11institutetext: School of Mathematics, University of Bristol,
Woodland Road, Bristol BS8 1UG, U. K. 22institutetext: Quantum Engineering
Center for Doctoral Training, University of Bristol,
Tyndall Avenue, Bristol BS8 1FD, U. K.

# The Page curve from the entanglement membrane

Mike Blake 1, 2 and Anthony P. Thompson

###### Abstract

We study entanglement dynamics in toy models of black hole information built
out of chaotic many-body quantum systems, by utilising a coarse-grained
description of entanglement dynamics in such systems known as the
‘entanglement membrane’. We show that in these models the Page curve
associated to the entropy of Hawking radiation arises from a transition in the
entanglement membrane around the Page time, in an analogous manner to the
change in quantum extremal surfaces that leads to the Page curve in semi-
classical gravity. We also use the entanglement membrane prescription to study
the Hayden-Preskill protocol, and demonstrate how information initially
encoded in the black hole is rapidly transferred to the radiation around the
Page time. Our results relate recent developments in black hole information to
generic features of entanglement dynamics in chaotic many-body quantum
systems.

##  1 Introduction

#####

The past few years have seen significant new developments in understanding the
fate of quantum information during the evaporation of a black hole
Hawking:1975vcx ; Hawking76 . In particular, it was appreciated by Page in the
early 1990s that if black hole evaporation is unitary then the entanglement
entropy of the emitted Hawking radiation should follow the Page curve
Page_1993a ; Page_1993b , increasing until the black hole has approximately
half evaporated before decreasing to zero. Remarkably, it was established in
penington2020entanglement ; Almheiri_2019 ; Almheiri_2020  that the Page curve
could be obtained from a computation in semi-classical gravity, by adapting
the traditional quantum extremal surface prescription Ryu_2006 ; Hubeny_2007 ;
Engelhardt_2015  to include the contributions of so-called ‘islands’. The
existence of such entanglement islands can be derived from the Euclidean
gravitational path integral using the replica trick penington2020replica ;
Almheiri_2020Replica , where it is related to the existence of ‘replica
wormhole’ configuration in which different replicas of the gravitational
system are coupled in a non-trivial manner111See Marolf_2021a ; Marolf_2021b
for a Lorentzian interpretation of replica wormholes.. Whilst these
developments have led to significant progress, see e.g. Almheiri_2020HigherDim
; Almheiri:2019yqk ; Akers_2020 ; Hollowood:2020cou ; Rozali_2020 ; Zhao_2020
for a selection of early work and Almheiri_2021  for a review, they rely on
complex details of the gravitational path integral. As such a clear physical
understanding of entanglement islands and replica wormholes remains lacking.

#####

A closely related question concerns the thermalization of a pure state
$\ket{\psi}$ in a chaotic many-body quantum system. At late times such a state
is expected to thermalize in the sense that measurements of local observables
agree with those computed via a thermal density matrix $\rho$. Nevertheless,
unitarity of quantum mechanics ensures that the system remains globally in a
pure state, which has non-trivial consequences for the behaviour of non-local
observables such as the von Neumann and Renyi entropies of a subregion $A$.
For instance, for a globally pure state the entanglement entropies of $A$ and
its complement must be equal, which lies at the heart of the original
arguments for the Page curve Page_1993b . It is therefore natural to suggest
that the recent insights into the unitarity of black hole evaporation are
closely related to generic features of entanglement dynamics in chaotic many-
body quantum systems.

#####

A particular microscopic connection between the Page curve and many-body
quantum dynamics has been established in liu2019void ; Liu_2021 .
Specifically, Liu_2021  studied simple models of evaporating and eternal black
holes built out of chaotic quantum many-body systems. They found that various
probes of entanglement dynamics in these models, including the Page curve and
the Hayden-Preskill protocol Hayden_2007 , can be understood as arising
microscopically from a dynamical mechanism known as ‘void formation’
liu2019void . Whilst these works represent a significant development, it is
not straightforward to relate the microscopic process of void formation in
such models to the recent computations in semi-classical gravity. With this
motivation in mind, the purpose of this paper is to provide an alternative way
of describing entanglement dynamics in similar toy models of evaporating and
eternal black holes that more closely resembles the (quantum) extremal surface
prescription of semi-classical gravity.

#####

In order to do this, we will make use of an effective theory222A related
general theory of information spreading in chaotic systems has been presented
in Couch:2019zni . for the coarse-grained dynamics of von Neumann and Renyi
entropies in chaotic many-body quantum systems with local interactions known
as the ‘entanglement membrane’ Nahum_2017 ; jonay2018coarsegrained ; Zhou_2019
; PhysRevX.10.031066 . This description of entanglement dynamics can be
explicitly derived for the case of 1+1 dimensional random quantum
circuits333Numerical evidence in support of the entanglement membrane
description for higher dimensional random circuits has recently been provided
in sierant2023entanglement . with a large local Hilbert space dimension
Zhou_2019 , and holographic quantum field theories dual to classical gravity
Mezei_2018 ; Mezei_2020 444In this context the entanglement membrane exists in
the space-time of the dual non-gravitational theory, and can be understood as
the projection of the bulk HRT surface into the boundary.. It has been
conjectured to apply to a wide class of chaotic many-body quantum systems,
although we note that at present the full extent of systems described by such
an approach remains an important open question555In particular, it has been
established that the simplest formulation of the entanglement membrane we will
discuss in this paper does not describe the behaviour of higher ($n\geq 2)$
Renyi entropies for random circuits with diffusive conserved charges for which
the entanglement entropy following a quench grows as$~{}\sqrt{t}$
PhysRevLett.122.250602 ; Huang_2020 . Other effects of conserved charges on
entanglement dynamics have recently been studied in e.g. PhysRevB.107.045102 ;
Majidy_2023 ..

#####

Within this effective description Nahum_2017 ; jonay2018coarsegrained ;
Zhou_2019 ; PhysRevX.10.031066 ; Mezei_2018 ; Mezei_2020 , the entanglement
entropies of a subregion of the chaotic many-body quantum system can be
computed by minimising the entropy cost associated to a co-dimension one
surface, the ‘entanglement membrane’, in a slab of space-time. We will see
that, when applied to models of evaporating and eternal black holes, the Page
curve arises from a transition in the minimal membrane around the Page time,
in an analogous manner to the change in the quantum extremal surface in semi-
classical gravity. We will also use the entanglement membrane to probe the
Hayden-Preskill protocol, and find that information encoded in the black hole
is rapidly emitted around the Page time as anticipated in Hayden_2007 .

#####

The remainder of this paper is organised as follows. In Section 2 we briefly
review the entanglement membrane description of entanglement dynamics in local
chaotic many-body quantum systems. In Section 3 we study a model of an
evaporating black hole built out of a local chaotic many-body quantum system,
and demonstrate how the entanglement membrane framework reproduces both the
expected form of the Page curve and the Hayden-Preskill protocol in this
model. In Section 4 we apply the entanglement membrane to study these probes
of entanglement dynamics in a model of an eternal black hole coupled to a
bath. Our results are consistent with those derived explicitly in
gravitational systems, and also the microscopic calculations performed in
related toy models using void formation in liu2019void ; Liu_2021 . Finally,
in Section 5 we review important open questions, in particular commenting on
the connection between the entanglement membrane and permutation degrees of
freedom used in Vardhan_2021  to connect the entanglement entropies of
thermalized states to replica wormholes.

##  2 Review of the entanglement membrane

#####

We begin by reviewing a coarse-grained description of entanglement dynamics in
certain chaotic many-body quantum systems based on an ‘entanglement membrane’
Nahum_2017 ; jonay2018coarsegrained ; Zhou_2019 ; PhysRevX.10.031066 . The
entanglement membrane is most easily motivated by results for random unitary
circuits, but has a wider validity in more general chaotic systems including
an appropriate limit of holographic quantum field theories Mezei_2018 ;
Mezei_2020 . This description can be derived analytically for ‘large-$q$’
random circuits with a large local Hilbert space dimension Zhou_2019 ;
PhysRevX.10.031066 , and is consistent with numerical results for certain
chaotic spin chains and Floquet models666In systems with conserved charges the
simplest formulation of the entanglement membrane jonay2018coarsegrained  we
will discuss here assumes that after coarse-graining the system is in local
equilibrium. We also note the even starting from homogeneous initial states,
the presence of conserved charges can have non-trivial consequences for the
behaviour of higher Renyi entropies. In particular, in random local quantum
circuits with a diffusive conserved charge and finite Hilbert space dimension
the higher Renyi entropies with $n>1$ grow as $\sqrt{t}$ and are not described
by the simplest formulation of the entanglement membrane we discuss here
PhysRevLett.122.250602 ; Huang_2020 . Nahum_2017 ; jonay2018coarsegrained .

#####

In particular we consider a homogeneous many-body quantum system, following a
quench from a pure state $\ket{\psi(0)}$ at time $t=0$, with time evolution
determined by chaotic unitary dynamics. At late times the system is expected
to thermalise to an equilibrium configuration with entropy density $s_{eq}$.
Our goal is to compute the entanglement entropy $S(A,t)$ of a subregion $A$ of
the many-body quantum system during thermalization. In the entanglement
membrane picture (valid on large lengthscales and timescales) the entropy
$S(A,t)$ can be computed by minimising a functional describing the ‘entropy
cost’ associated to each choice of a co-dimension one membrane in a space-time
slab of height $t$ Nahum_2017 ; jonay2018coarsegrained ; Zhou_2019 ;
PhysRevX.10.031066 .

