aainstitutetext: Stanford Institute for Theoretical Physics, Stanford
University, 382 Via Pueblo Mall, Stanford, CA 94305-4060,
U.S.A.bbinstitutetext: Perimeter Institute for Theoretical Physics, Waterloo,
Ontario N2L 2Y5, Canada

# Non-local computation and the black hole interior

Alex May a Michelle Xu<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In a two sided black hole, systems falling in from opposite asymptotic regions
can meet inside the black hole and interact. This is the case even while the
two CFTs describing each asymptotic region are non-interacting. Here, we
explore these behind the horizon interactions in planar black holes, where we
can relate them to non-local quantum computations. This gives a quantum
circuit perspective on these interactions, which applies in arbitrary
dimension. We further revisit the connection discussed earlier between bulk
interactions in one sided AdS geometries and non-local computation, and
recycle some of our techniques to offer a new perspective on making that
connection precise.

##  1 Introduction

In the AdS/CFT correspondence, local bulk interactions are supported by local
interactions in a lower dimensional boundary description. How this can be
possible is puzzling, in a way that is most sharp in the context of two sided
black hole solutions. There, excitations originating in each of the two
asymptotic regions can meet and interact behind the black hole horizon. For
instance, we can imagine an observer, Alice, who is created by acting on the
left CFT, and a second observer Bob who is created by acting on the right CFT.
While the CFTs don’t interact, the bulk description of their time evolution
involves Alice and Bob meeting and interacting behind the horizon. After being
initially emphasized by Marolf and Wall marolf2012eternal , how to understand
these behind the horizon interactions as emerging from a boundary perspective
has remained puzzling.

In this article, we relate behind the horizon interactions to a circuit
construction in quantum information theory known as a non-local quantum
computation, illustrated in figure 1. Our discussion is closely related to
earlier work which argued bulk interactions in spacetimes with a single
asymptotically AdS boundary are supported by non-local computations in the CFT
may2019quantum ; may2020holographic ; may2021holographic ; dolev2022holography
. Our main contribution is to extend this to the setting of two sided black
hole geometries, where the tension between having local bulk interactions and
two boundary theories which do not interact becomes particularly sharp. Our
construction also works in arbitrary dimensions, extending earlier results
which apply to $2+1$ bulk dimensions.

The basic setting we study is of a planar AdSd+1 black hole, dual to two CFTs
on spatial $\mathbb{R}^{d-1}$ and placed in the thermofield double state. In
this setting, the entanglement wedge of a region defined by $X_{1}>0$ (and
including points from both CFTs) will include a portion of the black hole
interior hartman2013time . We leverage this geometrical observation and
entanglement wedge reconstruction to argue behind the horizon interactions can
be reproduced in the form of a non-local computation. More quantitatively, we
are interested in understanding how much entanglement is necessary to support
interaction within a bulk region of a given size. While our planar black hole
solutions have infinite entanglement between the two CFTs, we use end-of-the-
world brane geometries, dual to pairs of entangled BCFTs, to truncate the
geometry and render the entanglement finite. This also fixes the size of the
bulk subregion where interaction can happen. For large enough BCFTs, extremal
surfaces enclosing half spaces still reach into the black hole, but small
BCFTs have extremal surfaces that sit outside the hole and instead attach to
the ETW branes. In bulk $2+1$ dimensions we can solve for this transition,
which gives a quantitative perspective on how much entanglement is necessary
to support bulk interactions within a scattering region of a given size.

$\mathbf{U}$ (a)

$\mathbf{V}^{L}$$\mathbf{V}^{R}$$\mathbf{W}^{R}$$\mathbf{W}^{L}$ (b)

$\mathbf{V}^{L}$$\mathbf{V}^{R}$$\mathbf{W}^{R}$$\mathbf{W}^{L}$ (c)

Figure 1: (a) Circuit diagram showing the local implementation of a unitary in
terms of a unitary $\mathbf{U}$. (b) Circuit diagram showing the non-local
implementation of a unitary $\mathbf{U}$. The boundary description implies
that interactions in the interior of certain black hole solutions must be
reproducible in this form, using an entangled state with mutual information
set by the black hole area. c) For AdS geometries with a single asymptotic
boundary, bulk computations are naively related to the augmented non-local
computation shown here. Extra systems enter the circuit in the second round.

It is useful to compare our discussion to the earlier one may2019quantum ;
may2020holographic ; may2021holographic  addressing the single sided setting.
There, interactions in the bulk (figure 1(a)) were most easily related to
boundary processes with the form shown in figure 1(c), which we refer to as an
augmented non-local computation. Notice that compared to a standard non-local
computation, extra quantum systems appear in the second round operations.
While we could now choose to consider the (slightly more complicated) boundary
description, these ‘extra regions’ actually make the boundary circuit much
more difficult to understand and constrain. In particular, these circuits can
support interactions while having the reduced density matrix describing the
state held at the two input locations be unentangled dolev2022holography .
This obscures the role entanglement is playing in supporting local bulk
interaction. In the two sided setting, this complication is naturally removed,
and bulk interactions are directly related to the standard non-local
computation scenario.

To relate interactions in the one-sided setting to non-local computation,
may2019quantum ; may2020holographic ; may2021holographic  argued on heuristic
grounds that the (non-augmented) non-local computation was the correct
boundary model, and dolev2022holography  argued this can be made precise in a
lattice regularization of the CFT.111In more detail, the authors assume a
lattice approximation of the holographic CFT exists, and that this lattice
approximation captures bulk physics up to a radial cut-off set by the lattice
spacing. One interesting recent attempt to construct such lattice descriptions
is given in kohler2019toy ; apel2022holographic . Applied in AdS2+1, our
construction reaches the same conclusion by assuming the AdS/BCFT
correspondence rather than introducing a lattice. Both constructions can be
used to relate bulk computation to non-local computation exploiting finite
dimensional entangled states: dolev2022holography  inherits this from the
lattice description, while our approach uses the relation between smooth max
entropy and von Neumann entropy in holographic states bao2019beyond  to
justify a finite dimensional approximation.

With the relationship between bulk interaction and non-local computation in
hand, we can revisit a number of questions raised around this connection,
which we can now pose in the black hole setting and in arbitrary dimensions.
In particular, this connection implies that constraints on non-local
computation imply constraints on bulk interaction.222Unfortunately,
constraints on non-local computation are so far poorly understood, but it
remains an active area in quantum information theory. See may2022complexity
for a recent summary. This has potential, but not fully explored, implications
for constraints on computation in holographic spacetimes may2022complexity .
Conversely, whatever interactions can happen in the bulk within spacetime
regions of a given size must, assuming the AdS/CFT correspondence, be
implementable non-locally with an amount of entanglement set by the size of
the region. This has potential, but not understood, implications for non-local
computation. While we focus in this article on establishing the non-local
computation and bulk interaction in new settings, we briefly comment on these
directions in the discussion.

Examining the connection between non-local computation and bulk interactions
from the perspective of black holes and ETW branes also highlights an issue
relating to sub-AdS locality. In both the two-sided and one-sided cases, our
boundary description of the bulk interaction also includes a description of a
super-AdS sized subregion of the bulk, even when the interaction is localized
to a small region. In our context, this is enforced by a change in the minimal
extremal surface homologous to certain boundary subregions. In this sense, our
quantum circuit model of behind the horizon interactions can’t (at this stage)
isolate the boundary description of interactions happening within sub-AdS
scale bulk regions. The construction in dolev2022holography  also suffers from
this limitation. This is analogous to tensor network models of spatial slices
of the bulk geometry, which also struggle to capture sub-AdS locality
bao2015consistency ; pastawski2015holographic ; hayden2016holographic ;
yang2016bidirectional . However, the heuristic arguments of may2019quantum ;
may2021holographic ; may2022connected  suggest that there should be a non-
local computation description of sub-AdS scale bulk subregions.333In
particular, the connected wedge theorem of may2019quantum ; may2021holographic
; may2022connected  gives that the boundary entanglement is consistent with
small bulk subregions having a non-local computation description. This
motivates looking for constructions that can related the sub-AdS sized regions
to (non-augmented) non-local computation. We comment in the discussion on some
extensions of our setting that may allow this.

Outline of our article

In section 2.1 we describe the AdS/BCFT correspondence, our prescription for
relating entangled BCFT states to bulk AdS geometries. In section 2.2 we
review an earlier framework for relating quantum information processing tasks
in the bulk and boundary perspectives, into which we can fit our construction.

In section 3 we discuss behind the horizon interactions in the planar BTZ
black hole. Section 3.1 gives our general construction relating behind the
horizon interactions and non-local computation in the context of planar black
holes. Section 3.2 describe a more concrete setting, the planar BTZ black
hole, where more quantitative details can be determined.

In section 4 we point out that the entanglement between our BCFTs can be well
approximated by a state in a finite dimensional Hilbert space. This allows a
stronger connection between existing bounds on non-local computation given in
the quantum information literature, typically proven for finite dimensional
systems, and constraints on interactions happening in a holographic geometry.

In section 5 we revisit the global AdS2+1 setting, discussed in earlier works
may2019quantum ; may2020holographic ; may2021quantum ; dolev2022holography .
Here, we point out that the same ETW brane techniques allow a new perspective
on the relationship between bulk interactions and non-local computation. The
principle claim is that interactions happening in the bulk can be reproduced
as non-local computations, using entanglement related to the size of the
region in which they occur. For large regions, the entanglement needed
approaches the area of the region, up to an additive constant. In 5.4 we
briefly discuss how to extend the connection between non-local computation and
bulk interaction to global AdSd+1 for $d>2$.

In section 6 we conclude with some remarks and open questions.

Summary of notation:

Spacetime notation:

  * •

We use italic capital letters $\mathcal{A},\mathcal{B},\mathcal{C},\dots$ for
boundary spacetime regions.

  * •

The entanglement wedge of a boundary region $\mathcal{A}$ is denoted by
$E_{\mathcal{A}}$.

  * •

The quantum extremal surface associated to region $\mathcal{A}$ is denoted
$\gamma_{\mathcal{A}}$.

