δ)SBH, there will be plenty of space for the image of the code subspace in
the energy band M − ω ≤ E ≤ M + δM − ω.
Of course, instead of looking at a random subspace of states within the energy window, we
can alternatively make δM so small that the entire code space of states in the energy band has
a single state-independent reconstruction. To do so, it would be necessary to have

δM = O(e−δSBH).
(105)

If we insisted that the energy band M ≤ E ≤ M + δM be mapped invertibly into the energy
band M − ω ≤ E ≤ M + δM − ω, we would still have a paradox, because the latter band is still
strictly smaller than the former.
This would be far too strong a requirement however. More reasonably, we should only expect
the energy of a state to decrease by ω plus non-perturbatively small corrections.
However,
since δM is itself non-perturbatively small, this non-perturbatively small uncertainty in ω can
significantly increase the size of the energy window. We can therefore avoid the paradox.

4
Discussion

4.1
Summary of Results

In this paper, we have argued that the key expected features of unitary black hole evaporation
in AdS/CFT can be derived from the bulk semiclassical description of an evaporating black hole,
so long as we assume entanglement wedge reconstruction. We now review those arguments.

• We studied entanglement wedge reconstruction in an evaporating black hole, formed from
collapse, where the Hawking radiation was extracted out of the AdS space containing the
black hole, with boundary Hilbert space HCFT, and into an auxiliary Markovian reservoir
Hrad. Importantly, we assumed that the bulk description of the evaporation was semiclas-
sical, and so the bulk entanglement between the Hawking radiation and the interior of the
black hole continued to grow, even after the Page time.

• Using the maximin prescription, we saw that the quantum Ryu-Takayanagi surface, asso-
ciated to the entire boundary Hilbert space HCFT, must become non-empty at the Page
time. Since the overall state |ψ⟩ ∈ HCFT ⊗ Hrad is pure, this non-empty RT surface will
also be the RT surface for Hrad.

• If the Hawking radiation is extracted into Hrad from deep inside the zone, near the horizon,
the greybody factors will all be either zero or one, depending on whether a given angular
momentum mode is extracted or not.
The location of the non-empty quantum Ryu-
Takayanagi surface can then be calculated explicitly. It lies at a radius

r = rs −
cevapGN

3(d − 1)rd−2
s
Ωd−1
= rhor −
cevapGN

6(d − 1)rd−2
s
Ωd−1
,
(106)

51


where rs is the radius of the classical apparent horizon of the black hole, rhor is the radius
of the event horizon of the black hole and cevap is the number of modes that are extracted
into the reservoir Hrad. The infalling time of the quantum extremal surface is given by

v = − β

2π log SBH

cevap
+ O(β),
(107)

where v = 0 is the current boundary time. A large part of the black hole interior lies in the
entanglement wedge of the Hawking radiation reservoir Hrad, rather than the boundary
Hilbert space HCFT.

• A small diary thrown into the black holes early in the evaporation can therefore be re-
constructed from the Hawking radiation immediately after the Page time. A small diary
thrown into the black hole after the Page time can be reconstructed after waiting for the
scrambling time. These two results constitute the Hayden-Preskill decoding criterion [8].

• If the number of angular momentum modes cevap extracted into the Hawking radiation is
large, there is a small, logarithmic decrease in the delay before the diary can be decoded
from the radiation. This decrease is consistent with a heuristic picture of fast scrambling
where a perturbation spreads exponentially through the degrees of freedom.

• More generally, we showed that, in any evaporating black hole after the Page time, the
RT surface lies at an infalling time

v = − β

2π log
1

GN
+ O(β),
(108)

where the subleading corrections depend on the details of the evaporation and, in general,
cannot be analytically calculated, because of greybody factors. We can therefore derive
the Hayden-Preskill decoding criterion, up to unknown, but subleading, corrections, even
when non-trivial greybody factors are present.

• Given the location of the Ryu-Takayanagi surface, it is an immediate consequence of the
Ryu-Takayanagi formula that the entanglement between the CFT and the reservoir is
given by the Page curve. Moreover, entanglement wedge reconstruction explains how the
entanglement entropy ends up decreasing (and how the AMPS firewall paradox is avoided).
It decreases because the outgoing radiation is entangled with interior modes that are in
the entanglement wedge of, and so are encoded in, Hrad.

• If we consider the change in bulk entanglement entropy from transferring a small amount
of Hawking radiation from HCFT to Hrad, we find exact quantitative agreement with the
change in entropy given by the Ryu-Takayanagi formula, i.e. the Page curve.

• This quantitative agreement does not only exist in the simple cases where we can calculate
the Ryu-Takayanagi surface explicitly. It is a general consequence of the fact that the
quantum Ryu-Takayanagi surface is an extremum of the generalised entropy. As with the
Hayden-Preskill decoding criterion, we are therefore able to derive the Page curve, and
avoid the firewall paradox, even when there are greybody factors.

• As argued in [38], based on results about approximate operator algebra quantum error
correcting codes derived in [39–41], state-independent entanglement wedge reconstruction
is only possible for a given code space if the bulk operator is contained in the entanglement
wedge of the boundary region for all states, both pure and mixed that have support only
within the code space.

52


• State-dependent entanglement wedge reconstruction is possible so long as the bulk operator
is contained in the entanglement wedge of the boundary region for all pure states in the
code space.

• Using these results, we were able to derive the correct state dependence for Hayden-Preskill
reconstructions. Immediately after the Page time, the diary can only be reconstructed if
the exact initial black hole microstate is known. As the black hole continues to evaporate,
less state dependence is required, in exact agreement with toy models. Specifically, a single
reconstruction will work for a large code space of possible initial black hole states, with
entropy Scode, so long as

Scode < Srad − SBH,
(109)

where Srad is the bulk entanglement entropy between the Hawking radiation and the
interior and SBH is the Bekenstein-Hawking entropy of the black hole.

• Similarly, before the Page time, reconstructions of the interior on the boundary CFT
become increasingly state dependent as the black hole evaporates. The entropy Scode of
the code space of allowed initial states must satisfy

Scode < SBH − Srad.
(110)

Eventually, at the Page time, the reconstruction is only possible if the exact initial black
hole state is known.

• The state dependence in the encoding of interior partners of Hawking modes in the early
radiation (after the Page time) explains how the final (microscopic) state of the combined
early and late radiation can depend on the initial state of the black hole (i.e. how we can
avoid information loss) despite the state of the late Hawking mode and its interior partner
having no dependence on the initial state.

• These results are only consistent because entanglement wedge reconstruction is only ap-
proximate. Tiny, non-perturbatively small errors build up and have O(1) effects.