Figure 1: Illustration of the entanglement membrane as a coarse-grained
description of entanglement dynamics in random quantum circuits and chaotic
many-body quantum systems.

#####

Whilst the entanglement membrane can be formulated in any dimension, the
simplest setting in which to illustrate its use is an infinite one dimensional
many-body quantum system with spatially homogeneous dynamics. We are
interested in the entropy $S(x_{1},t)$ across a cut in the system at position
$x_{1}$, i.e. the entropy of the subregion $A=\\{x:-\infty<x\leq x_{1}\\}$. To
do this, one considers all curves $x(t^{\prime})$ in a slab of spacetime
$(x,t^{\prime})$, where $t^{\prime}\in[0,t]$ and $x(t^{\prime})$ obeys the
boundary conditions $x(t)=x_{1}$ and $x(0)=x_{0}$ with $x_{0}$ left free (see
Figure 1). The entropy can then be computed by minimising the following
functional

$S(x_{1},t)=\mathrm{min}_{x(t^{\prime})}\bigg{(}\int_{0}^{t}s_{eq}{\cal
E}(\dot{x}(t^{\prime}))dt^{\prime}+S(x_{0},0)\bigg{)},$ (1)

where $s_{eq}$ is the equilibrium entropy density of the final state and
$S(x_{0},0)$ is the entropy of the region $\\{x:-\infty<x\leq x_{0}\\}$ in the
initial state $\ket{\psi(0)}$. The above prescription can naturally be
modified to compute entanglement entropies of more/complicated subregions or
in finite size systems. For our purposes it will be particularly useful to
note the following jonay2018coarsegrained :

  * •

For a finite sized system (i.e. where $x\in[0,L]$) one must also consider
membranes $x(t^{\prime})$ that escape the space-time slab through either of
the left or right hand edge of the system. For such membranes there is no
contribution to the entropy cost from the initial state (i.e. one does not
have the $S(x_{0},0)$ term above).

  * •

For a finite subregion $R=\\{x:x_{1}\leq x\leq x_{2}\\}$ one includes
membranes $x_{L}(t^{\prime})$, $x_{R}(t^{\prime})$ originating from each of
the boundaries of $R$, i.e. satisfying $x_{L}(t)=x_{1}$ and $x_{R}(t)=x_{2}$.
Such membranes may intersect (i.e. join) in the space-time slab777If the
curves $x_{L}(t^{\prime})$ and $x_{R}(t^{\prime})$ intersect the initial state
there is a contribution $S(R_{0},0)$ to the entanglement cost of the membrane
from the entropy of the region $R_{0}$ enclosed by the curves in initial
state..

  * •

For disjoint subregions $R=R_{1}\cup R_{2}$ one includes curves emanating from
the boundaries of both $R_{1}$ and $R_{2}$ at time $t$.

#####

The precise form of the line tension ${\cal E}(\dot{x})$ depends both on the
microscopic model and on the particular entanglement entropy being computed
(e.g. the von Neumann $S_{vN}$ or second Renyi entropy $S_{2}$). Nevertheless,
it can be argued that the line tension satisfies the following general
properties jonay2018coarsegrained ; Zhou_2019

${\cal E}(0)=v_{E},\hskip 56.9055pt{\cal E}(v_{B})=v_{B},\hskip 56.9055pt{\cal
E}^{\prime}(v_{B})=1,$ (2)

with $v_{E}$ the entanglement velocity of the entropy being computed and
$v_{B}$ the butterfly velocity of the system of interest. Furthermore, we have
the following inequalities

${\cal E}(v)\geq v,\hskip 56.9055pt{\cal E}^{\prime\prime}(v)\geq 0,\hskip
56.9055pt{\cal E}(v)\leq v_{E}+|v|\bigg{(}1-\frac{v_{E}}{v_{B}}\bigg{)}.$ (3)

We note that in minimising (1) it is sufficient to consider only curves inside
the lightcone $|\dot{x}|\leq v_{B}$. For a generic system in which these
inequalities are satisfied strictly for $0<v<v_{B}$ we have that both ${\cal
E}(v)$, ${\cal E}^{\prime}(v)$ are monotonically increasing functions in this
domain. Finally we note that for 1+1 dimensional systems with homogeneous
unitary dynamics then convexity of the line tension means that it is
sufficient to only consider piece-wise linear curves jonay2018coarsegrained .

#####

For certain microscopic models the line tension can be computed analytically.
For instance if one considers a one dimensional chain with one spin of local
Hilbert space dimension $q$ per unit length, then under random unitary
dynamics a generic state $\ket{\psi(0)}$ thermalizes to an infinite
temperature state with $s_{eq}=\ln q$. For a circuit built by applying random
unitarities to bonds in a Poisson fashion (with rate 1), then as $q\to\infty$
one finds that the von Neumann entropy is governed by the line tension888In
fact in the $q\to\infty$ limit the line tension is independent of the entropy
being computed. jonay2018coarsegrained

${\cal E}(v)=\frac{1}{2}(1+v^{2}),$ (4)

from which we can read off $v_{E}=1/2$, $v_{B}=1$ in such a model. In a second
model with unitary dynamics built from a random brickwork circuit, then at
large but finite $q$ (ignoring subleading $1/q$ corrections) one finds the
second Renyi entropy is governed by the line tension999At strictly infinite
$q$ the line tension ${\cal E}_{2}(v)$ becomes flat with ${\cal E}_{2}(v)=1$.
This results in a highly degenerate minimal membrane problem. This degeneracy
is resolved by working at large but finite $q$. jonay2018coarsegrained

${\cal
E}_{2}(v)=\log_{q}\frac{q^{2}+1}{q}+\frac{1+v}{2}\log_{q}\frac{1+v}{2}+\frac{1-v}{2}\log_{q}\frac{1-v}{2},$
(5)

The line tension can also be computed for the von Neumann entropy of certain
holographic quantum field theories Mezei_2018 ; Mezei_2020 .

#####

Before applying the entanglement membrane to models of black hole evaporation,
it will be useful to review how the entanglement membrane reproduces the
expected results for the thermalization of a sub-region of a one dimensional
system. In particular, consider the entanglement entropy $S(x,t)$ in a system
of length $L$ starting from a product state (such that we can ignore any
contribution to the entanglement cost from the initial state). At early times,
the entropy is then minimised by a straight line dangling down vertically from
the cut which gives $S(x,t)=s_{eq}{\cal E}(0)t=s_{eq}v_{E}t$. The entropy
therefore initially increases linearly with time as expected. However, we can
also consider membranes emanating from the cut at $(x,t)$ with slopes $v=\pm
v_{B}$. For sufficiently large $t$ these will escape the system via the
left/right edges of the space time slab and have a time-independent entropy
cost. Assuming $x<L/2$ the cut escaping the left hand edge has the smaller
entropy cost given by $S(x,t)=s_{eq}x$. For $t>x/v_{E}$ this is smaller than
the entropy cost of the vertical membrane, and the entropy saturates at this
thermalized value.

##  3 Model of an evaporating black hole

#####

In the remainder of this paper we will apply the entanglement membrane to
study entanglement dynamics in various models of black hole information. In
this section we begin with a very crude model of information dynamics for an
evaporating black hole. Whilst this model is over-simplified in many respects,
it is instructive to see how the entanglement membrane reproduces both the
expected form of the Page curve and the Hayden-Preskill protocol Hayden_2007
in this setting.

#####

In particular our model of an evaporating black hole consists of a homogeneous
chaotic many-body quantum system of finite length $L$, with spatially local
interactions. We will assume that the system is described by an entanglement
line tension ${\cal E}(v)$, and thermalizes to an equilibrium configuration
with uniform entropy density $s_{eq}$. We now divide the system into two
parts. At each time $t$, we refer to the region $0\leq x\leq x_{B}(t)$ where
$x_{B}(t)=L-\alpha t$ as the black hole subsystem $B(t)$. Here $\alpha\ll
v_{E}$ parameterises the rate at which the black hole evaporates, and
$v_{E}={\cal E}(0)$ is the entanglement velocity of the underlying chaotic
many-body system101010We caution the reader that in this section the
entanglement velocity $v_{E}$ and butterfly velocity $v_{B}$ refer to the
entanglement/butterfly velocities of the underlying local many-body system,
and should not be viewed as those one would compute for a black hole in
holography.. The remaining degrees of freedom from $x_{B}(t)\leq x\leq L$ are
referred to as the radiation subsystem $R(t)$. Note that the black hole and
radiation systems are non-trivially coupled by the underlying local dynamics.
A very similar model, in which the underlying microscopic system was a
large-$q$ spin chain, was studied in Section II of Liu_2021  using void
formation.

Figure 2: Entanglement membranes responsible for the Page curve in a toy model
of an evaporating black hole.