  * •

We use plain capital letters $A,B,C,\dots$ to refer to bulk spacetime regions.

  * •

We use $J^{\pm}(A)$ to denote the causal future or past of region $A$ taken in
the bulk geometry, and $\hat{J}^{\pm}(\mathcal{A})$ to denote the causal
future or past of $\mathcal{A}$ within the boundary spacetime.

Quantum notation:

  * •

We use capital letters to denote quantum systems $A,B,C,...$

  * •

We use boldface, script capital letters for quantum channels,
$\mathbfcal{N}(\cdot)$, $\mathbfcal{T}(\cdot)$,…

  * •

We use boldface capital letters to denote unitaries or isometries,
$\mathbf{U},\mathbf{V},...$

##  2 Preliminaries

###  2.1 AdS/BCFT primer

Our constructions exploit the AdS/BCFT correspondence
takayanagi2011holographic ; fujita2011aspects , which relates holographic CFTs
defined on manifolds with a boundary to AdS geometries ended by end-of-the-
world branes. One way to understand why these solutions appear in our
constructions is that we will consider the state on subregions of a CFT, then
want some controlled way to consider a purification of that subregion with a
well defined bulk geometry. Introducing CFT boundary conditions at the edge of
the subregion we are considering is one way to do this. In this brief section
we introduce the needed elements of AdS/BCFT.

A BCFT is a conformal field theory living on a manifold with boundary, along
with a conformally invariant boundary condition. For appropriate BCFTs, the
AdS/BCFT correspondence conjectures a bulk dual description, which consists of
an asymptotically AdS region along with an extension of the CFT boundary into
the bulk as an ETW brane. To avoid confusion with the bulk-boundary language
of the AdS/CFT correspondence, we will refer to the CFT boundary as the
_edge_. The bulk spacetime and brane are described by an action

$\displaystyle I_{\text{bulk}}+I_{\text{brane}}$ $\displaystyle=\frac{1}{16\pi
G_{N}}\int d^{d+1}x\,\sqrt{g}(R-2\Lambda+L_{\text{matter}})$
$\displaystyle\qquad+\frac{1}{8\pi
G_{N}}\int_{\mathcal{B}}d^{d}y\,\sqrt{h}(K+L_{\text{matter}}^{\mathcal{B}})\;,$
(1)

where $L_{\text{matter}}$ and $L_{\text{matter}}^{\mathcal{B}}$ are matter
Lagrangians for fields in the bulk and brane respectively. $R$ is the Ricci
curvature and $\Lambda$ the bulk cosmological constant, while $K$ is the trace
of the extrinsic curvature of the brane,

$K_{ab}=\nabla_{a}n_{b}\;,$ (2)

for outward normal $n_{j}$ to $\mathcal{B}$, and $a,b$ refer to brane
coordinates $y^{a}$. This action leads to Einstein’s equations in the bulk,
along with the boundary condition

$\displaystyle-\frac{1}{8\pi G_{N}}(K_{ab}-Kh_{ab})=T_{ab}^{\mathcal{B}}\;.$
(3)

This AdS/BCFT model should be understood as a bottom-up model of concrete
holographic dualities. For example one can set conformally invariant boundary
conditions on $\mathcal{N}=4$ SYM and study a holographic dual aharony2011near
; assel2011holographic ; bachas2018massive ; van2021finding . In these models
the bulk geometry has compact dimensions that degenerate somewhere, which
corresponds in the bottom-up model to the placement of the ETW brane. We will
be most interested in on-shell action and entropy calculations, where the
bottom-up model reproduces universal CFT results takayanagi2011holographic .
More refined probes of these AdS/BCFT models may be more problematic
reeves2021looking , though our arguments don’t rely on the models being
correct for those observables.444In more detail, reeves2021looking  studies
Lorentzian BCFT correlation functions that probe the (putative) brane geometry
in the bottom-up model given here. They find the singularity structure of
these correlators doesn’t match that expected from bulk causality unless an
apparently unnatural condition is imposed on the BCFT spectrum. In contrast,
our setting involves a bulk process that remains far away from the branes, and
we only rely on the bottom up model to determine the placement of minimal
surfaces and to understand when a particular geometry is the relevant bulk
saddle.

The Ryu-Takayanagi formula gives a method for computing boundary entropy in
terms of bulk degrees of freedom. In one of its modern forms, this can be
expressed as engelhardt2015quantum

$\displaystyle
S(A)=\min_{\gamma_{ext}}\text{ext}_{\gamma\in\text{Hom}(A)}\left(\frac{\text{area}(\gamma)}{4G_{N}}+S_{bulk}(E_{\gamma})\right).$
(4)

The set $\text{Hom}(A)$ is the set of codimension $2$ spacelike surfaces such
that there exists a codimension 1 spacelike surface $E_{\gamma}$ satisfying

$\displaystyle\partial E_{\gamma}=\gamma\cup A.$ (5)

We say that $\gamma\in\text{Hom}(A)$ are _homologous_ to $A$. The quantity
$S_{bulk}(E_{\gamma})$ is the von Neumann entropy of the bulk subregion
$E_{\gamma}$. In AdS/BCFT, the Ryu-Takayanagi formula continues to calculate
the entropy of boundary subregions, provided the homology condition is
appropriately adapted sully2021bcft . The appropriate definition of homologous
in the presence of ETW branes is that there needs to exist a spacelike
codimension 1 surface $E_{\gamma}$ such that

$\displaystyle\partial E_{\gamma}=\gamma\cup A\cup b$ (6)

where $b$ is allowed to be any portion of the ETW brane.

Given a subregion of the boundary $A$, it is natural to ask if a subregion of
the bulk is recorded into $A$. To make this question more precise, we should
introduce a choice of bulk subspace, which we refer to as the code-space and
label $\mathcal{H}_{code}$. The subspace $\mathcal{H}_{code}$ might for
instance be specified by a particular choice of bulk geometry, along with some
qubits distributed spatially across the bulk. Then, assume we are told the
bulk degrees of freedom are in a state within $\mathcal{H}_{code}$, and we are
given the degrees of freedom on subregion $A$. What portion of the bulk
degree’s of freedom can we recover?

Answering this question is related closely to the RT formula, in both AdS/CFT
and AdS/BCFT In particular, the portion of the bulk we can recover if we know
the bulk state in $\mathcal{H}_{code}$ is given by hayden2019learning ;
akers2019large

$\displaystyle E_{A}\equiv\bigcap_{\psi\in\mathcal{H}_{code}}E_{\gamma_{A}}.$
(7)

That is, for each state in the code space we find where the RT surface
$\gamma_{A}$ sits, and define the corresponding bulk subregion
$E_{\gamma_{A}}$. Then, we define the intersection of all such surfaces,
considering all states in the code-subspace. Note that in this procedure we
should include mixed states of the code-space. The resulting region is the
portion of the bulk degrees of freedom we can recover, if we know nothing
about which state in the code-space the full bulk is in. This region is
sometimes referred to as the _reconstruction wedge_ of region $A$, defined
relative to the code-space $\mathcal{H}_{code}$.

Given that it is possible to recover information inside the reconstruction
wedge, we can also ask what explicit operation recovers the code space from
the CFT degrees of freedom. Given a global map from the bulk subspace
$\mathcal{H}_{code}$ to the boundary Hilbert space, it was understood in
cotler2019entanglement  how to construct such a recovery channel. Note that in
this construction, a single choice of recovery channel works correctly for the
entire code-space.

$\Theta$ Figure 2: Geometry dual to the vacuum state of a BCFT on a half line.

To illustrate the AdS/BCFT correspondence, we show a simple AdS/BCFT solution
in figure 2. There, the BCFT is defined on $\mathbb{R}^{-}\times\mathbb{R}$,
though we supress the timelike $\mathbb{R}$ direction in the figure. We have
taken the bulk matter Lagrangian to be zero, and the brane matter Lagrangian
to be

$\displaystyle L^{\mathcal{B}}_{matter}=-T.$ (8)

The parameter $T$ is referred to as the tension. The solution can be described
by

$\displaystyle ds^{2}=\frac{\ell^{2}}{z^{2}}(dz^{2}-dt^{2}+dx^{2})$ (9)

with an ETW brane along an $AdS_{1+1}$ slice $\frac{x}{z}=\tan\Theta$, where
the angle $\Theta$ is related to the tension parameter as

$\displaystyle\sin\Theta=T\ell.$ (10)

Notice that consistent solutions require

$\displaystyle-\frac{1}{\ell}\leq T\leq\frac{1}{\ell}.$ (11)

In the CFT, the tension parameter is related to the boundary entropy, a
measure of the number of degrees of freedom localized to the edge. The
boundary entropy is expressed as $S_{b}=\log g$, and is related to the tension
by

$\displaystyle\log g_{B}=\frac{\ell}{4G_{N}}\text{arctanh}(T\ell).$ (12)

Qualitatively, increasing the boundary entropy adds degrees of freedom to the
CFT, and corresponding increases the angle $\Theta$, adding a portion of bulk
spacetime.

###  2.2 Holographic quantum tasks with input and output regions

In this section we review an operational perspective on AdS/CFT introduced in
may2019quantum  and subsequently developed in may2020holographic ;
may2021quantum ; may2022connected . The basic idea is to consider quantum
information processes that can be interpreted in the bulk and boundary
perspectives, and to reason about how the two descriptions of these processes
inform each other. An up to date and more complete description of this
perspective can be found in may2022connected . Here we briefly review a few
ingredients which contextualize our discussion.

We will consider information processing tasks that have inputs and outputs
distributed in spacetime. To describe this task succinctly, it is helpful to
introduce two agencies, Alice and Bob, who interact to carry out the
task.555Note that introducing Alice and Bob is only a linguistic convenience:
we could rephrase our settings in the language without any notion of observer
or agent. We will label the input locations by $\mathcal{C}_{i}$ where Bob
gives Alice quantum systems $A_{i}$. These may be entangled with some
reference $R$, which we take to be held by Bob. Alice carries out the task by
manipulating the input systems and preparing a set of output systems $B_{i}$,
which she returns to Bob at another set of spacetime locations
$\mathcal{R}_{i}$. Bob then makes a measurement on the $B=B_{1}B_{2}...B_{n}$
system with elements $\\{\Lambda_{B},\mathcal{I}_{B}-\Lambda_{B}\\}$ and we
declare Alice successful if she obtains outcome $\Lambda_{B}$. Following Kent
kent2012quantum , who initially formalized this notion, we refer to these
scenarios as _relativistic quantum tasks_.