• When a heavy diary is thrown into the black hole, the Ryu-Takayanagi surface stops track-
ing along the horizon and instead asymptotes to a location approximately one scrambling
time before the diary was thrown in. The Hawking radiation will contain no further in-
formation until the new Page time is reached, at which point the Ryu-Takayanagi surface
will jump forwards in time and the diary can be decoded from the Hawking radiation.

• If the entropy of the diary is large, as well as its energy, we have to wait even longer
before it can be decoded. Specifically, we have to wait until the generalised entropy of
the earlier quantum extremal surface, where the diary is not in the entanglement wedge of
Hrad, is greater than the generalised entropy of the later quantum extremal surface plus
the entropy of the diary code space.

• All our results about Hayden-Preskill reconstructions of large diaries are consistent with
random unitary toy models [38].

• If we assume that the lower bound, derived in [38], on errors in entanglement wedge
reconstruction is saturated up to a linear coefficient, we find that the errors in Hayden-
Preskill reconstructions are consistent with toy models.

53


• Finally, our arguments suggest that, even when a black hole has not evaporated at all, its
interior can only be reconstructed with ‘minimal’ state dependence. Such state dependence
is beyond the framework of quantum error correction, but it provides a natural resolution
to the AMPSS typical state firewall paradox.

In appendices,

• We give a simple pedagogical example of the importance of the coordinate dependence of
cut-offs in entanglement entropy calculations.

• We calculate the location of the Ryu-Takayanagi surface explicitly when the infalling modes
are replaced by thermal modes or by pure modes of constant energy density. In each case
we find that information stops escaping in the Hawking radiation (from the perspective of
an observer with access either only to the reservoir, or to the reservoir and a purification of
the thermal infalling modes) exactly at the moment that no information escaping becomes
consistent with boundary unitarity.

• We show that the Kourkoulou-Maldacena state-dependent interior reconstruction in the
SYK model can be trivially extended to give minimally state-dependent reconstructions.

4.2
Entanglement Wedge Reconstruction in Toy Models

Throughout this paper, we have found that bulk calculations, using entanglement wedge re-
construction and the Ryu-Takayanagi formula, agree perfectly with random unitary and fast
scrambling toy models of the boundary dynamics.
Not only do our results agree with the Page curve and the Hayden-Preskill decoding criterion,
but we also found exactly the right state dependence, for both reconstructions acting on the
CFT before the Page time and reconstructions acting on the reservoir after the Page time. Our
results for large diaries were consistent with toy models as a function of the both the energy and
the entropy of the diary, as were the reconstruction errors so long as we assumed that the lower
bound from [38] was approximately saturated.
This seems either to be a somewhat remarkable coincidence, or to involve some deep magic of
quantum gravity. In fact, it is neither of these things. We simply have the direction of causation
in reverse. Rather than random unitary models determining the consequences of entanglement
wedge reconstruction, entanglement wedge reconstruction determines the behaviour of random
unitary toy models.
A random unitary toy model of black hole evaporation is an exceptionally trivial example
of a random tensor network [70]. An iterated random isometry model like that in Figure 9 is
another, more complicated, example.
However, random tensor networks are well known to obey the Ryu-Takayanagi formula and
hence have entanglement wedge reconstruction [70]. It is therefore inevitable that random uni-
tary models agree with results derived from Ryu-Takayanagi and entanglement wedge recon-
struction. Indeed, one of the most popular methods to prove results about error correction in
random unitaries, the so-called decoupling approach [71,72], essentially involves deriving entan-
glement wedge reconstruction from the Ryu-Takayanagi formula.
Of course, random tensor networks do not have all the properties of holographic spacetimes.
In particular, they are not covariant. A tensor network corresponds to a single timeslice of a
bulk spacetime; RT surfaces, in a tensor network, are minimal, not extremal, surfaces.
For many of the calculations in this paper, for example the scrambling time delay in the
Hayden-Preskill criterion, the extremality of the surface, or equivalently the maximisation over

54


Figure 16: After the black hole has completely evaporated, the bulk encoded in the CFT will
be in the vacuum state. However, we can choose a (disconnected) bulk Cauchy slice where the
radiation in the reservoir Hrad is still be entangled with the pinched-off interior wormhole, which
lies in its entanglement wedge. The Ryu-Takayanagi surface is empty and has zero generalised
entropy; the two systems HCFT and Hrad are therefore unentangled.

Cauchy slices in the maximin prescription, was crucial in deriving the correct results. In partic-
ular, the covariant Ryu-Takayanagi surface somehow knows about the fast scrambling dynamics
of the boundary theory. If there is any magic going on, it seems to be here.

4.3
The Post Evaporation State and the Bulk-to-Boundary Map

While we have studied evaporating black holes both before and after the Page time in this paper,
we have not discussed the final state of the system, after the evaporation is complete. We make
some comments about this state now.
When the black hole has nearly completely evaporated, the horizon curvature will become
large and so stringy and Planckian effects will become important.
We can no longer trust
semi-classical calculations.
However it is reasonable to expect that, after an indeterminate, but short, time, there will
no longer be any sort of smooth connected geometry between the wormhole and the AdS space
that previously contained the black hole. The mouth of the wormhole will have closed; the black
hole will have completely evaporated. Let us assume we have extracted all the remaining energy
out of the AdS space and into the reservoir Hrad, so that the original bulk AdS space lies in the
vacuum state. This is shown schematically in Figure 16.
Because the closed wormhole geometry has no boundary, on its own, the Ryu-Takayanagi
surface is no longer sufficient to define an entanglement wedge, and hence a generalised entropy.
For example, if the RT surface is empty, we need to further specify whether the wormhole is
in the entanglement wedge of the reservoir or of the CFT, before the generalised entropy is
well-defined. In this case, it is obvious that the RT surface should be empty, with the wormhole
in the entanglement wedge of the reservoir, since this gives zero generalised entropy.59

59Recall that the empty surface is contained in every Cauchy slice, and so will be the RT surface if there is
any Cauchy slice for which it has minimal generalised entropy.