#####

We are interested in computing the entanglement entropy of the black hole
region $B(t)$. For simplicity we will assume that the initial state of the
system at $t=0$ is a product state, such that the initial entropy profile
across a cut is trivial. Then for $\alpha\ll v_{E}$ the behaviour of the
entanglement entropy $S(B(t))$ can be understood from a competition between
two minimal surfaces of slopes $|\dot{x}|=v_{B}$. In particular, the relevant
membranes consist of:

  1. 1.

A curve from $x_{B}(t)$ to the right boundary at $x=L$ with slope
$\dot{x}=-v_{B}$. This has entropy $\alpha s_{eq}t$.

  2. 2.

A curve from $x_{B}(t)$ to the left boundary at $x=0$ with slope
$\dot{x}=v_{B}$. This has entropy $s_{eq}(L-\alpha t)=S_{BH}-\alpha s_{eq}t$
where we have introduced $S_{BH}=s_{eq}L$.

These surfaces are illustrated in Figure 2. Note that for this set up a
vertical membrane emanating from $x_{B}(t)$ has entropy cost
$v_{E}s_{eq}t\gg\alpha s_{eq}t$ and hence is never minimal for a slowly
evaporating black hole.

#####

Precisely the same membranes are also relevant for the computation of the
radiation subregion $R(t)$. We therefore find the entropies:

$\displaystyle S(B(t))=S(R(t))$ $\displaystyle=\alpha s_{eq}t,$ $\displaystyle
t\leq t_{P},$ $\displaystyle=S_{BH}-\alpha s_{eq}t,$ $\displaystyle t\geq
t_{P},$ (6)

which we can identify as the Page curve with Page time $t_{P}=S_{BH}/(2\alpha
s_{eq})$. The sharp change in the behaviour of the entropy around the Page
time arises from a change in the minimal membrane at $t=t_{P}$, analogous to
the transition between bulk extremal surfaces responsible for the Page curve
in the gravitational computation of entanglement entropy. Furthermore, we note
that in the above computation we have not needed to specify the precise
entropy being computed - the above results apply to both von Neumann and Renyi
entropies, provided that the entanglement membrane prescription is valid for
the quantity of interest.

###  3.1 Information transfer from an evaporating black hole

#####

As a more refined probe of information dynamics during black hole evaporation,
we now turn to study the Hayden-Preskill protocol Hayden_2007 . In particular,
we are interested in understanding how information originally inside a (young)
black hole is transferred to its Hawking radiation during evaporation.

#####

In order to study this in the context of our toy model, we introduce an
additional reference system $Q$ of length $p\ll L$, consisting of the same
degrees of freedom as those of the black hole/radiation subsystem $B(t)\cup
R(t)$. The reference system $Q$ is not coupled by any interaction to the
evaporating black hole system, however we assume it has non-trivial
entanglement with the black hole degrees of freedom at $t=0$. In particular we
assume the reference system $Q$ is entangled with an identical subsystem
$P\subset B(0)$ (i.e. of length $p\ll L$) such that the entropy density in $P$
at $t=0$ is given by $s_{eq}$ 111111Here we are choosing the entropy density
of $P$ in the initial state to be the same as that of the state the system
ultimately thermalizes to. Other choices for the initial entropy density can
also be considered and give qualitatively similar results.. The remaining
degrees of freedom of $B(0)$ are taken to be initially in a product state. We
are then interested in how the information in $Q$ initially encoded in the
black hole subsystem through this entanglement is spread to the radiation
subsystem as the black hole evaporates. According to the Hayden-Preskill
protocol, this information is expected to be rapidly transferred to the
radiation around the Page time Hayden_2007 .

Figure 3: Extremal membranes relevant for $S(R(t))$ for a young black hole
entangled with a reference system.

#####

In order to study this, we will use the entanglement membrane to compute the
(Renyi or von Neumann) mutual information between $Q$ and the subsystem $R(t)$

$\displaystyle I(Q,R(t))$ $\displaystyle=$ $\displaystyle S(Q)+S(R(t))-S(Q\cup
R(t)),$ (7) $\displaystyle=$ $\displaystyle
S_{Q}+S(R(t))-S(B(t)),\;\;\;\;\;\;\;\;\;\;\;\;S_{Q}=s_{eq}p,$

where we have used the overall system $Q\cup R(t)\cup B(t)$ is in a pure
state. An analogous formula for $I(Q,B(t))$ can be obtained by interchanging
$R(t)$ and $B(t)$ in (7).

#####

In order to proceed, it is necessary to adapt the entanglement membrane
prescription to apply in the presence of a reference system $Q$. To do this,
it is useful to consider the subsystem $Q$ to lie to the left hand side of the
black hole-radiation system, i.e. in the region $-p\leq x<0$. Since there are
no interactions coupling $Q$ to the region $B(t)\cup R(t)$, we now consider
applying the entanglement membrane in a geometry in which there is a thin gap
in the space-time slab at $x=0$ (see Figure 3). We further will assume that
any piece of the entanglement membrane in this gap in the space-time slab
gives rise to no additional cost to the entropy (i.e. ${\cal E}(v)=0$ in the
gap), except for a possible contribution from the initial state. Such a
prescription is natural from the ‘minimal cut’ picture of the entanglement
membrane in random circuits in which the entropy cost is associated with the
number of non-trivial gates in the circuit cut by the membrane Nahum_2017 .

#####

The computation of $S(R(t))$ can then be understood from the following
(candidate) minimal surfaces (illustrated in Figure 3).

  1. 1.

A surface that emanates from $x_{B}(t)$ and crosses to $x=L$ with slope
$\dot{x}=-v_{B}$. This has entropy $\alpha s_{eq}t$.

  2. 2.

A surface that emanates from $x_{B}(t)$ and crosses to $x=0$ with
$\dot{x}=v_{B}$, before dropping down vertically to $t=0$ through the gap
between $Q$ and $B(t)$. This gives a contribution to the entropy of
$S_{BH}-\alpha s_{eq}t$ from the surface passing through $B(t)$ and a
contribution $S_{Q}=s_{eq}p$ from the membrane cutting the initial state at
$x=0$.

The entropy is given by the minimum of these two surfaces, i.e.

$\displaystyle S(R(t))$ $\displaystyle=\alpha s_{eq}t,$ $\displaystyle t\leq
t_{P}+\Delta t,$ $\displaystyle=S_{Q}+S_{BH}-\alpha s_{eq}t,$ $\displaystyle
t\geq t_{P}+\Delta t,$ (8)

where $t_{P}=L/2\alpha$ is the Page time and $\Delta t=p/2\alpha$. We can
compute $S(B(t))$ in an entirely analogous way, with the relevant surfaces now
shown in Fig. 4. Computing the entropy cost of each membrane and choosing the
smaller we find

Figure 4: Extremal membranes for $S(B(t))$ for a young black hole entangled
with a reference system.

$\displaystyle S(B(t))$ $\displaystyle=S_{Q}+\alpha s_{eq}t,$ $\displaystyle
t\leq t_{P}-\Delta t,$ $\displaystyle=S_{BH}-\alpha s_{eq}t,$ $\displaystyle
t\geq t_{P}-\Delta t.$ (9)

The above results precisely match equations (2.25) and (2.26) of Liu_2021 ,
when applied to the microscopic model of a large-$q$ spin chain considered
there (for comparison in the notation of Liu_2021  one has $\alpha=1$,
$s_{eq}=\ln q$, $S_{BH}=k$). They imply the mutual informations have the form:

$\displaystyle I(Q,R(t))$ $\displaystyle=0,$ $\displaystyle t\leq t_{P}-\Delta
t,$ $\displaystyle=S_{Q}-S_{BH}+2\alpha s_{eq}t,$ $\displaystyle t_{P}-\Delta
t\leq t\leq t_{P}+\Delta t,$ $\displaystyle=2S_{Q},$ $\displaystyle t\geq
t_{P}+\Delta t,$ (10)

and similarly

$\displaystyle I(Q,B(t))$ $\displaystyle=2S_{Q},$ $\displaystyle t\leq
t_{P}-\Delta t,$ $\displaystyle=S_{Q}+S_{BH}-2\alpha s_{eq}t,$ $\displaystyle
t_{P}-\Delta t\leq t\leq t_{P}+\Delta t,$ $\displaystyle=0,$ $\displaystyle
t\geq t_{P}+\Delta t.$ (11)

Figure 5: Mutual information between reference $Q$ and radiation/black hole
subsystems in a model of an evaporating black hole. Here the parameters have
been set to $p=5$, $L=100$, $\alpha=0.1$, $t_{P}=500$.

These results show how the information about the reference system $Q$
initially contained in the black hole subsystem is rapidly transferred to the
radiation subsystem around the Page time in a time of order $\Delta t$ (see
Figure 5). As mentioned above, this is the expected behaviour implied by the
arguments of Hayden and Preskill for a young121212Here young refers to the
fact the the information is encoded in the black hole significantly before the
Page time. black hole Hayden_2007 . Here we see how, in the context of our toy
models, this transfer of information is associated to the transition between
different minimal membranes in the computations of $S(B(t))$ and $S(R(t))$.