The input and output locations will be extended regions in spacetime, though
we sometimes idealize these as spacetime points if we wish to take the regions
to be small. We will consider relativistic quantum tasks in the bulk and
boundary perspectives in AdS/CFT. By comparing the two perspectives, we can
gain interesting insights into the AdS/CFT correspondence and into quantum
information processing.

The key ingredient in relating bulk and boundary perspectives on relativistic
quantum tasks is the notion of the reconstruction wedge, which we reviewed
briefly in the last section. In particular, given a task defined in the bulk
geometry with input or output region ${X}_{i}$, this will correspond to a task
in the boundary which has an input or output region containing ${X}_{i}$ in
its reconstruction wedge. To make this more precise, consider a task
$\mathbf{T}$ defined in the bulk with input regions $\\{{C}_{i}\\}_{i}$ and
output regions $\\{{R}_{i}\\}_{i}$. In the boundary, consider a task with
input regions $\\{\mathcal{C}_{i}\\}_{i}$, $\\{\mathcal{R}_{i}\\}_{i}$ such
that

$\displaystyle C_{i}$ $\displaystyle\subseteq E_{\mathcal{C}_{i}},$
$\displaystyle R_{i}$ $\displaystyle\subseteq E_{\mathcal{R}_{i}}.$

Note that in general there can be many such choices of boundary regions, we
can take any one of them. The boundary task with the input and output regions
defined in this way is taken to have the same input and output systems
$A_{i}$, $B_{i}$, and the same choice of measurement that defines the success
or failure of the task. We call the resulting boundary task
$\hat{\mathbf{T}}$.

The tasks $\mathbf{T}$ and $\hat{\mathbf{T}}$ are related in a simple way. In
particular, we can notice that a strategy for completing the task in the bulk
that succeeds with probability $p_{suc}(\mathbf{T})$ implies the existence of
a corresponding strategy in the boundary that succeeds with the same
probability. This is because given access to boundary regions
$\mathcal{C}_{i}$, $\mathcal{R}_{i}$, Alice can encode her inputs into bulk
regions $C_{i}$, allow boundary time evolution (which in the bulk picture
implements the strategy for completing the task), then recover the outputs
$B_{i}$ from output regions $\mathcal{R}_{i}$. We can summarize this as

$\displaystyle p_{suc}(\mathbf{T})\leq p_{suc}(\hat{\mathbf{T}}).$ (13)

This is the starting point for the various implications for AdS/CFT that can
extracted from studying relativistic quantum tasks. This can also be
strengthened in an interesting way, as pointed out in may2022connected ,
although we won’t use the stronger version here.

##  3 Computation inside black holes

In this section, we show that computations happening inside of planar black
holes can be reproduced as non-local quantum computations. More precisely,
computations inside of a particular subregion of the black hole, defined
below, can be reproduced in the non-local form of figure 1(b) using a resource
system with mutual information given by the black hole area.

###  3.1 General construction

Figure 3: Penrose diagram of the planar black brane. $d-1$ planar spatial
directions are supressed. An extremal surface (blue) anchored to the boundary
at $x_{0}=0$ threads through the black hole.

The metric for the AdSd+1 planar black brane can be written as hartman2013time

$\displaystyle ds^{2}=-(\ell g(\rho))^{2}dt^{2}+\ell^{2}d\rho^{2}+(\ell
h(\rho))^{2}\sum_{i=1}^{d-1}dX^{2}_{i}$ (14)

where

$\displaystyle h(\rho)$
$\displaystyle=\frac{2}{d}\left(\cosh\left(\frac{d\rho}{2}\right)\right)^{2/d}$
$\displaystyle g(\rho)$
$\displaystyle=h(\rho)\tanh\left(\frac{d\rho}{2}\right).$ (15)

These coordinates cover one exterior region for $\rho\geq 0$. The temperature
of the black hole has been fixed at $1/2\pi$. A Penrose diagram showing the
maximally extended geometry is shown in figure 3. The boundary dual is
understood to be copies of a holographic CFT, placed in the thermofield double
state. Each CFT lives on a $d$ dimensional Minkowski space.

We will consider the entanglement wedges of half spaces of the combined CFTs,
defined by $X_{1}>0$, $X_{1}<0$. As discussed in hartman2013time , the
corresponding minimal surfaces thread through the black hole. The explicit
form of this surface is discussed there. We will give explicit formula for
this surface in the low dimensional case treated below, but for now only need
to note that this extremal surface exists. We will call this the connected
extremal surface. rangamani2017holographic

$X_{1}$$\phi$ (a)

(b)

Figure 4: Euclidean gravity solutions corresponding to the thermofield double
state of two BCFTs on intervals. (a) When the interval is short, the ETW brane
connects the two BCFT edges. (b) When the interval is long, the ETW brane is
in two pieces, and separately attached to each edge. We Wick rotate the
angular $\phi$ coordinate. The solution with two disconnected branes Wick
rotates to a planar black hole.

We will also consider truncated versions of this geometry, corresponding to
the metric 14 but with $X_{1}$ taken in a finite interval. To find the dual
CFT description, Wick rotate $t\rightarrow i\phi$ to obtain a Euclidean
geometry. This is dual to a Euclidean BCFT on
$S^{1}\times[-X_{1}^{0},X_{1}^{0}]\times\mathbb{R}^{d-2}$. We assume boundary
conditions at $\pm X_{1}^{0}$ that have zero boundary entropy for simplicity.
There are two possible bulk saddles dual to the path integral preparing this
state, which we illustrate in figure 4, that feature zero tension branes in
two possible configurations. For our construction to succeed, we will need to
be in the regime where each BCFT edge is dual to its own bulk ETW brane, as in
figure LABEL:fig:disconnectedbranes. In this solution, analytically continuing
the $\phi$ coordinate back to $it$ we obtain two sided Lorentzian planar black
brane geometries, whereas in the opposite saddle (figure
LABEL:fig:connectedbranes) we obtain two disconnected geometries. We will
assume that for large enough $X_{1}^{0}$ the saddle with two branes will be of
minimal action. The zero tension brane trajectory for the connected geometry
is defined by $X_{1}=\pm X_{1}^{0}$, as is easy to verify. In the next section
we study this transition explicitly for AdS2+1. Notice that the black hole
area in the truncated geometry is now $2X_{1}^{0}$, and in particular is
finite.

$\hat{X}$$\hat{t}$ Figure 5: The planar BTZ black hole, obtained by Wick
rotating the $\phi$ coordinate in the Euclidean geometry of figure
LABEL:fig:disconnectedbranes. The planar direction extends over
$X\in[-X_{1}^{0},X_{1}^{0}]$, where it is ended by ETW branes.

$A_{L}$$A_{R}$$B_{-}$$B_{+}$$CFT_{L}$$CFT_{R}$ (a)

$B_{-}$$B_{+}$$CFT_{L}$$CFT_{R}$$X_{1}<0$$X_{1}>0$ (b)

Figure 6: Illustration of our thought experiment reproducing bulk interactions
as non-local computations. AliceL and AliceR hold the left and right BCFTs of
the solution shown in figure 5. (a) The Alice’s receive the inputs to the
computation, and throw their respective systems into the black hole. The
systems interact inside of the black hole interior. (b) After the two CFTs
have evolved to time $T$, AliceL and AliceR use a single, simultaneous round
of communication to redistribute systems so that AliceL holds the
$(0,X_{1}^{0}]$ of both CFTs, and AliceR holds both of the $[-X_{1}^{0},0)$
intervals. For sufficiently large $X_{1}^{0}$, the portion of the bulk that
can be recovered from the two $(0,X_{1}^{0}]$ intervals includes $B_{L}$, and
the systems recoverable from the two $[-X_{1}^{0},0)$ intervals includes
$B_{R}$. This construction shows computations happening in the black hole
interior can be implemented as a non-local computation.

With these geometrical comments in place we can move on to establish the
relationship between non-local computation and the black hole interior. The
most explicit setting will be in the planar BTZ black hole, but the
qualitative picture is the same in all dimensions. We first discuss the
general setting, making some qualitative statements, then in the next section
we give the more concrete BTZ case.

To begin, consider quantum systems $A_{L}$, $A_{R}$ which are initially stored
within the left and right asymptotic regions of the planar black hole,
respectively. These systems fall into the black hole, interact, and produce
output systems $B_{+}$, $B_{-}$. We then have these systems travel in the
$\hat{X}_{1}$ and $-\hat{X}_{1}$ directions, respectively. The overall process
is described in figure 6. We take all relevant systems to consist of fewer
qubits than the black hole entropy.

To describe this process in the boundary perspective, we first note that we
can take as input regions $\mathcal{C}_{L}$, $\mathcal{C}_{R}$ the entire left
and right CFTs. These then contain the inputs $A_{L}$ and $A_{R}$ in their
reconstruction wedges, which will be the left and right exterior regions. For
output regions, we choose a time $T$ in the CFT, and consider the CFT regions

$\displaystyle\mathcal{R}_{-}$ $\displaystyle=\\{p:t=T,X_{1}<0\\},$
$\displaystyle\mathcal{R}_{+}$ $\displaystyle=\\{p:t=T,X_{1}>0\\}.$ (16)

In words $\mathcal{R}_{-}$ is the $X_{1}<0$ portion of both CFTs taken at time
$T$, $\mathcal{R}_{+}$ is the $X_{1}>0$ portions of both CFTs taken at the
same time.