55


There is no entanglement between HCFT and Hrad; both states are pure. Entanglement
wedge reconstruction tells us that the state of the wormhole is encoded in the Hawking radia-
tion. Any information thrown into the black hole during the evaporation will be contained in the
entanglement wedge of Hrad, no matter how large the entropy of the initial state. The informa-
tion thrown into the black hole is therefore encoded in Hrad in a completely state-independent
way.60 All the information has been preserved.
Even though we know from the Ryu-Takayanagi formula that the state of the Hawking
radiation in Hrad is pure, it appears from a bulk perspective that it is still entangled with the
closed-off wormhole. How should we understand this seeming contradiction?
An entangled state of the Hawking radiation and wormhole can be written as

|ψ⟩ =
�

i

√pi |φi⟩ |χi⟩ ,
(111)

where the states |φi⟩ describe the radiation and the states |χi⟩ describes the wormhole. The
Hilbert space of a holographic theory is associated with its boundary. However, the wormhole
has no boundary. We therefore conclude (perhaps somewhat controversially) that its Hilbert
space is trivial; it is isomorphic to the complex numbers C. The states |χi⟩ are therefore simply
complex coefficients ci. The state

|ψ⟩ =
�

i

√pici |φi⟩
(112)

is therefore simply some complicated, pure state in Hrad.
Of course, there exists a perfectly valid bulk description of the same bulk state |ψ⟩ from
(112) where the Hawking radiation is simply in a bulk state that has a complicated entanglement
structure, but no wormhole.61 Some version of black hole complementarity, or the ER = EPR
duality [18], makes it equally valid to suppose that the wormhole still exists, or that it has
vanished leaving some complicated pure state of the Hawking radiation.
If we apply some complicated unitary operator to Hrad, we can transform the complicated
state |ψ⟩ into a simple state, say |ψ0⟩. From the perspective where the wormhole continues to
exist, we will still be in an ‘entangled state’
�

i

�

p′
i |ψi⟩ |χ′
i⟩ ,
(113)

where there are many non-zero p′
i. However, the states |χ′
i⟩ are now very complicated superpo-
sitions of the original ‘simple’ wormhole states |χi⟩. Each term in each superposition is simply a
complex number. For |χ′
0⟩ the coefficients in the superposition must add constructively, so that
|χ′
0⟩ is some large complex number c′
0. For |χ′
i⟩ with i ̸= 0, they must interfere destructively so
that |χ′
i⟩ = 0. The ‘entangled state’ really is just |ψ0⟩.
Just like in ordinary AdS/CFT, we have a linear ‘bulk to boundary’ map from the state
of bulk fields Hbulk on a fixed background spacetime (in this case the closed wormhole) to a
‘boundary’ state of the spacetime itself. However, since the ‘boundary’ Hilbert space is trivial,
up to normalisation, this map simply projects the bulk fields on the wormhole into a particular,
very complicated state.
The coefficients ci which describe the ‘boundary’ state associated to a particular wormhole
state are simply the coefficient of that state in the projected wormhole state. This story is,

60In contrast, the interior of the black hole, from before it began to evaporate, appears to be encoded in the
reservoir Hrad in exactly the same minimally state dependent way that it was originally encoded in the CFT.
61If it is not clear that the Markovian reservoir Hrad has a bulk description at all, recall that we can imagine
throwing each small chunk of Hawking radiation into its own copy of anti-de Sitter space.

56


in effect, simply the Horowitz-Maldacena final state postselection proposal [42]. From a bulk
perspective, Horowitz and Maldacena suggested that the projection happens at the singularity.
Our arguments suggest that, in the microscopic boundary description of the theory, it happens
the moment that the wormhole closes off, even if we choose a bulk Cauchy slice that includes
the wormhole.
In fact, the process that ends in this final state projection begins immediately after the Page
time, long before the black hole fully evaporates. At this point, there are, for the first time,
more ‘orthogonal’ states of the interior fields, that are needed to describe the entanglement with
the Hawking radiation, than there are microstates of the black hole according to the Bekenstein-
Hawking entropy.
As a result, even though each ‘orthogonal’ pair of bulk states will be almost orthogonal on
the boundary, there must exist very complicated superpositions of states of the interior fields
that are annihilated by the map to the boundary. Indeed, this is what allows the entanglement
entropy between the reservoir and the CFT to be much less than the bulk entanglement entropy
between the radiation and the interior, in accordance with the Ryu-Takayanagi formula.
As with the post-evaporation state, the combined state of the black hole and radiation, after
the Page time can be written as

|ψ⟩ =
�

i

√pi |φi⟩ |χi⟩ ,
(114)

where the probabilities pi are determined by the semiclassical bulk evaporation, the states |φi⟩
describe the Hawking radiation in the reservoir and the states |χi⟩ now describe black hole mi-
crostates, which are encoded as CFT states. If there was an isometry, or even an approximate
isometry, mapping each apparently distinct microstate |χi⟩ to a different CFT state, the entan-
glement entropy between the reservoir and the CFT would be equal to Srad. However, minimal
state dependence says that the bulk ‘code space’ of microstates |χi⟩ is too large for such an
isometry to exist. Since the map from bulk states to boundary states is not an isometry, there
is no inconsistency with the entanglement entropy between HCFT and Hrad actually being the
Bekenstein-Hawking entropy SBH.
It is often suggested that state dependence makes quantum mechanics nonlinear. However
the map from bulk states to boundary states is perfectly linear in this proposal; it just isn’t an
isometry. In effect, the naïve inner product on the bulk effective field theory is very different from
the pull back of the boundary inner product to bulk states, which defines the actual microscopic
inner product of the quantum theory. What then is the bulk inner product? The most natural
answer, which is consistent with recent work on JT gravity [73], is that the bulk inner product is a
statistical average of the boundary inner product over an ensemble of microscopic Hamiltonians,
such as a small range of couplings.
It is well known that final state projection models can lead to issues with describing mea-
surements done by an observer falling into the black hole [74–76]. While we leave a detailed
accounting of these issues to future work, the solution to at least some of these problems appears
to be quantum error correction.
So long as we only consider a sufficiently small code spaces of allowed states, for example the
code space of states describing the state of the observer jumping into the black hole, along with
her experimental apparatus, there is always an isometry from the bulk to the global boundary
(including the reservoir).
On the boundary, the evolution is described by standard quantum mechanics with no posts-
election. In the bulk, the observer is free to manipulate the unitary evolution of the experiment.
The state of the experiment can also decohere and become entangled with the state of the ob-

57


server; in more conventional phrasing, the observer can measure the experiment.62 The isometry
from the code space to global boundary states maps all these events to ordinary unitary quan-
tum mechanics with no postselection on the global boundary. The only change that happens as
the black hole evaporates is that the observer and the experiment end up being encoded in the
reservoir Hrad, rather than the CFT.