##  4 Model of an eternal black hole

#####

We now consider information dynamics using the entanglement membrane in the
context of a toy model of an eternal black hole coupled to a bath. The Page
curve in this setting was originally studied in Almheiri:2019yqk , and our
model is closely related to a toy dynamical model of an eternal black hole
studied in Liu_2021 . In particular, we wish to apply the entanglement
membrane to study information dynamics in an inhomogeneous system consisting
of two black hole regions $B_{1},B_{2}$ and two bath/radiation regions
$R_{1},R_{2}$ in the following one dimensional system131313The equilibrium
entropy of the bath $s_{eq}L$ should be viewed as parametrically larger than
the black hole entropy $S_{BH}=\tilde{s}_{eq}a$. In all cases we discuss we
take $S_{BH}$ sufficiently large that the Page time is parametrically larger
than the other timescales that appear in this section.:

  1. 1.

A bath region $R_{1}$ in the interval $[-(L+a),-a]$ and $R_{2}$ in $[a,L+a]$,
with $a\ll L$. The bath regions consist of a local chaotic many-body system
with equilibrium entropy density $s_{eq}$ and entanglement line tension ${\cal
E}(v)$.

  2. 2.

Two black hole regions $B_{1}$ in the interval $[-a,0]$ and $B_{2}$ from
$[0,a]$ consisting of a second local chaotic many-body system with a
parametrically larger equilibrium entropy density $\tilde{s}_{eq}\gg s_{eq}$
and an entanglement line tension $\tilde{{\cal E}}(v)$.

A concrete example to have in mind is a random circuit model, in which the
bath regions $R_{1},R_{2}$ consist of spins with a $q$-dimensional Hilbert
space with equilibrium entropy density $s_{eq}=\ln q$ and the black hole
regions $B_{1},B_{2}$ consist of spin with a $q^{N}$-dimensional Hilbert space
($N\gg 1)$ and equilibrium entropy density $\tilde{s}_{eq}=N\ln q$.

#####

We assume that the left(right)-hand regions $R_{1}(R_{2})$ and $B_{1}(B_{2})$
are coupled locally by the dynamics. However there are no interactions between
the left hand system $R_{1},B_{1}$ and the right hand system $R_{2},B_{2}$.
The geometry within which we can apply the entanglement membrane is then
illustrated in e.g. Figure 6, where we emphasise that the entanglement line
tension and equilibrium density is different in the black hole and bath
regions. Since we are interested in modelling an eternal black hole, we assume
that at $t=0$ the black hole regions $B_{1}$ and $B_{2}$ are in a thermofield
double state (or maximally entangled if a random circuit model) with
equilibrium density $\tilde{s}_{eq}$. In contrast for the purposes of the
discussion in the main text we will assume that the bath regions $R_{1},R_{2}$
are initially in a product state at $t=0$141414It is possible to also study
the entanglement dynamics when $R_{1}$ and $R_{2}$ initially have non-trivial
entanglement, as was done in the model studied in Section III of Liu_2021  and
we discuss in Appendix A..

#####

Note that in contrast to the related examples studied in Almheiri:2019yqk ;
Liu_2021 , where the black hole regions were 0+1 dimensional systems, our
black hole regions $B_{1},B_{2}$ have a finite width $a\ll L$ and local
dynamics in the black hole region. These assumptions are necessary in order to
be able to apply the entanglement membrane in these regions. However we will
see that the limit $a\to 0$ with $S_{BH}=\tilde{s}_{eq}a$ fixed can be freely
taken in our results for both the Page curve and Hayden-Preskill protocol
below - these do not depend on the details of the black hole dynamics, but
only the total entropy of the black hole region.

### Page curve for an eternal black hole

Figure 6: Surfaces contributing to the entropy of the bath $S(R,t)$ in our toy
model of an eternal black hole.

#####

We begin by computing the entanglement entropy of the bath region $R=R_{1}\cup
R_{2}$ in our model of an eternal black hole. At time $t=0$ this is zero since
we have assumed that $R_{1}$ and $R_{2}$ are initially in a pure state. The
relevant entanglement membranes for computing this entropy are shown in Figure
6. They consist of:

  1. 1.

A membrane which consists of two disconnected vertical lines near $x=\pm a$.
These lines should be viewed as lying infinitessimally inside the radiation
regions $R_{1}$ and $R_{2}$, as vertical lines inside the black hole regions
would have a larger entropy cost due to the parametrically larger entropy
density inside the black hole $\tilde{s}_{eq}\gg s_{eq}$151515Here we are
assuming the entanglement velocities for the bath and black hole regions are
comparable, as is the case for a random circuit model built of spin-$q$
(spin-$q^{N}$) degrees of freedom in the bath (black hole) regions.. The
entropy cost of this membrane is then $S(R,t)=2s_{eq}{\cal
E}(0)t=2s_{eq}v_{E}t$, with $v_{E}$ the entanglement velocity of the bath
region.

  2. 2.

A closed surface inside the black holes formed from straight lines of
velocities $\dot{x}=\pm\tilde{v}_{B}$, with $\tilde{v}_{B}$161616This should
not be viewed as the butterfly velocity of a holographic theory. the butterfly
velocity in the black hole region. This contributes an entropy $S=2S_{BH}$,
where $S_{BH}=\tilde{s}_{eq}a$.

Picking the entropy of the minimal membrane we then find

$\displaystyle S(R,t)$ $\displaystyle=2s_{eq}v_{E}t,$ $\displaystyle t\leq
t_{P},$ $\displaystyle=2S_{BH},$ $\displaystyle t\geq t_{P},$ (12)

where the Page time $t_{P}=S_{BH}/(s_{eq}v_{E})$ is when the two membranes
exchange dominance. Note that for the concrete case of a large $q$ random
quantum circuit model discussed above, then we have $s_{eq}=\ln q$ and
$S_{BH}=k\ln q$ where $k=aN$. The Page time would then evaluate to
$t_{P}=k/v_{E}$, where the entanglement velocity depends on the entropy being
computed and the precise circuit. Our results for $S_{2}$ for a large-$q$
brickwork circuit (for which $v_{E}=1$) then coincide with those obtained in
Liu_2021  for a related dynamical model of an eternal black hole using void
formation.

#####

Before moving on to discuss the Hayden-Preskill protocol in our model of an
eternal black hole, we note it was argued in Liu_2021  that void formation
leads to the generation of mutual information $I(R_{1},R_{2},t)$ between the
radiation regions around the Page time. In Appendix A we show how this physics
arises from the point of view of the entanglement membrane.

###  4.1 Information transfer from inside an eternal black hole

Figure 7: Illustration of the space-time set up for studying information
transfer in our model of an eternal black hole. The dotted line is to indicate
that the region $B_{1}$ has been extended by additional degrees of freedom
$P\subset B_{1}$.

#####

We now turn to study the Hayden-Preskill protocol in our model of an eternal
black hole. We begin by studying how information initially encoded inside the
black hole is transferred to the radiation/bath regions over time. Motivated
by the analysis of Liu_2021 , we add a small number of additional degrees of
freedom to the black hole $B_{1}$ by expanding its width to $a+p$, where $p\ll
a$. We will denote these additional black-hole degrees of freedom as $P\subset
B_{1}$. We then consider the initial state of $P$ to be entangled with a
reference system $Q$, such that $S(Q)=S(P)=\tilde{s}_{eq}p$ at $t=0$.
Otherwise, the set up is the same as in the previous subsection - i.e. the
remaining spins in $B_{1}$ are initially in a thermofield double state with a
second black hole region $B_{2}$ of width $a$ with entropy density
$\tilde{s}_{eq}$, and the radiation regions $R_{1}(R_{2})$, are taken to
initially be in product states.

#####

A diagrammatic representation of the space time slab in which we will apply
the entanglement membrane is shown in Figure 7. Note that the system consists
of three subsystems $(R_{1},B_{1})$, $(R_{2},B_{2})$ and $Q$ which are
decoupled (through interactions) but have non-trivial entanglement between
them. We can consider arranging these subsystems in any order - for the
purpose of drawing entanglement membranes we find it convenient to place $Q$
in between the left and right black hole/bath subsystems as shown in the
figures.

#####

As in Section 3.1 we are interested in computing the mutual informations
$I(Q,B,t)$ and $I(Q,R,t)$, where $B=B_{1}\cup B_{2}$ and $R=R_{1}\cup R_{2}$.
We have

$\displaystyle I(Q,R,t)$ $\displaystyle=$ $\displaystyle S(Q)+S(R,t)-S(Q\cup
R,t),$ (13) $\displaystyle=$ $\displaystyle S_{Q}+S(R,t)-S(B,t),\hskip
85.35826ptS_{Q}=\tilde{s}_{eq}p$

and an identical expression for $I(Q,B,t)$ by exchanging $R$ and $B$. We again
just need to compute the entropies $S(R,t)$ and $S(B,t)$.

Figure 8: Extremal surfaces for entropy of bath regions $S(R,t)$ in our model
of an eternal black hole.

#####

The behaviour of $S(R,t)$ can be understood from the entanglement membranes in
Fig. 8, consisting of:

  1. 1.

Two vertical surfaces at the edges of each bath each contributing an entropy
$s_{eq}v_{E}t$. As in the previous section these surfaces should be viewed as
lying infinitessimally inside the bath regions.