The boundary of the reconstruction wedges of $\mathcal{R}_{+}$ and
$\mathcal{R}_{-}$ is defined by the extremal surface anchored to the two
boundary cuts $\gamma_{+}$, $\gamma_{-}$, defined at $X_{1}=0,t=T$ in the two
CFTs. For $X_{1}^{0}$ large enough, we expect this extremal surface is the
connected surface that threads through the black hole. For small $X_{1}^{0}$
however, a new extremal surface can become the minimal one. Rather than thread
the black hole, this alternative surface runs from the asymptotic region to
one of the ETW branes. This extremal surface cannot cross the black hole
horizon.666One way to see this is to note that if it did, operators in the
corresponding entanglement wedge could be used to signal the opposite
asymptotic region (if it crosses the past horizon) or to receive signals from
the opposite asymptotic region (if it crosses the future horizon). Thus in the
disconnected case, the reconstruction wedges of $\mathcal{R}_{-}$ and
$\mathcal{R}_{+}$ do not see into the black hole interior. We will assume that
we have chosen $X_{1}^{0}$ large enough that the connected extremal surface is
minimal, so that the wedge of $\mathcal{R}_{-}$ and $\mathcal{R}_{+}$ include
a portion of the black hole interior.

To relate the bulk interaction to a non-local computation, we exploit that the
reconstruction wedges of our two half spaces reach into the interior. In
particular, consider an interaction that takes place in the bulk subregion

$\displaystyle J_{T}=J^{-}(E_{\mathcal{R}_{+}})\cap
J^{-}(E_{\mathcal{R}_{-}})\cap J^{+}(E_{\mathcal{C}_{L}})\cap
J^{+}(E_{\mathcal{C}_{R}})$ (17)

which we can also express as

$\displaystyle J_{T}=J^{-}(E_{\mathcal{R}_{+}})\cap
J^{-}(E_{\mathcal{R}_{-}})\cap I$ (18)

where $I$ denotes the black hole interior. We refer to $J_{T}$ as the
_scattering region_. If the interaction occurs inside this region, then the
output systems $B_{-}$ and $B_{+}$ will each enter one of the wedges
$E_{\mathcal{R}_{-}}$ and $E_{\mathcal{R}_{+}}$. Notice that if the input
systems are too large, or the interaction in the bulk is associated with a
large backreaction, the minimal extremal surface may transition from a
connected surface threading through the black hole to a disconnected, brane
anchored surface.

So long as the connected surface remains the minimal one, interactions
happening within $J_{T}$ can be reproduced as non-local computations. To see
this, consider the non-local computation circuit shown in figure 1(b).
Relating that picture to our setting, the left and right ends of the shared
entangled state represent the left and right CFTs. The local operations on the
left and right at the first time step can be used to insert the inputs $A_{L}$
and $A_{R}$ into the bulk, and allow the left and right CFTs to evolve under
time evolution. Then, we consider recovery operations that act on
$\mathcal{R}_{\pm}$. Since systems $B_{\pm}$ have been brought into the
corresponding reconstruction wedges $E_{\mathcal{R}_{\pm}}$, these systems can
be extracted from the bulk by acting on $\mathcal{R}_{\pm}$. The swap
operation at the middle level then collects both the $+$ halves and both the
$-$ halves of the CFTs, so that the final round operations can be taken to be
the necessary recovery operations. Thus we can reproduce the bulk computation
in the form of a non-local computation. Further, the needed entangled state
was exactly the thermofield double, which has a black hole’s entropy worth of
mutual information.

We can also express the relationship to non-local computation in more
operational language. We consider two agents, AliceL and AliceR, who initially
hold the left and right CFTs respectively. AliceL acts on her CFT to insert
$A_{L}$ into the bulk, AliceR acts on her CFT to insert $A_{R}$ into the bulk.
Both locally apply the time evolution operators for their CFTs. Next, AliceL
and AliceR divide their CFT degrees of freedom and exchange a round of
communication to bring the $\mathcal{R}_{+}$ degrees of freedom to AliceR and
$\mathcal{R}_{-}$ degrees of freedom to AliceL. They then each apply the
appropriate recovery operations to produce the $B_{-}$ and $B_{+}$ systems.

Beyond the circuit 1(b) having the form of a non-local computation, our
construction also shows it has an additional property. In particular, a non-
local circuit capturing a bulk interaction can always be taken to have fixed
recovery channels as its second round operations. This means bulk computation
must be reproducible in a stricter form than the usual form of a non-local
computation. This property also appears in the construction in
dolev2022holography , though isn’t noted explicitly. We comment on this
property further in the discussion.

Notice that in relating bulk interaction and non-local computation we used two
features of our geometries: that ETW brane configuration shown in figure
LABEL:fig:disconnectedbranes is minimal at large $X_{1}^{0}$, and that the
connected extremal surface that threads through the black hole is minimal at
large $X_{1}^{0}$. Both these statements must hold in the limit
$X_{1}^{0}\rightarrow\infty$, but to better understand this we find the
explicit phase transitions for AdS2+1 below. This in particular lets us study
how much entanglement (as set by the black hole area) is required to support
interactions within a given scattering region, and further how small of
regions can be studied in this way.

###  3.2 Non-local computation in the planar BTZ solution

Our construction can be made more explicit in the planar BTZ black hole. We
take the planar direction to be be a finite interval $X\in[-X^{0},X^{0}]$.
This is dual to a pair of BCFTs living on the spatial interval
$[-X^{0},X^{0}]$. For simplicity, we take the bulk ETW branes to be zero
tension.

To construct our BTZ black hole solution, we begin with the Euclidean path
integral on a cylinder $[0,2\pi)\times[-X^{0},X^{0}]$. Note that by taking the
angular direction to run from $0$ to $2\pi$ we are fixing the black hole
temperature to be $1/2\pi$. The bulk solution is global Euclidean AdS,

$\displaystyle
ds^{2}=\ell^{2}\left(\frac{\rho^{2}}{\ell^{2}}+1\right)dX^{2}+\frac{d\rho^{2}}{\frac{\rho^{2}}{\ell^{2}}+1}+\rho^{2}d\phi^{2}.$
(19)

As studied in cooper2019black 777See equation A.24 there. Note that they
perform a different Wick rotation, so our black hole occurs for disconnected
branes, while the black hole in cooper2019black  appears for connected branes.
The same transition can also be extracted from fujita2011aspects ., at zero
tension the ETW branes are in the disconnected configuration of figure
LABEL:fig:disconnectedbranes when $X^{0}\geq\pi/2$, and otherwise the two
branes connect. Explicitly, the branes sit at $X=\pm X^{0}$. From here
forward, we take $X^{0}\geq\frac{\pi}{2}$.

Considering the metric above, Wick rotate $\phi\rightarrow it$. The result is
the thermofield double state with temperature $1/2\pi$, defined on two BCFTs
on intervals $[-X^{0},X^{0}]$. The resulting metric

$\displaystyle
ds^{2}=\ell^{2}\left(\frac{\rho^{2}}{\ell^{2}}+1\right)dX^{2}+\frac{d\rho^{2}}{\frac{\rho^{2}}{\ell^{2}}+1}-\rho^{2}d\phi^{2}$
(20)

covers the exterior region $\rho>0$. We can extend this geometry to the full
Lorentzian black hole geometry, obtaining

$\displaystyle
ds^{2}=\frac{\ell^{2}}{\cos^{2}(w)}\left(-ds^{2}+dw^{2}+\cos^{2}(s)dX^{2}\right).$
(21)

The two end of the world branes are still located at $X=\pm X^{0}$. The
asymptotic regions sit at $w=\pm\pi/2$, and the black hole horizons at $s=\pm
w$. The coordinate change relating this metric to 20 is given in appendix A.

In relating bulk interaction to non-local computation, consider the extremal
surface anchored to the points $X=0,s=T,w=\pm\pi/2$. There are two candidate
extremal surfaces: a connected surface that threads through the black hole,
and a disconnected surface, that connects each of the boundary anchor points
to the ETW brane. The form of the connected surface $\gamma_{T}$ is simple, it
is just

$\displaystyle w$ $\displaystyle=\arcsin(\tanh\lambda),$ $\displaystyle s$
$\displaystyle=T.$ (22)

In appendix C, we find the areas of this and the brane-anchored surfaces. We
find the connected surface has area

$\displaystyle A_{c}[T]=2\ell\log\left(\frac{2}{\epsilon}\right),$ (23)

and the disconnected one has area

$\displaystyle
A_{d}[T]=2\ell\log\left(\frac{2\sinh|X^{0}|\cos(T)}{\epsilon}\right).$ (24)

These give that the area difference is

$\displaystyle\Delta A=2\ell\log\left(\sinh|X^{0}|\cos T\right)$ (25)

and the connected surface is minimal when

$\displaystyle\sinh X^{0}\geq\sec T.$ (26)

Referring to the general discussion in the last section, we see that imposing
condition 26 along with $X^{0}\geq\pi/2$ suffice for bulk interactions to be
reproduced as non-local computations.

##  4 Finite dimensional model

We argued above that, so long as we don’t induce a large backreaction,
computations which happen in the region $J_{T}$ can be implemented as non-
local computations, using the thermofield double state as the entangled
resource system. While the entanglement as measured by the mutual information
of the thermofield double is finite in our setting, the CFT Hilbert spaces are
infinite dimensional. However results on non-local computation are typically
proven in a finite dimensional setting. If we would like to apply them to
holography then, we need to understand if the protocol carried out by the
Alice’s in the last section can be approximated by one involving only finite
dimensional systems.

To do this, a key ingredient is that in many holographic states the max
entropy is close to the von Neumann entropy. Since the von Neumann entropy of
our two CFTs is finite, this will let us show that the initial resource
system, consisting of the entangled state of two CFTs, can be approximated
closely by a finite dimensional system.

In more detail, we study the following general form of a non-local computation
protocol, which we argue below captures the holographic setting.

######  Definition 1.

A non-local computation protocol takes the following form.

  1. 1.

The inputs are recorded into Hilbert spaces $\mathcal{H}_{A_{L}}$,
$\mathcal{H}_{A_{R}}$, each consisting of $n$ qubits.

  2. 2.

The resource system $\ket{\phi}_{C_{L}C_{R}}$ lives in a Hilbert space
$\mathcal{H}_{C_{L}}\otimes\mathcal{H}_{C_{R}}$, where each tensor factor
consists of $E$ qubits.

  3. 3.