4.4
The Peak of the Page Curve

The other part of the evaporation that we skipped, in the interests of avoiding speculative
discussion, was the period of time, very close to the Page time, when the Page curve peaks and
begins to decline.
In a simple random unitary toy model of black hole evaporation, the entanglement entropy
is almost exactly equal to the number of qubits in the radiation, until the black hole is within an
O(1) number of qubits of half its original size. There is then an O(1) correction to the entropy at
the peak of the curve, before the entropy becomes approximately equal to the number of qubits
describing the black hole an O(1) number of qubits later [60].
For an actual black hole, at times within O(tpage/√SBH) of the Page time, O(√GN) fluctu-
ations in the horizon area of the black hole and O(
√

S) fluctuations in the total energy of the
Hawking radiation mean that there will not be a single well-defined Ryu-Takayanagi surface. At
such times, we should therefore expect that the entanglement entropy will neither increase as
fast as the bulk entropy of the radiation, nor decrease as fast as the Bekenstein-Hawking entropy
of the black hole.
However, we can write the total state of the black hole and Hawking radiation as a superpo-
sition of O(
√

S) states, each of which has only O(GN) fluctuation in the horizon area and where
the entanglement spectrum of the Hawking radiation, in each state, has O(1) width.63

For almost all of the states in such a superposition, there is still a well defined Ryu-Takayanagi
surface. Indeed, if we believe that the lower bound on the reconstruction error from [38] is
approximately saturated, then, at any given time, the quantum extremal surface prescription
will be valid with very small error for all but an O(1) number of the states in this superposition.
For some fraction f of the states in the superposition, the interior will be encoded in the
reservoir Hrad, while for almost all the rest, it will be encoded in the CFT. As the black hole
evaporates, the fraction f will increase, until eventually the fraction f approaches one and the
interior can be reconstructed using only the reservoir, with only a very small error. From the
fraction f, as a function of time, it should, in principle, be possible to calculate the shape of the
peak of the Page curve.
Because the fluctuations in the evaporation rate smear out the Page time over an O(β√SBH)
time window, it seems plausible that, with sufficient work, one could calculate the entropy, up
to an O(1/√SBH) error, at all times. In other words, the fluctuations in the area of the black
hole and the energy of the Hawking radiation may make it feasible to calculate the Page curve
much more precisely than would otherwise be possible.

4.5
Explicit Interior Reconstruction

While we were able to make very precise statements in this paper about when and where infor-
mation was encoded in an evaporating black hole, we said comparatively little about how the

62It is important to note that causality prevents this decoherence from escaping the black hole. No measurement
by an interior observer will ever ‘have happened’ from the perspective of an observer that remains outside the
black hole.
63We cannot reduce the area fluctuations by more than this without substantially altering the bulk geometry,
see [77,78], for example, for details.

58


information was encoded. In particular, one would ideally want to have explicit, even if not
necessarily practical, reconstructions of the interior operators.
There has been considerable work done in recent years on understanding how entanglement
wedge reconstruction can be done explicitly [24,79,80]. However, these approaches often assume
knowledge of some global reconstruction, such as HKLL [81]. For interior operators, it is not
clear even what global reconstruction to use as a starting point. Infalling modes can, of course,
simply be evolved back in time to give exterior operators, but this is not possible for outgoing
interior modes.
This is not a new problem; it has been a major focus of research in AdS/CFT for many
years. In particular, there are some credible suggestions of ways in which one can construct
a state-dependent interior operator, given a particular choice of microstate [32, 33, 37]. If we
believe the arguments from Section 3.5, however, reconstructions should in principle be possible
for much larger code spaces.
In Appendix C, we give a simple generalisation of the Kourkoulou-Maldacena state-dependent
interior reconstruction for the SYK model that works for code spaces with entropy almost as
large as the Bekenstein-Hawking entropy.
We can also make the following more general argument for extending state-dependent re-
constructions that work for a single microstate to minimally state-dependent reconstructions.
Suppose we assume the existence of a single unknown reconstruction φcode of an interior operator
φ for a large code space Hcode with orthonormal basis |i⟩. Moreover, suppose we also assume
that for any state |i⟩ in this basis, we know an explicit reconstruction φi, which is consistent
with the unknown reconstruction φcode. In other words, for all |i⟩,

φcode |i⟩ ≈ φi |i⟩ .
(115)

But then for any state
|ψ⟩ =
�

i
ci |i⟩

we have
�

i
φi |i⟩ ⟨i|ψ⟩ =
�

i
ciφi |i⟩ ≈ φcode |ψ⟩ .
(116)

Hence

˜φcode =
�

i
φi |i⟩ ⟨i|
(117)

is an explicit reconstruction that approximates φcode when acting on any state in the code space.

4.6
The Information Paradox Beyond AdS/CFT

This paper is entirely about AdS/CFT. However, by understanding the information paradox
in AdS/CFT, we hope to eventually learn something about the information paradox in more
general quantum gravity. Does information escape black holes in our universe (in the absence
of the cosmic microwave background etc.) and if so how does it do so?
So far, entanglement wedge reconstruction, and the Ryu-Takayanagi formula, are only un-
derstood in the context of AdS/CFT. However, there is no obvious reason to think that they are
specific to spacetimes with a negative cosmological constant. In particular, for asymptotically
flat spacetimes, one can anchor a ‘Cauchy’ slice at some ‘boundary’ surface in asymptotic future
infinity, and then calculate quantum extremal surfaces based on this. Indeed, in this case, one

59


does not even need to do anything special to get absorbing boundary conditions. There is no
timelike boundary for modes to reflect from. Instead, early modes will simply automatically not
be included if they reach the asymptotic infinity at an earlier outgoing time than the time at
which we anchored our ‘Cauchy’ slice.
For de Sitter spacetimes, which most resemble our universe, there is no timelike or lightlike
asymptotic region that we can use to anchor spacelike slices. However, one would still hope that
the basic conceptual ideas of this paper – essentially the fact that there is a state-dependent
encoding of the black hole interior in the early Hawking radiation after the Page time – might
be relevant.
As an intermediate step, consider the case of black holes in AdS/CFT that are small enough
to be microcanonically unstable. These black holes are so small that we do not need to extract
Hawking radiation into an auxiliary system for the black hole to evaporate; the black hole will
have already evaporated by the time the Hawking radiation can reach the boundary and come
back.
If we don’t extract the Hawking radiation, there is no entanglement wedge that can show us
that the interior is encoded in the Hawking radiation after the Page time. The Hawking radiation
entirely surrounds the black hole horizon; there is no boundary region whose entanglement wedge
contains the radiation, but not the black hole.
However, if we do extract the Hawking radiation, which we can do using non-local boundary
dynamics, it is clear from entanglement wedge reconstruction that the interior is indeed encoded
in the Hawking radiation, just like for larger AdS black holes. The interior must still have been
encoded in the Hawking radiation before we extracted the radiation; we just had no way to learn
this using only entanglement wedge reconstruction.
To directly see that the interior was encoded in the radiation, even before we extracted the
radiation, we would need a way to distinguish the microscopic degrees of freedom encoding a
small neighbourhood of the black hole from the microscopic degrees of freedom describing the
Hawking radiation further out. This would require understanding holography beyond asymptotic
boundaries. There has been considerable recent progress in that direction using T ¯T deformations
of conformal field theories [82–86].