  2. 2.

A surface that cuts across the black hole regions ${B}_{1}$, ${B}_{2}$ with
velocities $\dot{x}=\pm\tilde{v}_{B}$ and then drops down to $t=0$ through the
gaps in the space-time slab around the reference system $Q$. This surface
contributes an entropy $2S_{BH}+2S_{Q}$. Note one factor of $S_{Q}$ is arising
due to the addition of the extra degrees to freedom to $B_{1}$, and the second
factor of $S_{Q}$ from the fact the membrane encloses the region $Q$ in the
initial state.

Picking the minimal surface gives

$\displaystyle S(R,t)$ $\displaystyle=2s_{eq}v_{E}t,$ $\displaystyle t\leq
t_{P}+\Delta t,$ $\displaystyle=2S_{BH}+2S_{Q},$ $\displaystyle t\geq
t_{P}+\Delta t,$ (14)

where $t_{P}=S_{BH}/(s_{eq}v_{E})$ is the Page time and $\Delta
t=S_{Q}/(s_{eq}v_{E})\ll t_{P}$.

#####

Likewise we find the behaviour of $S(B,t)$ from the following entanglement
membranes (see Fig. 9):

Figure 9: Extremal surfaces for the entropy of black hole region $S(B,t)$ in
our model an eternal black hole.

  1. 1.

A disjoint surface consisting of four vertical surfaces near the edges of
$B_{1}$ and $B_{2}$. These surfaces should be viewed as two vertical surfaces
infinitesimally inside the bath regions, each contributing an entropy
$s_{eq}v_{E}t$, and two interior surfaces that run down to the initial state
through the gaps in the space time slab. The total entropy is then
$2s_{eq}v_{E}t+S_{Q}$, with the factor of $S_{Q}$ coming from the region
enclosed by the membrane in the initial state.

  2. 2.

The join of two surfaces that cut across the interiors of the regions
${B}_{1}$ and ${B}_{2}$ with $\dot{x}=\pm\tilde{v}_{B}$ contributing a total
entropy $2S_{BH}+S_{Q}$.

We therefore find

$\displaystyle S(B,t)$ $\displaystyle=$ $\displaystyle
2s_{eq}v_{E}t+S_{Q},\hskip 105.2751ptt\leq t_{P},$ (15) $\displaystyle=$
$\displaystyle 2S_{BH}+S_{Q},\hskip 113.81102ptt\geq t_{P}.$

From which we obtain the mutual informations

$\displaystyle I(Q,R,t)$ $\displaystyle=$ $\displaystyle 0,\hskip
176.407ptt\leq t_{P},$ (16) $\displaystyle=$ $\displaystyle
2s_{eq}v_{E}t-2S_{BH},\hskip 105.2751ptt_{P}\leq t\leq t_{P}+\Delta t,$
$\displaystyle=$ $\displaystyle 2S_{Q},\hskip 162.18062ptt\geq t_{P}+\Delta
t,$

and similarly

$\displaystyle I(Q,B,t)$ $\displaystyle=$ $\displaystyle 2S_{Q},\hskip
162.18062ptt\leq t_{P},$ (17) $\displaystyle=$ $\displaystyle
2S_{Q}+2S_{BH}-2s_{eq}v_{E}t,\hskip 73.97733ptt_{P}\leq t\leq t_{P}+\Delta t,$
$\displaystyle=$ $\displaystyle 0,\hskip 176.407ptt\geq t_{P}+\Delta t.$

From which we see that the information in $Q$ is transferred rapidly (in a
time $\Delta t$) from the black hole to the bath/radiation around the Page
time. Note our results precisely match those of Liu_2021  when applied to the
case of $S_{2}$ for a brickwork large-$q$ quantum circuit. This can be seen by
noting that in the context of the model in Liu_2021  one has $s_{eq}=\ln q$,
$v_{E}=1$, $S_{BH}=k\ln q$ and hence $t_{P}=k$, $\Delta t=p$.

###  4.2 Information transfer from outside an eternal black hole

Figure 10: Simple diagrams contributing to entropy of bath $S(R,t)$ during
transfer of information from outside an eternal black hole.

#####

We now turn to consider what happens if the information contained in the
reference system $Q$ is entangled with bath degrees of freedom outside the
eternal black hole. We will see that the coupling between the bath and the
black hole means that some of this information is rapidly transferred to the
black hole region, and the precise details of this information transfer have
some model dependence. However in the cases we study we find this information
is rapidly recovered around the Page time, consistent with the expectations of
Hayden_2007 .

#####

To this end we consider extending the bath region $R_{1}$ by length $p$, and
again refer to these new degrees of freedom as subsystem $P$. We will take the
region $P$ to be located a distance $l$ away from the interface between
$R_{1}$ and the black hole region $B_{1}$. We then again consider an initial
state in which the new bath degrees of freedom in $P$ are entangled with a
reference system $Q$, such that initially $S(P)=S(Q)=s_{eq}p$ and $P$ has
uniform entropy density $s_{eq}$. As before, we take the the bath degrees of
freedom outside of $P$ to initially be in a product state, and the black hole
regions $B_{1},B_{2}$ to be initially in a thermofield double state with
density $\tilde{s}_{eq}$.

#####

We are interested in computing the mutual information $I(Q,R,t)$. We will see
that the early time behaviour, describing the transfer of information to the
black hole region, is sensitive to two particular timescales. In particular
let us first define $t_{\mathrm{signal}}=(l+p)/v_{B}$, with $v_{B}$ the
butterfly velocity in the bath region. This corresponds to the time required
for a signal emitted from the edge of $B_{1}$ to reach the left hand side of
the region $P$. Further, we define $t_{\mathrm{sat}}=l/v_{E}$. We will see the
early time behaviour of the mutual information differs depending on which of
these timescales is larger.

####  4.2.1 $t_{\mathrm{sat}}<t_{\mathrm{signal}}$

Figure 11: Membranes contributing to $S(R,t)$ at intermediate times during the
Hayden-Preskill protocol for our model of an eternal black hole. The membranes
are shown for three qualitatively different times satisfying (a)
$t_{1}<t_{\mathrm{signal}}$, (b) $t_{\mathrm{sat}}<t_{2}<t_{\mathrm{signal}}$,
(c) $t_{3}>t_{\mathrm{signal}}$.

#####

First, we will consider the case when $t_{\mathrm{sat}}<t_{\mathrm{signal}}$,
i.e. we have $l/v_{E}<(l+p)/v_{B}$. In this case there are four different
types of membrane to consider that are relevant for the entropy $S(R,t)$. The
first two are shown in Figure 10, and are analogous to cases we have studied
previously. They contribute $S_{Q}+2s_{eq}v_{E}t$ (shown in red, which gives
the early time behaviour of $S(R,t)$) and $S_{Q}+2S_{BH}$ (shown in blue,
which gives the late time saturation value of $S(R,t)$).

#####

However, at intermediate time-scales there are additional membranes that
become minimal, due to the inhomogeneous entropy profile of the initial state
due to the entanglement between $P$ and $Q$. The first intermediate time
membrane is shown as the orange membrane in Figure 11, shown for three
qualitatively different times. This membrane corresponds to a straight line
with gradient $\dot{x}=v_{B}$ from the interface between $B_{1}$ and $R_{1}$
into the bath region. This results in a cut along the bottom of our spacetime
a distance $v_{B}t$ away the black hole region. The contribution of this curve
is different depending on the value of $t$, and we find it becomes minimal for
$l/v_{E}<t<(l+p)/v_{B}$. In particular, focusing for simplicity on the
contributions to the entropy from the part of the membrane in the left hand
system $(R_{1},B_{1})$ we have:

  1. 1.

For $t<l/v_{B}$ this part of the membrane contributes an entropy
$s_{eq}v_{B}t$ from the slope part plus an entropy $S_{Q}$ from the fact the
membrane cuts the initial state to the right of $P$. This gives a total
entropy $S_{Q}+s_{eq}v_{B}t>S_{Q}+s_{eq}v_{E}t$ and hence this cut is not
minimal.

  2. 2.

For $l/v_{B}<t<(l+p)/v_{B}$ we again have a contribution $s_{eq}v_{B}t$ from
the slope, but now the cut across the initial state lies in the region $P$.
The entropy across the cut is now given by $S_{Q}-s_{eq}(v_{B}t-l)$ and hence
the total contribution of this surface is $S_{Q}+s_{eq}l$ which is $t$
independent. For $t>l/v_{E}$ this is less than $S_{Q}+s_{eq}v_{E}t$ coming
from a vertical line and hence this becomes part of the minimal surface.

  3. 3.

For $t>(l+p)/v_{B}$ there is a contribution $s_{eq}v_{B}t$ from the slope, but
now the cut lies to the left of region $P$ and there is no contribution to the
entropy cost from the initial state. This part of the membrane therefore has
an entropy cost $s_{eq}v_{B}t$.