In the first round an isometry $\mathbf{V}^{L}_{A_{L}C_{L}\rightarrow
C_{L,L}C_{L,R}}\otimes\mathbf{V}^{R}_{A_{R}C_{R}\rightarrow C_{R,L}C_{R,R}}$
is applied.

  4. 4.

In the second round isometries
$\mathbf{W}^{L}_{C_{L,L}C_{R,L}}\otimes\mathbf{W}^{R}_{C_{L,R}C_{R,R}}$ are
applied. A pair of the output subsystems are identified as $B_{L}$ and
$B_{R}$.

Notice that between the first and second rounds systems $C_{L,R}$ and
$C_{R,L}$ have been exchanged. This interchange of systems corresponds to the
communication round of the non-local computation. We illustrate this protocol
in figure 7.

$\mathbf{V}^{L}$$\mathbf{V}^{R}$$\mathbf{W}^{R}$$\mathbf{W}^{L}$$S_{L,R}$$S_{R,R}$$S_{L,R}$$S_{R,L}$$S_{L}$$S_{R}$$A_{L}$$A_{R}$$B_{R}$$B_{L}$
Figure 7: General form of a non-local computation, with systems labelled. Note
that $V^{L}$ and $V^{R}$ are isometries.

We would like to justify modelling the protocol described in our black hole
setting in this form, and in particular understand how large of Hilbert spaces
are needed to capture the power of the holographic protocol. To do this, first
note that the entropy of the left (or right) BCFT is

$\displaystyle S(L)=\frac{A_{bh}}{4G_{N}}=2\ell X^{0}.$ (27)

In particular, there is no UV divergence. Next recall that the smooth max
entropy renner2005simple  is defined as

$\displaystyle
S_{max}^{\epsilon}(A)_{\rho}=\min_{\sigma_{A}:||\rho_{A}-\sigma_{A}||_{1}\leq\epsilon}\log(\rank\sigma_{A})$
(28)

and that the smooth max entropy and von Neumann entropy are close for
holographic states bao2019beyond ,

$\displaystyle
S_{max}^{\epsilon}(A)=S(A)+a\frac{\log(1/\epsilon)}{\sqrt{G_{N}}}$ (29)

where $a$ is independent of $G_{N}$, but can depend on the state $\rho$. This
holds for holographic states as a consequence of the Renyi entropy taking the
form,

$\displaystyle S_{\alpha}=\frac{s_{\alpha}}{G_{N}}$ (30)

where $s_{\alpha}$ is independent of $G_{N}$, but is otherwise an arbitrary
function of $\alpha$ and the choice of state. While bao2019beyond  does not
explicitly consider holographic BCFT states, it is straightforward to see that
30 continues to hold for these states, and hence we recover 29 as well in our
setting.888One can see this by noting that the cosmic brane prescription of
dong2016gravity  applies to our setting, which yields a value of the Renyi
entropy of the form 30. A comment is that in our setting the cosmic brane will
anchor to the two ETW branes, so the Renyi entropy’s are UV divergence free.
We can also note that the form 30 has been found directly from a CFT
perspective in sully2021bcft , though only for intervals ending on one CFT
boundary (rather than anchored on both ends to CFT boundaries, as in our
setting).

Let the state of the two BCFTs appearing in our thought experiment be
$\ket{\Psi}_{C_{L}C_{R}}$. Consider the reduced density matrix $\Psi_{C_{L}}$.
Then as a consequence of 29, we have that there is a state
$\Psi^{\prime}_{C_{L}}$ with rank $S^{\epsilon}_{max}(C_{L})$ which is
$\epsilon$ close in trace distance to the state ${\Psi}_{C_{L}}$. Then using
the Fuchs-Van-de-Graff inequality,

$\displaystyle
1-\sqrt{F(\rho,\sigma)}\leq\frac{1}{2}||\rho-\sigma||_{1}\leq\sqrt{1-F(\rho,\sigma)}$
(31)

we have that $\sqrt{F}(\Psi_{C_{L}},\Psi_{C_{L}}^{\prime})\geq 1-\epsilon/2$.
Using Uhlmann’s theorem uhlmann1976transition  we have

$\displaystyle
1-\epsilon/2\leq\sqrt{F(\Psi_{C_{L}},\Psi_{C_{L}}^{\prime})}=\max_{\Phi_{C_{L}C_{R}}}|\innerproduct{\Phi}{\Psi}_{C_{L}C_{R}}|.$
(32)

Thus there exists a purification of $\Psi_{C_{L}}^{\prime}$, call it
$\ket{\Psi^{\prime}}_{C_{L}C_{R}}$, which has
$|\innerproduct{\Psi^{\prime}}{\Psi}_{C_{L}C_{R}}|\geq 1-\epsilon/2$, or using
the Fuchs van de Graff inequality again we have

$\displaystyle||\ket{\Psi^{\prime}}-\ket{\Psi}||\leq\sqrt{\epsilon/2}.$ (33)

Because this purifies $\Psi_{C_{L}}^{\prime}$ we know
$\ket{\Psi^{\prime}}_{C_{L}C_{R}}$ has Schmidt rank
$S_{max}^{\epsilon}(C_{L})\equiv E$, so

$\displaystyle\ket{\Psi^{\prime}}_{C_{L}C_{R}}=\sum_{i=1}^{2^{E}}\sqrt{\lambda_{i}}\ket{i}_{C_{L}}\ket{i}_{C_{R}}.$
(34)

Next, we define finite dimensional Hilbert spaces
$\mathcal{H}_{C_{L}^{\prime}}\otimes\mathcal{H}_{C_{R}^{\prime}}$ whose basis
vectors are identified with the Schmidt vectors of
$\ket{\Psi^{\prime}}_{C_{L}C_{R}}$. These each have dimension $2^{E}$.

In our original thought experiment, each of AliceL and AliceR act on their
inputs plus their respective BCFTs to insert the inputs into the bulk. In our
finite dimensional model, we should first locally prepare the BCFT states from
the finite dimensional state $\ket{\Psi^{\prime}}$. This is possible because
the finite dimensional entangled state shares the same Schmidt spectrum. Each
Alice then insert the inputs into the bulk. This is captured by the
application of the first round operations
$\mathbf{V}^{L}_{A_{L}C_{L}\rightarrow
C_{L,L}C_{L,R}}\otimes\mathbf{V}^{R}_{A_{R}C_{R}\rightarrow C_{R,L}C_{R,R}}$.
Next, The Alice’s divide the BCFTs into $X>0$ and $X<0$ portions, and
communicate half of the degrees of freedom. Naively this step presents a
challenge for our finite dimensional model to capture, since the entanglement
across the $X=0$ cut in the CFT will be infinite, so we can’t prepare states
sharing approximately the same spectrum in a finite dimensional Hilbert space.
However, a key feature of our finite dimensional model is that we allow
isometries in the first round, and in particular systems $C_{L,L}$, $C_{L,R}$
and $C_{R,L}$, $C_{R,R}$ can be arbitrarily large. Thus we can choose the
$X_{+}$ and $X_{-}$ regions to be separated by a small cut-off $\delta$, and
our protocol captures the entanglement across this cut-off arbitrarily well.
The dimension of the needed system will grow as $\delta\rightarrow 0$, but
because we allow isometries, this is captured by our protocol above. Note that
there is no dependence on the cut-off in the amount of entanglement used,
which is fixed by the (finite) black hole area.

##  5 Computation in global AdS2+1

In this section we study global AdS2+1, and show that a similar claim to the
two sided case holds: computation happening in the bulk of AdS can be
reproduced as a non-local quantum computation, using entanglement controlled
by the size of the scattering region.

Towards showing this the main obstruction is that interactions in the bulk of
global AdS2+1 are most naturally related to augmented non-local computation,
rather than non-local computation directly. This is related to the appearance
of the “side-regions” that we define below, from which the extra systems
discussed in figure 1(c) originate. To get around this, we approximate the
original CFT with a pair of entangled BCFTs. The BCFTs live on the same
geometry as the original CFT but with the side regions removed. We show that
the dual of the BCFTs includes enough of the bulk geometry that they can still
be used to perform the computation, and that they share similar entanglement
to the relevant subregions of the original CFT. Quantitatively, they share
entanglement that becomes close to the area of the scattering region as the
scattering region becomes large.999This justifies the assumption made in
may2022complexity  that the entanglement between the inputs is responsible for
supporting the computations happening inside the scattering region in the
limit where the side regions are becoming small. It also gives a quantitative
bound on how much more entanglement can be necessary in the setting of large
side regions.

###  5.1 Bulk and boundary geometry of $2\rightarrow 2$ task

$c_{1}$$c_{2}$$p$$r_{2}$$r_{1}$ (a)

$c_{2}$$\mathcal{V}_{1}$$c_{2}$$\mathcal{V}_{2}$$c_{1}$$r_{1}$$r_{2}$$\mathcal{X}_{2}$$\mathcal{X}_{1}$
(b)

(c)

$\mathcal{X}_{1}$$\mathcal{X}_{2}$$\mathcal{V}_{1}$$\mathcal{V}_{2}$ (d)

Figure 8: (a) Global AdS2+1 with an example choice of input and output points.
These points have a non-empty scattering region in the bulk, but an empty
scattering region in the boundary. (b) Boundary view of AdS2+1 with an
illustration of the input regions $\mathcal{V}_{1},\mathcal{V}_{2}$ and side
regions $\mathcal{X}_{1},\mathcal{X}_{2}$. (c) Causal structure present in the
bulk of AdS, with the choice of input and output regions shown in figure (a).
(d) Causal structure present in the boundary of AdS with the same choice of
input and output regions.

In this section we give the geometrical set-up of input and output points at
the boundary of AdS2+1. To begin, consider the picture in figure 8(a), showing
an AdS2+1 spacetime. In figure 8(b) we show the dual boundary spacetime. The
basic observation is that, choosing four points in the boundary, we can have
it happen that one can meet in the bulk but not in the boundary. To discuss
this more carefully, define the bulk _scattering region_ ,

$\displaystyle J[c_{1},c_{2}\rightarrow r_{1},r_{2}]\equiv J^{+}(c_{1})\cap
J^{+}(c_{2})\cap J^{-}(r_{1})\cap J^{-}(r_{2}).$ (35)

The observation is that we can have geometries and choices of points
$c_{1},c_{2},r_{1},r_{2}$ such that $J[c_{1},c_{2}\rightarrow r_{1},r_{2}]$ is
non-empty, while the object $\hat{J}[c_{1},c_{2}\rightarrow r_{1},r_{2}]$
obtained by replacing each bulk light cone with its boundary restriction is
empty.