5
Acknowledgements

I would like to thank Raphael Bousso, Netta Engelhardt, Daniel Harlow, Frances Kirwan, Lam-
pros Lamprou, Juan Maldacena, Don Page, Daniel Ranard, Phil Saad, Eva Silverstein, Jon
Sorce, Steve Shenker, Douglas Stanford, Alex Streicher, Lenny Susskind, Mae Teo and Aron
Wall for valuable discussions. In particular, I would like to thank my advisor, Patrick Hay-
den, for his invaluable support throughout, Edward Witten, for first suggesting that arguments
similar to those in [38] could potentially explain the black hole information paradox and for
detailed feedback on this manuscript, and Ahmed Almheiri, for explaining his paper [31] to me
and for other very valuable discussions. This work was supported in part by AFOSR award
FA9550-16-1- 0082 and DOE award DE-SC0019380.

A
Cut-offs in Rindler Space

In this appendix, we study a simple pedagogical example of a situation where understanding
the coordinate dependence of cut-offs is vital, if we want to calculate entanglement entropies
correctly. The example is closely related to, but distinct from, the black hole extremal surface
calculations in Sections 2.2 and 2.4.

60


Consider the interval [0, r] of the vacuum state in some (1 + 1)-dimensional conformal field
theory. The entanglement entropy of this interval is given by

S = c

3 log
r
√ε1, ε2
(118)

where ε1 and ε2 are the cut-offs at each end of the interval [57,58]. We therefore find that the
derivative of the entanglement entropy

dS
dr = c

3r.
(119)

We can also do this calculation in Rindler space, where it corresponds to finding the derivative
of the entropy of an infinite half-line of a thermal state at inverse temperature β = 2π. The
entropy of a long interval of a thermal state of a CFT is equal to

S = πc l

3β
(120)

where l is the length of the interval [57,58]. We therefore find that

dS
dr∗ = c

6
(121)

where the Rindler position r∗ = log r. Hence

dS
dr = 1

r
dS
dr∗ = c

6r.
(122)

However, this differs from (119) by a factor of two. We apparently have a contradiction.
We can also look at the entanglement entropy of the interval [r, ∞]. In the Minkowski space
calculation, the gradient of the entropy of an interval tends to zero as the length of the interval
tends to infinity. However, for the thermal state in Rindler space, the gradient is simply the
negative of (122).
The resolution of the paradox is, of course, the cut-offs.
Implicitly, we assumed in the
Minkowski space calculation that the cut-off was constant in units of r, while in the Rindler
space calculation we assumed that it was constant in terms of the Rindler position r∗. However,
r and r∗ are nonlinearly related. So the cut-off cannot be constant in both units.
Let us assume that we actually wanted the cut-off to be constant in terms of the Rindler
position r∗. Recall that a constant cut-off in units of r∗ really means that the cut-off is equal to

ε0
∂
∂r∗ ,

for some constant ε0. Since

∂
∂r∗ = r ∂

∂r
(123)

the cut-off ε2 in (118), which is in units of r, is r ε0. We therefore find that

dS
dr = c

6r,
(124)

while the derivative of the entropy for the interval [r, ∞] is −c/6r. The results now agree with
the Rindler space calculation.

61


B
Finite Temperature Infalling Modes

In this appendix, we generalise the explicit calculation of the location of the non-empty quantum
extremal surface from Section 2.2 to thermal infalling modes at finite temperature, and to pure
infalling modes with constant energy density and without long range entanglement. As in Section
2.2, we assume that the outgoing modes are extracted from close to the horizon and so there
are no greybody factors. We assume throughout the section that we are after the Page time,
and so the non-empty quantum extremal surface is the Ryu-Takayanagi surface.64 Throughout
this section, we shall work in Eddington-Finkelstein coordinates, as in Section 2.2. However, the
calculations can also easily be done in Kruskal-like coordinates, as in Section 2.4.
We begin by studying the case of thermal infalling modes at a temperature T ′ that may be
different from the black hole temperature T. We assume that the infalling modes are purified
by an auxiliary Hilbert space Hpur and hence, importantly, are unentangled with the outgoing
Hawking radiation. We consider both the information learned by an observer with access to
Hrad ⊗ Hpur, and an observer with access only to Hrad.
Taking into account the new infalling thermal flux, we find that (20) becomes

dM
dv = cevapπ

12
(T ′2 − T 2).
(125)

Hence we have
drs
dv =
cevapπGN

3T(d − 1)rd−2
s
Ωd−1
(T ′2 − T 2),
(126)

and, using (16), the event horizon is at

rhor = rs +
cevapGN

6(d − 1)rd−2
s
Ωd−1

T ′2 − T 2

T 2
.
(127)

Using (18), the classical maximin surface lies on the classical apparent horizon rs at

v = − β

2π log
SBHT 2

cevap (T 2 − T ′2) + O(β).
(128)

As the infalling radiation temperature T ′ approaches the black hole temperature T, the location
of the classical maximin surface diverges into the infinite past because drs/dv → 0. In contrast,
we shall see that the non-empty quantum extremal surface for the CFT remains well-behaved
at this temperature.
We first calculate the Ryu-Takayanagi surface associated to HCFT. (Since the overall tri-
partite state is pure, this is also the Ryu-Takayanagi surface for Hrad ⊗ Hpur.) The entropy
of the outgoing modes is the same as (31), but, because the ingoing modes are now at finite
temperature, the entropy of the infalling modes in the entanglement wedge of the CFT is now

Sin = −cevapπT ′v

6
+ . . . ,
(129)

where, as usual, we have dropped constant terms and we have assumed that the cut-off is
independent of position in units of v.65 The total bulk entropy is therefore

Sbulk = cevap

6
log (rlc(v) − r) − cπv

6 (T + T ′) + . . .
(130)

64The exception is when the temperature, or energy density of the infalling modes is sufficient to prevent the
black hole ever reaching the Page time. As we shall see, in those cases, there does not exist a non-empty quantum
extremal surface at all.
65We are also, as usual, ignoring the corrections associated with the finite infalling time range of the infalling
modes because these corrections should vanish in the semiclassical limit.