For $t>(l+p)/v_{B}$ however, the above membrane is never minimal, as can be
seen from considering the entropy of a membrane beginning at the edge of
$B_{1}$ at time $t$ and ending on the left hand edge of $P$ at $t=0$ (shown in
magenta in Figure 11). Such a membrane has a velocity
$v_{*}(t)=(l+p)/t<v_{B}$, which contributes an entropy $s_{eq}{\cal
E}(v_{*}(t))t$. Note that for ${t_{\mathrm{sat}}}<{t_{\mathrm{signal}}}$ it is
simple to show that convexity and monotonicity of the line tension ensure that
$s_{eq}{\cal E}(v_{*}(t))t$ is less than the contributions of all other
surfaces considered above until the Page time, as we show in Appendix B.

#####

Taking into account that for all membranes there is an additional vertical
piece of the membrane in the right hand subsystem contributing $s_{eq}v_{E}t$
to the entropy cost, we then find that the entropy of the bath region is given
by the following expressions:

$\displaystyle S(R,t)$ $\displaystyle=S_{Q}+2{s_{eq}}v_{E}t,$ $\displaystyle
0\leq t\leq t_{\mathrm{sat}},$ $\displaystyle={s_{eq}}(p+l+v_{E}t),$
$\displaystyle t_{\mathrm{sat}}\leq t\leq t_{\mathrm{signal}},$
$\displaystyle={s_{eq}}\mathcal{E}(v_{*}(t))t+{s_{eq}}v_{E}t,$ $\displaystyle
t_{\mathrm{signal}}\leq t\leq t_{P}+\Delta t/2,$
$\displaystyle=2S_{BH}+S_{Q},$ $\displaystyle t\geq t_{P}+\Delta t/2.$ (18)

The entanglement entropy of the black hole is simpler to compute, receiving
contributions from surfaces analogous to those in Figure 10, without the
vertical cut shown near the reference system $Q$ at the left hand edge of
diagram. We find

$\displaystyle S(B,t)$ $\displaystyle=2{s_{eq}}v_{E}t,$ $\displaystyle 0\leq
t\leq t_{P},$ $\displaystyle=2S_{BH},$ $\displaystyle t\geq t_{P}.$ (19)

From which we can deduce the mutual information between the bath and the
reference system is given by171717Strictly speaking the behaviour for
$t_{P}<t<t_{P}+\Delta t/2$ is given by
$S_{Q}+{s_{eq}}v_{E}t+{s_{eq}}\mathcal{E}(v_{*}(t))t-2S_{BH}$. However for
$t>t_{P}$ we have $v_{*}\approx 0$, giving the results stated.

$\displaystyle I(Q,R,t)$ $\displaystyle=2S_{Q},$ $\displaystyle t\leq
t_{\mathrm{sat}},$ $\displaystyle=2S_{Q}+{s_{eq}}(l-v_{E}t),$ $\displaystyle
t_{\mathrm{sat}}\leq t\leq t_{\mathrm{signal}},$
$\displaystyle=S_{Q}+{s_{eq}}(\mathcal{E}(v_{*}(t))-v_{E})t,$ $\displaystyle
t_{\mathrm{signal}}\leq t\leq t_{P},$
$\displaystyle=S_{Q}+2{s_{eq}}v_{E}t-2S_{BH},$ $\displaystyle t_{P}\leq t\leq
t_{P}+\Delta t/2,$ $\displaystyle=2S_{Q},$ $\displaystyle t\geq t_{P}+\Delta
t/2.$ (20)

#####

To illustrate the above behaviour we can consider the von Neumann entropy
$I_{vN}(Q,R,t)$ of a large $q$ random circuit with Poisson dynamics, for which
we have ${\cal E}(v)=(1+v^{2})/2$ implying that $v_{E}=1/2$ and $v_{B}=1$.
Then $\mathcal{E}(v_{*}(t))t-v_{E}t=(l+p)^{2}/t$. The mutual information is
then shown in Figure 12 for a generic choice of parameters
$l=10,p=15,t_{P}=400$. We see that the approximately half the mutual
information initially inside the bath is rapidly transferred to the black hole
after $t_{\mathrm{sat}}$. The initial decay of $I(Q,R,t)$ is linear for
$t_{\mathrm{sat}}<t<t_{\mathrm{signal}}$ but then becomes a power law
asymptoting to $S_{Q}$ for $t>t_{\mathrm{signal}}$. In particular, for
timescales $t\gg t_{\mathrm{signal}}$ approximately half the information has
been transferred to the black hole. This information however is then rapidly
returned to the radiation around the Page time, over a timescale of order
$\Delta t$.

#####

Similar behaviour can be seen if one computes the second Renyi entropy
$I_{2}(Q,R,t)$ for a large-$q$ brickwork random circuit, noting that in this
case ${\cal E}(v)\to 1$, $v_{E}\to 1$, $v_{B}\to 1$ as $q\to\infty$. The
results are qualitatively similar to those of the von Neumann entropy for the
Poisson model, although in this case the flatness of the line tension means
that the mutual information is flat $I_{2}(Q,R,t)=S_{Q}$ for
$t_{\mathrm{signal}}<t<t_{P}$. Our results for $I_{2}(Q,R,t)$ agree precisely
with those presented in Liu_2021 , which studied the Hayden-Preskill protocol
in a related model using void formation, and are illustrated in the right hand
panel of Figure 12 for $l=10,p=15,t_{P}=400$.

Figure 12: Mutual information between reference system $Q$ and bath $R$ during
the Hayden-Preskill protocol in models of an eternal black hole built from
random large $q$ circuits. The left hand plot shows $I_{vN}(Q,R,t)/s_{eq}$ in
a Poisson circuit for $l=10,p=15,t_{P}=400$. The right hand plot shows
$I_{2}(Q,R,t)/s_{eq}$ in a brickwork circuit for $l=10,p=15,t_{P}=400$.

####  4.2.2 ${t_{\mathrm{signal}}}<{t_{\mathrm{sat}}}$

#####

We now consider the case ${t_{\mathrm{signal}}}<{t_{\mathrm{sat}}}$, i.e.
$(l+p)/v_{B}<l/v_{E}$. In this case, the orange membrane above is never
minimal since its entropy only become less than the red membrane for
$t>{t_{\mathrm{sat}}}$, by which time we are now guaranteed that the magenta
membrane exists and has a smaller entropy. The behaviour of the mutual
information is now determined by a competition between the red and magenta
membranes. For a generic line tension ${\cal E}(v)$ we then find the behaviour

$\displaystyle S(R,t)$ $\displaystyle=S_{Q}+2{s_{eq}}v_{E}t,$ $\displaystyle
0\leq t\leq t^{*},$
$\displaystyle={s_{eq}}\mathcal{E}(v_{*}(t))t+{s_{eq}}v_{E}t,$ $\displaystyle
t^{*}\leq t\leq t_{P}+\Delta t/2,$ $\displaystyle=2S_{BH}+S_{Q},$
$\displaystyle t\geq t_{P}+\Delta t/2,$ (21)

and

$\displaystyle I(Q,R,t)$ $\displaystyle=2S_{Q},$ $\displaystyle 0\leq t\leq
t^{*},$ $\displaystyle=S_{Q}+s_{eq}(\mathcal{E}(v_{*}(t))-v_{E})t,$
$\displaystyle t^{*}\leq t\leq t_{P},$
$\displaystyle=S_{Q}+2{s_{eq}}v_{E}t-2S_{BH},$ $\displaystyle t_{P}\leq t\leq
t_{P}+\Delta t/2$ $\displaystyle=2S_{Q},$ $\displaystyle t>t_{P}+\Delta t/2.$
(22)

were $t^{*}$ is the timescale at which the red and magenta membranes exchange
dominance. This is the smallest timescale such that

$s_{eq}(\mathcal{E}(v_{*}(t))-v_{E})t\leq S_{Q},$ (23)

For the special case of the von Neumann entropy of a large-$q$ random circuit
with Poisson dynamics, i.e. with line tension $\mathcal{E}(v)=(1+v^{2})/2$,
one has $t^{*}=(l+p)^{2}/2p$ and the mutual information is given by

$\displaystyle I_{vN}(Q,R,t)$ $\displaystyle=2S_{Q},$ $\displaystyle 0\leq
t\leq t^{*},$ $\displaystyle=S_{Q}+{s_{eq}}(l+p)^{2}/2t,$ $\displaystyle
t^{*}\leq t\leq t_{P},$ $\displaystyle=S_{Q}+{s_{eq}}t-2S_{BH},$
$\displaystyle t_{P}\leq t\leq t_{P}+\Delta t/2,$ $\displaystyle=2S_{Q},$
$\displaystyle t\geq t_{P}+\Delta t/2.$ (24)

We therefore see that there is a non-trivial crossover between the magenta and
red surfaces at the intermediate timescale $t^{*}$, which is responsible for
the transfer of information to the black hole. That information is then
recovered from the black hole rapidly around the Page time. Qualitatively
similar behaviour is obtained for $S_{2}$ in the brickwork quantum circuit
using the line tension (5).