We introduce some definitions to capture relevant aspects of the boundary
geometry. Define the _input regions_

$\displaystyle\mathcal{V}_{1}$
$\displaystyle=\hat{J}^{+}(c_{1})\cap\hat{J}^{-}(r_{1})\cap\hat{J}^{-}(r_{2}),$
(36) $\displaystyle\mathcal{V}_{2}$
$\displaystyle=\hat{J}^{+}(c_{2})\cap\hat{J}^{-}(r_{1})\cap\hat{J}^{-}(r_{2}).$
(37)

Further, define the _side regions_ ,

$\displaystyle\mathcal{X}_{1}$
$\displaystyle=\hat{J}^{-}(r_{1})\cap\bar{[\mathcal{V}_{1}\cup\mathcal{V}_{2}]},$
(38) $\displaystyle\mathcal{X}_{2}$
$\displaystyle=\hat{J}^{-}(r_{2})\cap\bar{[\mathcal{V}_{1}\cup\mathcal{V}_{2}]},$
(39)

where $\bar{\mathcal{A}}$ denotes the spacelike complement of spacetime region
$\mathcal{A}$. See figure 8(b) for an illustration of these regions.

Figure 8(c) shows the causal features of the bulk schematically, in the form
of a causal network. The central vertex represents the (non-empty) scattering
region. In figure 8(d), we show the causal network describing the boundary.
Notice that in the boundary, we don’t have exactly the usual network
describing a non-local quantum computation. This is because of the regions
$\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ that sit between the regions
$\mathcal{V}_{1}$ and $\mathcal{V}_{2}$. We call this the _augmented NLQC_
scenario. This complicates the discussion, as it is only correlation between
$\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ that we will have control over, while
e.g. $I(V_{1}X_{1}:V_{2}X_{2})$ is divergent.

Figure 9: A scattering region in AdS2+1. The lower edge is the ridge, $r$.

We recall some further geometric facts established in may2020holographic .
First, an interesting observation was made that the scattering region sits
inside of the entanglement wedge of $\mathcal{V}_{1}\cup\mathcal{V}_{2}$. In
notation,

$\displaystyle J[c_{1},c_{2}\rightarrow r_{1},r_{2}]\subseteq
E_{\mathcal{V}_{1}\mathcal{V}_{2}}.$ (40)

Second, we will measure the size of the scattering region in terms of the area
of its lower edge, which we call the _ridge_ , defined by

$\displaystyle r=\partial J^{+}(c_{1})\cap\partial J^{+}(c_{2})\cap
J^{-}(r_{1})\cap J^{-}(r_{2}).$ (41)

We illustrate the scattering region and the placement of the ridge in figure
9. For the CFT vacuum state, may2020holographic  showed that the area of this
surface has a simple boundary expression,

$\displaystyle I(V_{1}:V_{2})_{\Psi}=\frac{\text{area}(r)}{4G_{N}}.$ (42)

This becomes $\geq$ away from the vacuum.

Our goal below will be to relate computations happening inside the scattering
region to (non-augmented) non-local computations. This connection was argued
for based on the heuristic of ignoring the side regions in may2019quantum ;
may2020holographic , but here we address this more carefully. Following our
discussion of the two sided black hole case, our strategy is to set boundary
conditions at the edge of the $\mathcal{V}_{1}$, $\mathcal{V}_{2}$ regions,
and remove the side region degrees of freedom from our system. This replaces
our initial CFT with two, entangled BCFTs that (in a sense we make precise)
approximate the original CFT, and in particular share much of the original
bulk geometry. In fact, we will see that it shares enough of the original
geometry so as to still support the same bulk computations.

###  5.2 ETW brane solutions

(a) Figure 10: Lorentzian global AdS2+1 showing a solution with two ETW
branes. These solutions can be prepared via the Euclidean path integral, as
studied in appendix D. The solution shown is the minimal action one when the
intervals cut out by the brane have angular radius below $24^{\circ}$.

The particular solutions we need can be prepared via a Euclidean path
integral. Recall that pure global AdS2+1 is prepared by the Euclidean path
integral on the infinite cylinder. After a Wick rotation, and using global
coordinates, the resulting solution is described by the metric

$\displaystyle
ds^{2}=\frac{\ell^{2}}{\cos^{2}(r)}\left(-dt^{2}+dr^{2}+\sin^{2}(r)d\phi^{2}\right).$
(43)

We modify the Euclidean path integral by cutting out two (roughly) disk shaped
regions, centered at $t=0$ and located antipodally. In appendix D we solve for
the brane solution explicitly.101010These brane solutions were also obtained
in may2021interpolating , although there multi-brane solutions are obtained by
sewing together single brane Lorentzian solutions, and no explicit path
integral giving the multi-brane solution is given. We prepare the two brane
solution to allow studying the transition between a connected and disconnected
geometry. For $T=0$, we find the trajectory of each brane is described by

$\displaystyle\cos(t-t_{0})=\frac{1}{\cos(\Delta\phi)}\cos(\phi-\phi_{0})\sin
r$ (44)

and the interior geometry is a portion of the pure AdS geometry described by
equation 43. We construct these solutions in detail in appendix D. Similar
solutions for general values of $T$ exist. The parameter $\Delta\phi$ is the
angular radius of the brane at $t=t_{0}$. We can intersect the interior
regions of several branes, placed at different $\phi_{0}$ and potentially with
different widths, to construct multibrane solutions.

Note that if we make the BCFT intervals too small, a new solution will
dominate the path integral where the branes connect in the opposite way, so
that the two BCFTs are not connected through the bulk geometry. We will assume
we are in the configuration shown, with a connected geometry. In appendix D,
we find that we are in the connected geometry whenever the branes occupy
intervals with angular radius $\Delta\phi\lesssim 24^{\circ}$.

An important view on this brane is its intersection with the $t=\pi/4$ slice.
This is given by

$\displaystyle 1=\frac{1}{\cos(\Delta\phi)}\cos(\phi-\phi_{0})\sin(r).$ (45)

We can observe that this is the same trajectory as the RT surface anchored to
the same endpoints.111111Compare e.g. to 6.1.26 in rangamani2017holographic ,
after the coordinate transformation $\rho=\tan r$. This is true for zero
tension branes and when comparing to extremal surfaces in pure AdS, but not in
general. Further, we can observe that the brane sits inside the causal future
of this $t=t_{0}$ surface — this follows from a simple calculation showing
that the constant $\theta$ curves, which foliate the full brane, are
everywhere timelike or null.

The trajectory of the branes endpoint on the boundary is found by setting
$r=\pi/2$ in 44, which gives

$\displaystyle\cos(t-t_{0})=\frac{1}{\cos(\Delta\theta)}\cos(\phi-\phi_{0}).$
(46)

We are interested specifically in a setting with two branes, one centered at
$t=\pi/4$, $\phi=\pi/2$, the other at $t=\pi/4,\theta=\pi/2$. We find that
these intersect at

$\displaystyle b_{1}=(t=3\pi/4,\phi=0),$ (47) $\displaystyle
b_{2}=(t=3\pi/4,\phi=\pi).$ (48)

We show the boundary brane trajectories in figure 11.

$c_{2}$$\mathcal{V}_{1}$$\mathcal{V}_{1}$$\mathcal{V}_{2}$$c_{2}$$c_{1}$$b_{2}$$b_{1}$$b_{1}$
Figure 11: Boundary picture, where the $\mathcal{X}_{i}$ regions have been
removed by replacing them with boundary conditions on the $\mathcal{V}_{i}$
regions. The brane trajectories are shown in blue. Note that the region behind
the branes is empty — no spacetime or degrees of freedom are associated with
it. These two boundary spacetime regions are connected through the bulk
geometry.

These BCFT geometries were studied in van2020spacetime ; may2021interpolating
, where it was pointed out that they approximate the original CFT state in a
precise sense. In particular, as the radius of the circles on which boundary
conditions have been set goes to zero, the reduced density matrix on the
remaining degrees of freedom approaches the density matrix of the
corresponding subsystems of the CFT. Because these BCFT states approximate the
original CFT state while having the side regions removed, they are natural
candidates to use to connect computation in the bulk to (un-augmented) non-
local computation. We take this up in the next section.

###  5.3 Non-local computation and the scattering region

In this section we give a protocol that allows whatever computations that can
happen in the scattering region to be performed non-locally, using a resource
system with mutual information controlled by the size of the scattering
region. This holds so long as the computation does not induce too large of a
backreaction, and the angular radius of the extra regions is smaller than the
threshold needed to obtain a connected brane geometry, $\Delta\phi\lesssim
24^{\circ}$.

To begin, suppose some quantum task $T$ can be completed in the bulk, with
some particular placement of input and output regions. This placement of
points defines spacetime regions $\mathcal{V}_{1}$, $\mathcal{V}_{2}$ in the
boundary. We take the background state to be the CFT vacuum, corresponding to
pure AdS in the bulk. The bulk picture has input systems $A_{1}$, $A_{2}$ meet
in the scattering region, interact, then exit the scattering region and travel
towards the output points. This can be arranged by acting with unitaries
$\mathbf{V}_{1}$, $\mathbf{V}_{2}$ on regions
$\mathcal{V}_{1},\mathcal{V}_{2}$ to insert the inputs and computing device
into the bulk, then allowing the CFT to time evolve.