62


Note that for T ′ = T, which corresponds to the Hartle-Hawking state, we expect that
the total entanglement entropy of ingoing and outgoing modes will agree with the Minkowski
vacuum formula for the entanglement entropy based on the proper distance between the quantum
extremal surface and the point where the outgoing modes are extracted. Using the Schwarzschild
time translation symmetry, or boost symmetry, of the Hartle-Hawking state, we can map this
interval to a small interval close to the bifurcation surface, where the Hartle-Hawking state
locally looks like the Minkowski vacuum.66 It can easily be verified that this is indeed the case.
Using (126) and (130), it is easy to calculate the location of the Ryu-Takayanagi surface of
HCFT. We find that

0 = ∂Sbulk

∂v
+
1

4GN

∂A
∂v ,
(131)

= ∂Sbulk

∂v
,
(132)

= ∂rlc/∂v

6(rlc − r) − π

6 (T + T ′),
(133)

(T + T ′)(rlc(v) − r) = 2T(rlc − rs(v)),
(134)

while (39) continues to be valid. Hence the quantum extremal surface lies at

r = rs + T ′ − T

T
GNcevap

3(d − 1)Ωd−1rd−2
s
= rhor − (T ′ − T)2
GNcevap

6(d − 1)rd−2
s
Ωd−1T 2 ,
(135)

and satisfies

rlc(v) = rs + T ′ + T

T
GNc

3(d − 1)Ωd−1rd−2
s
= rhor +
�
4 − (T ′ − T)2

T 2

�
GNcevap

6(d − 1)rd−2
s
Ωd−1
.
(136)

Thus

v = − β

2π log
SBH

cevap(4 − (T ′−T)2

T 2
)
+ O(β).
(137)

When the infalling modes are at the same temperature as the black hole, the Ryu-Takayanagi
surface lies exactly on the event horizon.67 At all other temperatures, it lies strictly inside the
event horizon. When T ′ = T, and only when T ′ = T, the total thermodynamic entropy of the
black hole and exterior radiation does not increase with time. The entropy of the new Hawking
radiation is cancelled by the loss of entropy from radiation falling into the black hole, while the
entropy of the black hole itself stays constant. Because all outgoing modes in the interior are in
the entanglement wedge of Hrad ⊗ Hpur, the Hawking radiation, even Hawking radiation far in
the future, is perfectly thermally entangled with Hrad ⊗ Hpur. Since the ingoing modes are also

66In contrast, the case where the infalling modes have zero temperature corresponds to the Unruh state, which
is singular at the white hole horizon and so does not locally look like the Minkowski vacuum near the bifurcation
surface.
67Since we only calculated the radius to O(GN), higher order corrections can potentially move the quantum
extremal surface outside the horizon. When calculating the entanglement wedge of the CFT plus all the future
ingoing thermal modes that will be thrown into the black hole, the RT surface cannot end up outside the event
horizon without creating a paradox. However this is not true for the entanglement wedge of the CFT alone.
In that case, corrections from only having a finite interval of infalling thermal modes pushes the RT surface an
O(G2
N) radial distance outside the horizon, as was shown in [87] after this paper first appeared on arXiv. (The
result follows most obviously as a consequence of the time-reversal symmetry of the state forcing the RT surface
to lie in the static slice.)

63


thermally entangled with Hpur and the horizon area is constant, the black hole entanglement
entropy stays constant.
This is consistent with the Page curve, which can also be derived using Ryu-Takayanagi
formula. Indeed, at any temperature T ′, the entanglement structure of the Hawking radiation
will be exactly consistent with the Page curve, because of our general argument from Section
2.3. It can easily be verified that this is indeed the case.
The Ryu-Takayanagi surface remains approximately one scrambling time in the past so long
as the temperature of the infalling modes is relatively low (the latest infalling time is obtained
at T ′ = T), but diverges into the past at T ′ = 3T. At higher temperatures, the radial distance
between the quantum extremal surface and the event horizon, required by (135), will be greater
than the radial distance between the quantum extremal surface and the outgoing lightcone,
required by (136).
Since the outgoing lightcone can never be inside the event horizon, no
quantum extremal surface can exist.
An observer with access to Hrad ⊗ Hpur will only ever learn the state of a diary thrown into
the black hole if the temperature T ′ < 3T. Interestingly, T ′ = 3T is exactly the temperature
at which thermal Hawking radiation, unentangled with Hrad ⊗ Hpur, becomes consistent with
unitarity. At this temperature,

1

4GN

dAhor

dv
= 2cevapπ

3β
= dSin

dv + dSrad

dv ,
(138)

where dSin/dv ≥ 0 is the entropy of the ingoing modes per unit infalling time and dSrad/dv ≥ 0
is the rate that entropy is produced in the Hawking radiation. The increase in the Bekenstein-
Hawking entropy is therefore just sufficient to purify both the outgoing Hawking radiation, and
the purification Hpur of the infalling modes.
If the observer only has access to Hrad, but not to Hpur, it will affect the information that
they learn about the black hole. To understand this, we need to calculate the location of the
Ryu-Takayanagi surface for Hrad. (Because the system is no longer in a bipartite pure state,
this is not the same as the Ryu-Takayanagi surface of HCFT.)
The entanglement wedge of Hrad contains the part of the interior inside the Ryu-Takayanagi
surface, as well as the outgoing modes that were extracted into the reservoir. Since the overall
state of the outgoing modes is pure, the outgoing entropy in the entanglement wedge of Hrad
for a given candidate RT surface is equal to the outgoing entropy in the entanglement wedge of
HCFT for the same candidate surface. (Since the Ryu-Takayanagi surfaces for Hrad and HCFT
will end up being different, they will have different entropies for the outgoing modes. However
as a function of the location of the surface, they are the same.)
This is not true for the ingoing modes, which are in a mixed state that is purified by Hpur.
The entanglement wedge of HCFT contains ingoing modes at infalling times that are later than
the Ryu-Takayanagi surface, while the entanglement wedge of Hrad contains at infalling times
that are earlier than the RT surface. Hence, instead of (130), we have

Sbulk = cevap

6
log (rlc(v) − r) + cπv

6 (T ′ − T) + . . .
(139)

We therefore find that the quantum extremal surface for Hrad lies at

r = rs − T ′ + T

T
GNcevap

3(d − 1)Ωd−1rd−2
s
= rhor − (T ′ + T)2
GNc

6(d − 1)rd−2
s
Ωd−1T 2 ,
(140)

and satisfies

rlc(v) = rs + T − T ′

T
GNc

3(d − 1)Ωd−1rd−2
s
= rhor +
�
4 − (T ′ + T)2

T 2

�
GNcevap

6(d − 1)rd−2
s
Ωd−1
.
(141)

64


Thus

v = − β

2π log
SBH

c(4 − (T ′+T)2

T 2
)
+ O(β).
(142)