#####

The above behaviour is generic to any line tension ${\cal E}(v)$ for which
there is a non-trivial cross-over between the red and magenta membranes at
time $t^{*}\ll t_{P}$. It is simple to check, that (23) will be satisfied at
sufficiently large $t$ provided that ${\cal E}(v)$ is analytic at $v=0$. This
can be seen from the fact that with these assumptions then at sufficiently
large $t$ (i.e. sufficiently small $v_{*}(t)$) the line tension can be Taylor
expanded as

${\cal E}(v_{*}(t))\approx v_{E}+{\cal E}^{\prime\prime}(0)(l+p)^{2}/2t^{2}$
(25)

where we have used that since ${\cal E}(v)$ is even we must have ${\cal
E}^{\prime}(0)=0$ provided the line tension is analytic at $v=0$. However it
is interesting to note that such a crossover would not be seen if the bath was
described by a non-analytic line tension of the form
$\mathcal{E}(v)=v_{E}+|v|(1-v_{E}/v_{B})$, as we show in Appendix B. Whilst we
are not aware of any microscopic system described by such a line
tension181818A system with such a line tension also saturates various bounds
on entanglement growth proposed in Mezei_2017 . it is noteworthy that in this
case we find that for ${t_{\mathrm{sat}}}>{t_{\mathrm{signal}}}$ the red
membrane remains minimal until the Page time, and as a result the information
encoded in the bath is never transferred to the black hole.

##  5 Discussion

#####

In this paper we have seen that the entanglement membrane provides a powerful
framework through which to understand entanglement dynamics in models of black
hole information built out of chaotic many-body quantum systems. Whilst the
models we studied were highly simplified descriptions of black hole dynamics,
we were able to reproduce the expected features of both the Page curve and the
Hayden-Preskill protocol. We emphasise that our results required very little
input as to the underlying nature of the microscopic physics, aside from the
assumptions of a system whose local chaotic unitary dynamics could be
describing using the entanglement membrane. The results we obtained for the
eternal black hole in Section 4 relied essentially only on this assumption and
a model of a black hole as a region of many, dense, degrees of freedom. As
such our results suggest these aspects of black hole information dynamics are
a consequence of generic features of entanglement dynamics in chaotic many-
body quantum systems.

#####

Furthermore, one key motivation for studying information dynamics in these
models using the entanglement membrane prescription is that it provides an
approach to studying the Page curve in these models that is reminiscent of
gravitational studies of entanglement dynamics using (quantum) extremal
surfaces Ryu_2006 ; Hubeny_2007 ; Engelhardt_2015 ; penington2020entanglement
; Almheiri_2019 ; Almheiri_2020 . Whilst we leave a detailed comparison to
future work, we wish to close this paper with some preliminary observations
about possible connections between the work in this paper and gravitational
approaches to studying entanglement dynamics.

#####

There already exists a well established connection between the entanglement
membrane and gravitational computations of entanglement entropies in
holographic quantum field theories using classical extremal surfacesMezei_2018
; Mezei_2020 . Specifically, the entanglement dynamics in the boundary of a
holographic quantum field theory admits a coarse-grained description in terms
of an entanglement membrane. In this context, the entanglement membrane
emerges as the projection of the classical bulk HRT surface into the boundary
theory. It has been established that in certain models of black hole
evaporation the Page curve can be derived simply from considering classical
HRT surfaces in a “doubly holographic” spacetime Almheiri_2020 ;
chen2020evaporating ; chen2020quantum1 ; chen2020quantum2 ; Geng:2020qvw ;
Geng:2020fxl . It would be interesting to study the projections of such
surfaces into the boundary, which could allow for a direct relationship
between the entanglement membrane studied here and the islands prescription in
gravity. Furthermore, currently the entanglement membrane has only been
derived from holography in the case where the bulk theory is classical. It
would be interesting to understand if one can provide a similar derivation of
the entanglement membrane in the case where the gravitational extremal surface
problem includes the effects of quantum corrections Engelhardt_2015 .

#####

Moreover, it would be interesting to explore the relationship between replica
wormholes and permutation degrees of freedom that are naturally connected to
the entanglement membrane proposal Vardhan_2021 ; PhysRevX.10.031066 ;
Zhou_2019 . In particular, in large-$q$ random quantum circuits it has been
established that the entanglement membrane can be understood as a domain wall
between regions of the circuit in which gates in different replica copies are
identified according to different permutations of the replicas
PhysRevX.10.031066 ; Zhou_2019 . Similar permutation degrees of freedom have
also been identified as governing the entanglement entropies of thermalized
states in general many-body systems including quantum field theories
Vardhan_2021 . The connection between the entanglement membrane and such
permutation degrees of freedom is most easily illustrated by considering the
computation of the second Renyi entropy

$S_{2}(A,t)=-\log\mathrm{Tr}_{A}(\rho^{2}_{A}(t)),\hskip
113.81102pt\rho_{A}(t)=\mathrm{Tr}_{\bar{A}}\ket{\psi(t)}\bra{\psi(t)},$ (26)

where $A$ is a subregion of the systems discussed in Sections 3 and 4 and
${\bar{A}}$ its complement.

#####

In order to compute such a Renyi entropy we require two copies of the density
matrix, which can be thought of as a state in a four copy Hilbert space with
time evolution generated by $U_{1}\otimes U_{1}^{\dagger}\otimes U_{2}\otimes
U_{2}^{\dagger}$, with $U_{i},U_{i}^{\dagger}$ built from independently drawn
random unitaries (gates). Then the insight of PhysRevX.10.031066 ; Zhou_2019
is that after averaging over random unitaries one obtains a description in
terms of a permutation degree of freedom $\sigma(t^{\prime},x)$, that
describes whether at each location in spacetime one identifies the gates
associated to different replicas pairwise through either
$U_{1}=U_{1}^{\dagger},U_{2}=U_{2}^{\dagger}$ or via
$U_{1}=U_{2}^{\dagger},U_{2}=U_{1}^{\dagger}$. We will refer to the first
configuration as identifying the gates using the identity permutation
$\sigma=e$ and the second via a swap configuration $\sigma=\eta$. The
entanglement membrane can then be understood as representing a domain wall
configuration across which the identification of unitaries changes from $e$ to
$\eta$ (or vice versa).

Figure 13: Permutations describing the identification between different
replicas in the computation of $S_{2}(R,t)$ in our model of an evaporating
black hole.

#####

It is instructive to illustrate the identification of the replicas in the
model of the evaporating black hole discussed in Section 3, where we take
$A=R(t)$ and $\bar{A}=B(t)$. As discussed in PhysRevX.10.031066 ; Liu_2021
the contractions implied by the trace in (26) imply boundary conditions at
$t^{\prime}=t$ on the permutation $\sigma$ such that $\sigma(t,x)=e$ in $B(t)$
and $\sigma(t,x)=\eta$ in $R(t)$. The permutations at times $t^{\prime}<t$ are
then determined dynamically by the entanglement membrane prescription. In
particular, the permutations associated with the two diagrams contributing to
the Page curve are shown in Figure 13. We note that the left hand diagram,
associated to the early time behaviour of the Page curve, corresponds to
identifying replicas at small $t^{\prime}$ according to $\sigma=e$, whilst the
late time behaviour is related to identifying them at small $t^{\prime}$ as
$\sigma=\eta$.

#####

The fact that the Page curve arises from a transition between different ways
of connecting multiple replicas is highly reminiscent of gravitational
computations of the second Renyi entropy, in which there is a transition
around the Page time between different ways of connecting the interiors of
black holes associated to each replica (so-called ‘replica wormhole’
geometries). Similar permutation degrees of freedom have been shown to govern
the entanglement entropies of thermalized states in generic many-body quantum
systems, and in this context a precise connection to replica wormholes has
been established in Vardhan_2021 . In our above discussion of an evaporating
black hole both the radiation and black hole systems are essentially
thermalized at all times, and the permutation degrees of freedom in Figure 13
can be identified with those obtained by the prescription of Vardhan_2021
191919Whilst we restricted to considering membranes inside the butterfly cone,
the membranes in Fig. 13 can be freely deformed outside the butterfly-cone
into horizontal membranes jonay2018coarsegrained . The permutations in Fig. 13
then become $e$ and $\eta$ respectively everywhere the bulk, which can be
interpreted as the terms in (2.34) of Vardhan_2021  for $S_{2}$. We are
grateful to Shreya Vardhan for discussions on this point..

#####

However the entanglement membrane prescription is more general in the sense
that it can be applied to non-thermal states, which was relevant for our
discussion of the Page curve in the eternal black hole. For instance whilst
Vardhan_2021  was able to apply techniques developed for thermalized states to
obtain the Page curve from permutation degrees of freedom in a model of an
eternal black hole, this could only be done for the special case where the
bath regions $R_{1},R_{2}$ were built from a many-body system with ‘maximally
efficient thermalization’ corresponding to $v_{E}=v_{B}=1$. For the more
generic models studied here in Section 4, with an arbitrary line tension
${\cal E}(v)$ in the bath regions, the techniques of Vardhan_2021  cannot be
applied at times $t<t_{P}$. It would therefore be extremely interesting to
perform a more detailed study of the relationship between permutation degrees
of freedom and gravitational computations of replica wormholes in non-
thermalized states such as this in the future202020Given that the entanglement
membrane is formulated in a Lorentzian setting the most natural starting point
for such a comparison may be provided by Marolf_2021a ; Marolf_2021b ..

Figure 14: Permutations describing the identification between different
replicas in the computation of $S_{2}(R,t)$ in our model of an eternal black
hole.