From this process, we build a non-local computation protocol that completes
the same task, in the non-augmented scenario. Our strategy is to use
$\ket{\tilde{\Psi}}_{{V}_{1}{V}_{2}}$ as the resource system, which we define
as the dual state to the two brane solution discussed in the last section. We
choose the size and placement of the branes such that the side regions
$\mathcal{X}_{i}$ are cut out, as shown in figure 11. We denote the spacetime
dual to $\ket{\Psi}_{{V}_{1}{V}_{2}{X}_{1}{X}_{2}}$ as $\mathcal{M}$, and the
spacetime dual to $\ket{\tilde{\Psi}}_{{V}_{1}{V}_{2}}$ as
$\tilde{\mathcal{M}}$. It will be convenient to view $\tilde{\mathcal{M}}$ as
a subset of $\mathcal{M}$, whose boundary is defined by the brane trajectory.

To describe the protocol, we will adopt the operational language briefly used
in section 3.1, and consider AliceL and AliceR, who will each initially hold
one of the BCFTs. To construct the protocol, first observe that
$\tilde{\mathcal{M}}$ contains the scattering region. To see this, notice that
the entangling surface $\gamma_{\mathcal{V}_{1}\mathcal{V}_{2}}$ in the
original geometry sits exactly on the two branes, along the moment of time
symmetry. Because the branes follow timelike trajectories, they sit outside
the domain of dependence of $E_{\mathcal{V}_{1}\mathcal{V}_{2}}$. Finally,
recall from section 5.1 that the scattering region sits inside of the
entanglement wedge $E_{\mathcal{V}_{1}\mathcal{V}_{2}}$, so that it also sits
inside of $\tilde{\mathcal{M}}$.

The first round of the protocol then is to take the state
$\ket{\tilde{\Psi}}_{{V}_{1}{V}_{2}}$ and act on it with
$\mathbf{V}_{1}\otimes\mathbf{V}_{2}$ — the same unitaries as in the original
state $\Psi$. In the bulk picture, the same computation happens inside of the
scattering region as in the original geometry, and the outputs begin moving
towards the output locations. In the communication round, Alice1 and Alice2
redistribute subsystems of the CFT according to the picture in figure 12.
Label the systems held by Alicei after the communication round by $W_{i}$, and
the corresponding subregions by $\mathcal{W}_{i}$.

$c_{2}$$\mathcal{V}_{1}$$\mathcal{V}_{1}$$\mathcal{V}_{2}$$c_{2}$$c_{1}$$b_{2}$$b_{1}$$b_{1}$
Figure 12: In the communication round of the protocol, degrees of freedom on
the left future boundary of $\mathcal{V}_{1}$ and right future boundary of
$\mathcal{V}_{2}$ are passed to Alice1, these are shown in red. Degrees of
freedom on the right future boundary of $\mathcal{V}_{1}$ and left future
boundary of $\mathcal{V}_{2}$ are passed to Alice2, these are shown in orange.

It remains to understand if, in the second round, the output systems $B_{i}$
can be recovered by Alicei. To understand this, we should ask where the
entanglement wedges of the $\mathcal{W}_{i}$ subregions sit. See figure 13 for
an illustration of the surfaces appearing in this paragraph. Considering
$\gamma_{\mathcal{W}_{1}}$, there are two candidate geodesics for this
surface: the line $\gamma_{\mathcal{W}_{1}}^{\prime}$ at constant time
connecting these two points straight through the bulk, or a surface that
attaches to the brane, call it $\gamma_{\mathcal{W}_{1}}^{\prime\prime}$.
Taking the brane radius $R$ sufficiently small, we expect
$\gamma_{\mathcal{W}_{1}}^{\prime}$ is minimal. Explicitly, in appendix D we
find that this happens whenever the angular radius of the extra regions
$\Delta\phi$ is less than $39^{\circ}$. Since this is larger than the
condition we already took to get a connected geometry, this adds no new
constraint.

$\gamma_{\mathcal{W}_{1}}^{\prime}$$b_{2}$$b_{1}$$\mathcal{W}_{2}$$\mathcal{W}_{1}$$\mathcal{W}_{2}$$\mathcal{W}_{1}$
(a)

$\gamma_{\mathcal{W}_{1}}^{\prime\prime}$$b_{2}$$b_{1}$$\mathcal{W}_{2}$$\mathcal{W}_{1}$$\mathcal{W}_{2}$$\mathcal{W}_{1}$
(b)

Figure 13: Connected (a) and disconnected (b) configurations of the minimal
surface enclosing the red system shown in figure 12. The orange subsystem has
a similar transition and entanglement wedge.

Given that the radial line $\gamma_{\mathcal{W}_{1}}^{\prime}$ defines the
wedge of $\mathcal{W}_{1}$ (and of its complement $\mathcal{W}_{2}$), there is
a simple way to describe its entanglement wedge. The wedge
$E_{\mathcal{W}_{1}}$ is just the wedge of the interval $(0,\pi)$ at time
$t=\pi/2$ taken in the spacetime $\mathcal{M}$, then restricted to
$\tilde{\mathcal{M}}$. This placement of entanglement wedges always ensures
$A_{1}$ enters the wedge of $\mathcal{W}_{1}$, and $A_{2}$ enters the wedge of
$\mathcal{W}_{2}$. To see this, observe that the two future boundaries of
$E_{\mathcal{V}_{1}\mathcal{V}_{2}}$ are the past boundaries of
$E_{\mathcal{W}_{1}}$ and $E_{\mathcal{W}_{2}}$. Thus, $A_{1}$, since it
travels to $r_{1}$, enters $E_{\mathcal{W}_{1}}$, and similarly $A_{2}$ enters
$E_{\mathcal{W}_{2}}$. We only need to see that this happens before these
systems leave $\tilde{\mathcal{M}}$ by colliding with the branes, which again
is true because the brane sits (strictly) outside of
$E_{\mathcal{V}_{1}\mathcal{V}_{2}}$. We illustrate this in figure 14.

$r_{2}$$r_{1}$$\gamma_{\mathcal{W}_{i}}$$b_{1}$$b_{2}$ Figure 14: Cross
section through the bulk of $\tilde{\mathcal{M}}$. Branes $b_{1}$, $b_{2}$
travel in a timelike direction, meeting somewhere to the future of
$E_{\mathcal{V}_{1}\mathcal{V}_{2}}$. The shaded region is a cross section of
the scattering region. Outputs pass through its two future boundaries,
travelling towards $r_{1}$ and $r_{2}$ respectively. The entangling surface
$\gamma_{\mathcal{W}_{1}}=\gamma_{\mathcal{W}_{2}}$ sits at the future edge of
$E_{\mathcal{V}_{1}\mathcal{V}_{2}}$, so these outputs pass into
$E_{\mathcal{W}_{1}}$, $E_{\mathcal{W}_{2}}$, and do so before reaching the
ETW branes.

Since $A_{1}$ sits in $E_{\mathcal{W}_{1}}$ and $A_{2}$ in
$E_{\mathcal{W}_{2}}$, Alice1 and Alice2 can use entanglement wedge
reconstruction to recover the output systems. Further, because we consider
input systems living in a subspace small enough not to move the entangling
surface, this can be done via a recovery channel which is universal for all
states of the input hayden2019learning .

Entanglement in the BCFT state

So far, we have established that the state $\ket{\tilde{\Psi}}_{V_{1}V_{2}}$
suffices to perform a computation in the standard non-local scenario whenever
it can be completed in the bulk scattering region. Next, we study how much
entanglement is available in $\ket{\tilde{\Psi}}_{V_{1}V_{2}}$, and how this
compares to the entanglement in the original CFT state
$\ket{\Psi}_{V_{1}V_{2}}$.

To relate these quantities it is helpful to recall the definition of the
_entanglement wedge cross section_ , and its relationship to the mutual
information. Informally, given two boundary subregions $\mathcal{A}$,
$\mathcal{B}$, the entanglement wedge cross section is defined by finding the
entanglement wedge of $\mathcal{A}\cup\mathcal{B}$, call it
$E_{\mathcal{A}\mathcal{B}}$, and the minimal area extremal surface $\gamma$
that divides $E_{\mathcal{A}\mathcal{B}}$ into a portion homologous to
$\mathcal{A}$ and portion homologous to $\mathcal{B}$. The entanglement wedge
cross section is then

$\displaystyle E_{W}(A:B)=\frac{\text{area}(\gamma)}{4G_{N}}.$ (49)

In dutta2021canonical , twice the entanglement wedge cross section was shown
to be equal to an entanglement quantity known as the reflected entropy, and
denoted $S_{R}(A:B)$, so that

$\displaystyle S_{R}(A:B)=2E_{W}(A:B).$ (50)

In hayden2021markov , it was proven that for holographic states the reflected
entropy and the mutual information are related by the inequality,

$\displaystyle S_{R}(A:B)-I(A:B)\geq\frac{\log(2)\ell}{2G_{N}}k$ (51)

where $k$ is the number of boundary points of $\gamma$. In the setting we
apply this we have $k=2$. This becomes an equality in the limit where
$\mathcal{A}$, $\mathcal{B}$ occupy the entire boundary.

$\gamma_{\mathcal{V}_{1}}$$V_{2}$$V_{1}$ Figure 15: A constant time slice of
AdS2+1 with two ETW branes (shown in gray). The minimal surface (blue)
homologous to $\mathcal{V}_{1}$, $\gamma_{\mathcal{V}_{1}}$, connects the two
branes.

Consider the entanglement wedge of $\mathcal{V}_{1}$ in the BCFT geometry
$\tilde{\mathcal{M}}$. The minimal surface enclosing this will be the one
shown in figure 15, that joins the two ETW branes. Because we’ve considered
zero tension branes, the brane trajectory corresponds to the minimal surfaces
that define $E_{\mathcal{V}_{1}\mathcal{V}_{2}}$ in the original geometry
$\mathcal{M}$. The area of the minimal surface in $\tilde{\mathcal{M}}$ is
then equal to the entanglement wedge cross section measured in the original
geometry $\mathcal{M}$. This leads to

$\displaystyle I(V_{1}:V_{2})_{\tilde{\Psi}}=2E_{W}(V_{1}:V_{2})_{\Psi}.$ (52)

This means that the mutual information in the BCFT state is equal to the
canonical entropy in the original CFT state, which from equation 50 is twice
the entanglement wedge cross section, giving

$\displaystyle I(V_{1}:V_{2})_{\tilde{\Psi}}=S_{R}(V_{1}:V_{2})_{\Psi}.$ (53)

Finally, from hayden2021markov  we have that for large regions
$\mathcal{V}_{1}$, $\mathcal{V}_{2}$, the canonical entropy and mutual
information become close, so as $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ become
large we have

$\displaystyle
I(V_{1}:V_{2})_{\tilde{\Psi}}=S_{R}(V_{1}:V_{2})_{\Psi}\rightarrow
I(V_{1}:V_{2})_{\Psi}+\frac{\log(2)\ell}{G_{N}}.$ (54)

We find that the mutual information in the original CFT state and the
introduced BCFT state are becoming close, up to an additive constant.