For T ′ < T, information thrown into the black hole will still reappear in the Hawking radiation.
However the location of the extremal surface diverges as T ′ → T; for T ′ ≥ T no information will
ever escape in the Hawking radiation for an observer with access only to Hrad.
As before, this is exactly the temperature at which the Hawking radiation can look thermal,
to an observer with access only to Hrad without violating unitarity. The total entropy of HCFT⊗
Hrad increases because thermal modes are being thrown into the black hole. The entropy of Hrad
can therefore increase at the same rate, and the new Hawking radiation can be unentangled with
Hrad, even while the horizon area, and hence the entropy, of the black hole remains constant.
Finally, the Ryu-Takayanagi surface of Hpur will always be empty – it will always be in a
thermal state. This is a necessary consequence of the boundary dynamics being unitary; the
state of Hpur is initially thermal, and this is unchanged when we throw its purification into a
black hole.
We can also calculate the location of the quantum extremal surface when the infalling modes
are in a pure state, with constant energy density η, but without long range entanglement, for
example, if there is a constant infalling particle flux. Again, it is important that the ingoing
modes are unentangled with the outgoing Hawking radiation. In this case, both (36) and (39)
will continue to be valid as in the vacuum case. However, instead of (20), we now have

dM
dv = −cevapπ

12β2 + η,
(143)

and

drs
dv = −
cevapπGN

3β(d − 1)rd−2
s
Ωd−1
+
4GNβη

(d − 1)rd−2
s
Ωd−1
.
(144)

This affects the radius rl.c.(v) of the past lightcone as a function of the infalling time v, which
is given in (17). Hence, while the Ryu-Takayanagi surface still lies at

r = rs −
GNcevap

3(d − 1)Ωd−1rd−2
s
,
(145)

its infalling time will now be given by

v = − β

2π log
SBH

1 − 4β2η/cevapπ + O(β).
(146)

As before, when sufficient energy, in this case η > cevapπ/4β2 are thrown into the black hole, no
quantum extremal surface exists. The event horizon, and thus the outgoing light cone, are at
too large a radius for (36) and (39) to be simultaneously satisfied.
Yet again, this is exactly the point at which it stops being necessary for information to escape
the black hole in order to preserve unitarity. At this energy density the rate of increase of the
Bekenstein Hawking entropy is equal to the rate of increase in the entropy of the radiation

1

4GN

dAhor

dv
= cevapπ

6β
= dSrad

dv .
(147)

Hence the Hawking radiation can remain thermal, and unentangled with Hrad, forever, without
exceeding the entanglement entropy exceeding the Bekenstein-Hawking entropy of the black
hole.

65


The ingoing energy flux at which information stops escaping the black hole is highest for
thermal infalling modes and an observer who has access to both the reservoir Hrad and the
purification Hpur of the ingoing modes. This is because there needs to be sufficient Bekenstein-
Hawking entropy in the black hole both to purify outgoing thermal Hawking radiation and to
purify the degrees of freedom in Hpur.
In contrast, when the observer only has acccess to Hrad, the required ingoing energy flux for
thermal infalling modes is much smaller. The ingoing entropy makes it easier for the Hawking
radiation to be unentangled with Hrad, because Hrad can be purified by Hpur as well as the black
hole.
Finally, when the ingoing modes are in a pure state with no long range entanglement, infor-
mation stops escaping at an intermediate ingoing energy flux. The increase in the Bekenstein-
Hawking entropy needs to be sufficient to purify the Hawking radiation in Hrad; there is no
ingoing bulk entropy to make this either harder or easier.

C
Minimal State Dependence in the SYK Model

In this appendix, we construct minimally state-dependent interior reconstructions in a simple
toy model of quantum gravity, known as the SYK model. This model has been studied in great
depth in the last few years [88–93]; here we provide only the bare minimum of background detail
necessary for our purposes.
The SYK model is a 0 + 1-dimensional quantum mechanical model that consists of N Ma-
jorana fermions ψi. These satisfy
{ψi, ψj} = δi,j.

Using a Jordan-Wigner transformation, it can be easily seen that there is a single qubit degree
of freedom associated with every pair of Majorana fermions. The Hilbert space therefore has
dimension 2N/2. The model has Hamiltonian

H =
�

iklm
jiklmψiψkψlψm,
(148)

where jiklm are independent Gaussian random couplings with ⟨j2
iklm⟩ = 6J2/N 3.
In the limit N → ∞, at fixed βJ ≫ 1, the SYK model appears to become holographic; it has
many features that resemble nearly-AdS2 gravity. Although the precise dual gravity description,
if one exists, remain unknown, both the SYK model and simple nearly-AdS2 gravity theories
such as Jackiw-Teitelboim gravity [73, 94–97] have an emergent reparameterisation symmetry
that is both spontaneously and explicitly broken, with the explicit symmetry-breaking term
proportional to the so-called Schwarzian action

S = αSN

J

�
dτ
�f′′

f′

�′
− 1

2

�f′′

f′

�2
,
(149)

where f(τ) is the reparameterisation and αS is a numerical constant. From a gravity perspective,
this action appears as a boundary term when we cut-off the nearly-AdS2 geometry at some fixed
dilaton value φb; for Jackiw-Teitelboim gravity, in particular, it describes the entire dynamics
of the theory, which can be interpreted as the dynamics of a boundary particle, describing the
location of the cut-off, in a rigid AdS2 background.
A complete basis for the entire Hilbert space of the SYK model is given by the states |Bs⟩,
satisfying

(ψ2k−1 − iskψ2k) |Bs⟩ = 0,
(150)

66


(a)
(b)

Figure 17: The states |Bs(β)⟩ have a gravity description as a one sided black hole, ending
on an end-of-the-world brane (black).
If the system is evolved using the unperturbed SYK
Hamiltonian, shown in Figure 17a, the bulk operator φ lies behind the black hole horizon (blue).
However if the Hamiltonian is perturbed in a state-dependent way, shown in Figure 17b, the
boundary (green) is pulled inwards and so the operator no longer lies behind a horizon. In the
original construction, the Hamiltonian was precisely tuned for a single state |Bs(β)⟩. However
one can easily adapt the Hamiltonian to work for a large set of states.

where for all k, we have sk = ±1. If we evolve these states in Euclidean time, we get a set of
states

|Bs(β)⟩ = e−βH/2 |Bs⟩ ,
(151)

which form an approximate, overcomplete basis for the low energy states of the theory. In fact,
if we allow arbitrary superpositions of these states, they still form a complete basis for the entire
Hilbert space, because the map e−βH is invertible. However, to create a high energy state, we
need a very finely tuned superposition

|ψ⟩ =
�

s
cs |Bs(β)⟩ ,
(152)

where

⟨ψ|ψ⟩ ≪
�

s
|cs|2 ⟨Bs(β)|Bs(β)⟩ .
(153)