###### Acknowledgements.

We are grateful to Hong Liu, Márk Mezei, Adam Nahum, Shreya Vardhan and Curt
von Keyserlingk for very helpful discussions. AT acknowledges support from UK
EPSRC (EP/SO23607/1).

##  Appendix A Mutual information of bath regions in eternal black hole

#####

In this appendix we wish to consider the mutual information $I(R_{1},R_{2},t)$
between the two bath regions in our model of an eternal black hole discussion
in Section 4. In particular, we note that Liu_2021  studied this (second
Renyi) mutual information in a similar dynamical model of an eternal black
hole using void formation, and found that the generation of mutual information
was sensitive to the entanglement structure of initial state of the bath
regions $R_{1},R_{2}$. Here we will see how the same physics arises using the
entanglement membrane.

#####

In particular, recall that the mutual information is defined as

$I(R_{1},R_{2},t)=S(R_{1},t)+S(R_{2},t)-S(R_{1}\cup R_{2},t).$ (27)

As in the main text we will take the black hole regions $B_{1}$ and $B_{2}$ to
be in a thermofield double state (or maximally entangled) with entropy density
$\tilde{s}_{eq}$. We now wish to compute (27) for two choices of initial state
in the bath region: (i) When $R_{1}$, $R_{2}$ are in a product state at $t=0$;
(ii) When $R_{1}$ and $R_{2}$ are in a thermofield double state with entropy
density $s_{eq}$.

#####

For the case $R_{1}$, $R_{2}$ are initially in a product state, we have
already determined $S(R_{1}\cup R_{2},t)$ in equation (12). Since by symmetry
we have $S(R_{1},t)=S(R_{2},t)$ we then only need to compute $S(R_{1},t)$. The
relevant entanglement membranes are

Figure 15: Candidate minimal surfaces for $S(R_{1},t)$.

  1. 1.

A vertical cut down from $x\approx-a$ to the initial state. This has entropy
$s_{eq}v_{E}t+S(-a,0)=s_{eq}v_{E}t$.

  2. 2.

A horizontal cut from $x=-a$ to $x=0$, joined to a vertical cut down to the
initial state through the gap in the space-time slab at $x=0$. This has
entropy $S_{BH}+S(0,0)=2S_{BH}$ where $S_{BH}=\tilde{s}_{eq}a$ as in the main
text.

Taking the minimal membrane one finds

$\displaystyle S(R_{1},t)$ $\displaystyle=s_{eq}v_{E}t,$ $\displaystyle t\leq
2t_{P},$ $\displaystyle=2S_{BH},$ $\displaystyle t\geq 2t_{P},$ (28)

with $t_{P}=S_{BH}/(s_{eq}v_{E})$ the Page time of the eternal black hole. The
mutual information $I(R_{1},R_{2},t)$ is then given by:

$\displaystyle I(R_{1},R_{2},t)$ $\displaystyle=0,$ $\displaystyle t\leq
t_{P},$ $\displaystyle=2s_{eq}v_{E}t-2S_{BH},$ $\displaystyle t_{P}\leq t\leq
2t_{P},$ $\displaystyle=2S_{BH},$ $\displaystyle t\geq 2t_{P}.$ (29)

#####

If we instead take $R_{1}$ and $R_{2}$ to initially be in a thermofield double
state, the computation of $S(R_{1}\cup R_{2},t)$ in equation (12) remains
unchanged, as the initial state entropy term continues to vanish for both the
surfaces contributing to (12). In contrast, the entropy cost of the surfaces
relevant for $S(R_{1},t)$ shown in Figure 15 are now modified to

  1. 1.

$s_{eq}v_{E}t+S(-a,0)=s_{eq}v_{E}t+s_{eq}L$ for the blue surface.

  2. 2.

$S_{BH}+S(0,0)=2S_{BH}+s_{eq}L$ for the red surface.

#####

For this initial state, the entropy of both these surfaces is now larger than
that of arising due to a membrane from $x=-a$ to the left hand edge of the
system with $\dot{x}=v_{B}$ (the yellow surface in Fig. 15). This has entropy
$s_{eq}L$ and remains the minimal membrane for all $t$. We therefore have a
constant entropy $S(R_{1},t)=s_{eq}L$, which results in the mutual information

$\displaystyle I(R_{1},R_{2},t)$ $\displaystyle=2s_{eq}L-2s_{eq}v_{E}t,$
$\displaystyle t\leq t_{P},$ $\displaystyle=2s_{eq}L-2S_{BH},$ $\displaystyle
t\geq t_{P}.$ (30)

The results (29) and (30) precisely match those in Section III of Liu_2021
for the appropriate entanglement velocity $v_{E}=1$.

##  Appendix B Minimal membranes for generic line tensions

#####

In this section we will justify various claims we made in our discussions of
the Hayden Preskill protocol in our model of the eternal black hole in Section
4.2.

#####

Firstly, we claimed that when ${t_{\mathrm{sat}}}<{t_{\mathrm{signal}}}$ the
behaviour of the entropy $S(R,t)$ and mutual information $I(Q,R,t)$ were
qualitatively the same for all forms of the line tension $\mathcal{E}(v)$. In
particular, we claimed that when $t>{t_{\mathrm{signal}}}$ the entropy of the
magenta membrane with $v_{*}(t)=(l+p)/t$ is always less than the orange or red
membranes. To show this we recall from the equations in (2) and (3) that the
line tension satisfies ${\cal E}(v)\leq{\cal E}(v_{B})=v_{B}$, where
generically the equality only holds for $|v|=v_{B}$. We therefore have that in
generic models $s_{eq}{\cal E}(v_{*}(t))t<s_{eq}v_{B}t$, implying that the
entropy of the magenta membrane is less than the orange membrane. Furthermore,
to compare the entropy of the magenta membrane to the vertical red membrane we
note that using the third inequality in (3) we have

$s_{eq}{\cal E}(v_{*}(t))t\leq
s_{eq}v_{E}t+s_{eq}(l+p)\bigg{(}1-\frac{v_{E}}{v_{B}}\bigg{)}=s_{eq}v_{E}t+S_{Q}+s_{eq}v_{E}({t_{\mathrm{sat}}}-{t_{\mathrm{signal}}}).$
(31)

Hence for ${t_{\mathrm{sat}}}<{t_{\mathrm{signal}}}$ we have that $s_{eq}{\cal
E}(v_{*}(t))t<s_{eq}v_{E}t+S_{Q}$, implying that the entropy of the magenta
membrane is less than the red membrane.

#####

Secondly, we noted for ${t_{\mathrm{sat}}}>{t_{\mathrm{signal}}}$ that if one
considers a non-analytic form of line tension the line tension
$\mathcal{E}(v)=v_{E}+|v|(1-v_{E}/v_{B})$ then for
${t_{\mathrm{sat}}}>{t_{\mathrm{signal}}}$ information initially encoded in
the bath is never transferred to the black hole. This follows from the fact
that for such a line tension we have that the inequality in (31) is saturated,
implying $s_{eq}{\cal E}(v_{*}(t))t>s_{eq}v_{E}+S_{Q}$ for
${t_{\mathrm{sat}}}>{t_{\mathrm{signal}}}$. As a result the red membrane
remains minimal until the Page time, and we find

$\displaystyle S(R,t)$ $\displaystyle=S_{Q}+2v_{E}{s_{eq}}t,$ $\displaystyle
t\leq t_{P},$ $\displaystyle=2S_{BH}+S_{Q},$ $\displaystyle t\geq t_{P},$

which implies that the mutual information $I(Q,R,t)=2S_{Q}$ for all time.

#####

Finally, throughout our discussion we argued that before the Page time the
minimal membrane is always one of the surfaces shown in the diagrams in
Section 4.2 (i.e. the red, orange or magenta curves). It is usually
straightforward to check that any other membrane would result in a larger
entropy cost than at least one of the membranes discussed in the main text.
However, the case where the membrane ends in the region $P$ requires a more
careful justification, due to the non-trivial entropy profile of the initial
state in this region. Therefore, suppose we consider a membrane of velocity
$v_{0}$ starting from the left hand edge of $B_{1}$ and intersecting $t=0$
somewhere in the region $P$. This requires $l/t\leq v_{0}\leq(l+p)/t$. The
entropy of such a membrane is

$s_{eq}{\cal E}(v_{0})t+s_{eq}(l+p-v_{0}t)=s_{eq}(l+p)+s_{eq}({\cal
E}(v_{0})-v_{0})t.$ (32)

For a generic line tension satisfying ${\cal E}^{\prime}(v_{B})=1$ and ${\cal
E}^{\prime\prime}(v)\geq 0$ (see equations (2) and (3)) we have that ${\cal
E}^{\prime}(v)<1$ for $v<v_{B}$. As such $({\cal E}(v_{0})-v_{0})$ is a
monotonically decreasing function, and so the smallest entropy cost is
obtained for a membrane with the largest possible choice of $v_{0}$. For we
therefore find that the minimal membrane intersecting $P$ at $t=0$ has
velocity $v_{B}$ for $t<{t_{\mathrm{signal}}}$ and velocity $v_{*}(t)$ for
$t>{t_{\mathrm{signal}}}$ as claimed.

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