However we will only allow ‘generic’ superpositions of the states |Bs(β)⟩ that do not satisfy
(153).
It was shown in [37] that the states |Bs(β)⟩ have a natural gravity interpretation as black
hole microstates with a smooth interior ending on an ‘end-of-the-world brane’ (Figure 17).
Excitations in the interior can be created by acting with additional boundary operators during
the Euclidean time evolution.
If the system is evolved with the unperturbed Hamiltonian H, then such excitations can
never reach the boundary. However, if we perturb the Hamiltonian by

δH = −εJ

N/2
�

k=1
skiψ2k−1ψ2k,
(154)

67


then, from a gravity perspective, the Schwarzian ‘boundary particle’ is pulled into the bulk of
the AdS′
2 space, as shown in Figure 17. By evolving the system with the perturbed Hamiltonian
H + δH, we can render the interior of the black hole visible to the boundary. Interior operators
can be reconstructed using boundary operators time-evolved using this perturbed Hamiltonian.
The perturbation δH was carefully adapted to the state |Bs(β)⟩. The reconstruction is there-
fore highly state-dependent. A natural question is whether we can reduce this state dependence,
and find a reconstruction that works for a larger class of microstates.
Instead of using the perturbation (154), suppose instead that we perturb the Hamiltonian
H by

δHf = −εJ

fN/2
�

k=1
skiψ2k−1ψ2k,
(155)

where 0 < f < 1 is a fixed O(1) fraction. Since the number of terms in δHf continues to be
O(N), the perturbation δHf will also make the interior of the black hole microstate |Bs(β)⟩
visible to the boundary.68

However this perturbation only depended on the first fN/2 spins sk of the microstate |Bs(β)⟩.
We have found a single reconstruction that works for 2(1−f)N/2 different microstates |Bs(β)⟩.
By linearity, the same reconstruction should also work for generic superpositions of these mi-
crostates.
Since the low temperature entropy of the SYK model is approximately

S0 ≈ 0.23N,
(156)

we have found a single reconstruction that is valid for more than eS0 microstates. Of course,
not all these microstates are independent. Instead, the effective size of the code subspace is
determined by the entropy of the mixed state

ρs1 = 2−(1−f)N/2 �

s2
|Bs1⊕s2(β)⟩ ⟨Bs1⊕s2(β)| ,
(157)

where the sum is over spins sk ∈ s2 for fN/2 ≤ k ≤ N/2 while the spins sk ∈ s1 for 1 ≤ k < fN/2
are held fixed.
How large is this entropy? Since

e−βH

Tr(e−βH) = 2−fN/2 �

s1
ρs1,
(158)

then the strict concavity of entropy means that

⟨S(ρs1)⟩s1 < S0,
(159)

where the expectation is taken over possible states s1 of the fixed spins. If ρs1 were a uniform
mixture of 2(1−f)N/2 randomly chosen microstates |Bs(β)⟩, we would expect that S(ρs−1) would

68The argument that the perturbation δH can be used to make the interior visible to a boundary observer is
given in Section 7 of [37]. For the full details, we simply refer readers to that work. However the basic strategy is to
assume that for ε ≪ 1 we can treat δH as a perturbation to the Schwarzian action (149) of the reparameterisation
modes. So long as we have O(N) terms, the perturbation will appear in the semiclassical action at large N. It can
then be shown that, at sufficiently large, but fixed, βJ, we can choose ε so that the semiclassical large N equation
of motion for the Schwarzian ‘boundary particle’ makes the entire black hole interior visible. The argument for
δHf is identical, except for the addition of the O(1) factor f in the perturbation to the large N semiclassical
Schwarzian action.

68


be very close to S0 for (1 − f)N/2 > S0. There would be no remaining space to encode the
interior degrees of freedom. However this will not be the case for the particular set of microstates
in (157).
At large N, there is an emergent O(N) symmetry of the SYK model. In particular there is
a Zn
2 subgroup of this symmetry group, called the flip subgroup, that acts transitively on the
set of states |Bs(β)⟩). This means that S(ρs1) depends, at leading order, only on the number of
spins that are held fixed, and not on the signs of those spins.
If no spins are held fixed, the ensemble is simply the canonical ensemble, which has entropy
S0 to leading order in 1/N for fixed βJ ≫ 1. Now suppose that we know the average entropy
Sf for a state ρs1 with a fixed fraction f of the spins held fixed. We then consider the ensembles
ρs1⊕1 and ρs1⊕−1 formed by fixing a single additional spin. These two ensembles therefore have
a fraction f + 2/N of their spins fixed. Note that

ρs1 = 1

2 (ρs1⊕1 + ρs1⊕−1) .
(160)

Hence

2S(ρs1) − S(ρs1⊕1) − S(ρs1⊕−1) = S(ρs1⊕1||ρs1) + S(ρs1⊕−1||ρs1)
(161)

≥ 1

4∥ρs1⊕1 − ρs1⊕−1∥2
1,
(162)

where we have used Pinsker’s inequality [98]. However, we can lower bound the trace distance
∥ρs1⊕1 − ρs1⊕−1∥1 by

∥ρs1⊕1 − ρs1⊕−1∥1 = max
O
|Tr [O (ρs1⊕1 − ρs1⊕−1)]|

∥O∥
≥ |Tr [ ψ2k−1ψ2k (ρs1⊕1 − ρs1⊕−1)]| , (163)

where k = Nf/2+1 labels the additional spin fixed in ρs1⊕±1, but not in ρs1. This last quantity
was shown to in [37] to be order one in the limit of large N (although it decays as a function of
βJ). Hence

Sf − Sf+2/N = O(1)
(164)

and thus

S0 − Sf = O(fN),
(165)

for any fixed f > 0. If the fraction f is small, the entropy is very close to S0 at leading order,
but there is still plenty of space left to encode the interior. We have found an explicit, minimally
state-dependent reconstruction.
Of course, we have only constructed minimally state-dependent operators for a very particu-
lar class of ensembles of microstates. Our arguments in Section 3.5 suggested that there should
exist minimally state-dependent reconstructions for any sufficiently small code subspace. For
the special ensembles that we have considered in this section, the reconstructions only involved a
simple perturbation to the SYK Hamiltonian; for arbitrary code subspaces, the reconstructions
would presumably be much more complicated. We gave a more general, but much less practi-
cal, procedure for constructing minimally state-dependent reconstructions out of reconstructions
that only work for individual states in Section 4.5; it seems reasonable to expect that such a
procedure should also work for the SYK model.

69


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