Recapping this discussion, we saw in the last section that the entanglement in
the BCFT state $\tilde{\Psi}$ suffices to reproduce in the non-local form any
computations happening inside the scattering region. The mutual information in
the BCFT state is expressed in terms of the original CFT state as the
reflected entropy $S_{R}(V_{1}:V_{2})_{\Psi}$. In the limit of large
scattering regions, $S_{R}(V_{1}:V_{2})_{\Psi}$ is equal to the mutual
information in the original CFT state, up to an additive constant, so in that
limit the mutual information in the original CFT state controls the needed
entanglement to support the computation happening inside the scattering
region.

###  5.4 Extension to higher dimensional global AdS

In higher dimensional global AdS we can also relate bulk interactions to non-
local computation, though (as happens with the two sided case in higher
dimensions) we lose some of our quantitative statements. For concreteness,
let’s consider AdS3+1/CFT2+1.

$\theta$$\phi$ (a)

(b)

Figure 16: (a) A constant time slice of global AdS2+1. (b) The CFT is replaced
with two BCFTs defined on caps. The brane is expected to take on a connected
configuration as shown when the caps are made large enough.

We show a single time slice of AdS3+1 in figure LABEL:fig:ads2+1global. A
basic obstruction to applying the ideas used above in this setting is that
whenever we have in the bulk

$\displaystyle J_{12\rightarrow 12}=J^{+}(c_{1})\cap J^{+}(c_{2})\cap
J^{-}(r_{1})\cap J^{-}(r_{2})\neq\emptyset$ (55)

we will also have $\hat{J}_{12\rightarrow 12}\neq\emptyset$. Thus in this
setting scattering in the bulk comes along with scattering in the boundary,
and we can’t immediately argue entanglement in the boundary is needed.
However, our understanding of the AdS/CFT dictionary suggests the correct
picture for how the boundary reproduces the bulk interaction is not that it is
implemented directly as a local interaction in the boundary scattering region.
Instead, since the bulk scattering region is recorded only into large boundary
regions, we still expect some entanglement based boundary process is
supporting this bulk physics. We can justify this intuition precise using a
brane construction.

To construct our brane solution, we replace the CFT with two BCFTs, defined on
caps restricted to $\phi<\pi/2-\epsilon$, $\phi>\pi+\phi_{0}$, and look for
brane solutions that end on $\phi=\pi/2\pm\epsilon$. This corresponds to a CFT
with a strip of angular width $2\epsilon$ around the equator of the sphere
removed. As the strip becomes thin enough we expect the bulk geometry connects
the two caps, leaving the bulk geometry connected, and that we see this
behaviour in any number of dimensions.121212This was also discussed in detail
in van2020spacetime . We will assume the strip is chosen small enough such
that this is the case, so that the brane solution takes on the topology shown
in figure LABEL:fig:ads3+1brane.

We can use this brane geometry to argue interactions happening in the bulk of
global AdS3+1 can be reproduced as non-local quantum computations as follows.
We take the two caps on which our BCFTs are defined, call them
$\mathcal{C}_{L}$ and $\mathcal{C}_{R}$, to be our input regions, and then
define output regions

$\displaystyle\mathcal{R}_{+}$ $\displaystyle=\\{q:0<\theta<\pi,t=T\\}$
$\displaystyle\mathcal{R}_{-}$ $\displaystyle=\\{q:-\pi<\theta<0,t=T\\}$ (56)

to be the output regions. We define the scattering region as before,

$\displaystyle J_{LR\rightarrow+-}=J^{+}(\mathcal{C}_{R})\cap
J^{+}(\mathcal{C}_{L})\cap J^{-}(\mathcal{R}_{+})\cap J^{+}(\mathcal{R}_{-})$
(57)

which captures the portion of the bulk where interactions can be reproduced in
the non-local form. Explicitly, to do this one inserts input systems into the
bulk by acting on the input regions, and has them interact and send outputs
towards the $+$ and $-$ regions. Using the communication round of the non-
local computation to exchange degrees of freedom, the second round operations
then act on the $+$ and $-$ regions to recover the outputs.

It remains to argue that, at least for $\epsilon$, $T$ small enough, the
scattering region is non-empty. To see this, first notice that if we make
$\epsilon$, $T$ small enough the extremal surface attached to $\theta=0$ and
$\theta=\pi$, $t=T$ will connect the two BCFTs. We claim that whenever this is
the case at least the point $p$ at $t=r=0$ is inside the scattering region. To
see this, note that the extremal surface enclosing the left (right) BCFT at
$t=0$ is the $\phi=\pi/2$ plane, which includes $p$, so

$\displaystyle p\in J^{+}(\mathcal{C}_{L})\cap J^{+}(\mathcal{C}_{R}).$ (58)

It remains to show $p$ is inside the past of the wedges $E_{\mathcal{R}_{-}}$,
$E_{\mathcal{R}_{+}}$. To see this, define the regions

$\displaystyle\mathcal{R}_{+}^{\prime}$
$\displaystyle=\\{q:0<\theta<\pi,t=-T\\}$
$\displaystyle\mathcal{R}_{-}^{\prime}$
$\displaystyle=\\{q:-\pi<\theta<0,t=-T\\}$ (59)

Notice that the entangling surface for $E_{\mathcal{R}_{+}}$ must sit at $t>0$
when it reaches $r=0$. If it didn’t, by symmetry it would cross the entangling
surface for $E_{\mathcal{R}^{\prime}_{+}}$, which can’t happen by causality.
Similarly, the entangling surface for $E_{\mathcal{R}_{-}}$ must sit at $t>0$
when it reaches $r=0$. But then both entangling surface have $p$ in their
past, so

$\displaystyle p\in J^{+}(\mathcal{C}_{L})\cap J^{+}(\mathcal{C}_{R})\cap
J^{-}(\mathcal{R}_{-})\cap J^{-}(\mathcal{R}_{+})$ (60)

and hence the scattering region is non-empty.

##  6 Discussion

A basic lesson of may2019quantum ; may2020holographic ; may2022connected  is
that boundary entanglement plays a necessary role in supporting bulk
interaction. Non-local computation provides a quantum information theoretic
way to understand this, and in particular to explore the many questions around
what interactions can be supported by entanglement in this way, and conversely
what holography implies about non-local computation. In this work, we extended
this basic lesson to higher dimensions and two sided black holes.

With briefly conclude with a few comments on future directions or
relationships to other work.

Complexity and the black hole singularity

There is a general expectation that high complexity unitaries should require
large entanglement to implement in the non-local form. Assuming this, our
construction would give that high complexity unitaries are forbidden from
being implemented inside reasonably small sized subregions of the black hole.
From a bulk perspective, this is a unsurprising claim: we might expect high
complexity operations to require large physical time to implement,131313For
another perspective on this, see kubicki2023constraints . and there is finite
time before reaching the singularity.

What is perhaps more interesting is that the non-local computation perspective
suggests a boundary understanding of the appearance of this finite time, and
hence on the appearance of this singularity. We suggest that the CFT is
limited by its finite entanglement to only allow low complexity computations
to happen, and the bulk dual “geometrizes” this constraint via the appearance
of the bulk singularity, which ends bulk time and limits computation.141414We
thank Steve Shenker for comments made to us in this direction.

Efficiency of non-local computation and holography

An interesting tension between the efficiency of the current best non-local
computation strategies and expectations of what it should be possible to
compute in a subregion of the bulk was raised in may2020holographic  and
explored in may2022complexity . One way out of this tension was, previously,
to recall that holography is most naturally related to ‘augmented’ non-local
computations, and that entanglement between the input regions is actually
unnecessary in these augmented non-local computations. While it was argued
earlier that the standard non-local computation scenario is the relevant one
on heuristic grounds in may2020holographic ; may2022complexity , and by going
to the lattice setting in dolev2022holography , our construction gives another
sharp setting where the efficiency of existing non-local computation protocols
is in tension with having reasonable bulk computations. Another route out of
this tension was previously to point to the connection between non-local
computation and holography being most precise in low dimensions, where it
would be less surprising if bulk interactions were severely limited. This
route out of the tension is now also closed.

Because non-local quantum computations are cheating strategies in the
cryptographic setting of position-verification buhrman2014position , the
suggestion from holography that many computations should be efficiently
implementable has potential practical implications. Conversely, an interesting
possibility is to use constraints on non-local computation to constrain bulk
physics, and in particular constrain the complexity of computation happening
in the bulk may2022complexity . Our construction further supports that such
entanglement constraints should constrain bulk physics.

Non-local computation with fixed second round

As noted in section 3.1, we related bulk interactions to non-local
computations where the second round is fixed, in the sense that the second
round doesn’t depend on the choice of interaction or on the inputs. Non-local
computation with a fixed second round has already appeared in the quantum
information literature. In junge2022geometry  the authors assumed a fixed
second round and derived interesting new constraints on non-local computation,
under some mathematical conjectures on type constants in Banach spaces. Fixed
second rounds also appear naturally elsewhere in cryptography
kawachi2021recent 151515In this reference the analogous idea is of “universal
reconstruction”.. We were not able to connect the constraints of
junge2022geometry  directly to holography and our setting, but this work does
suggest that adding that the second round is fixed may allow for stronger
constraints on bulk computation to be understood. Another comment is that all
non-local computation protocols devised so far do involve fixed second rounds,
and the observation here is just that protocols built from studying the
holographic setting also share this property.

Sub-AdS scale scattering regions

A basic limitation appearing here and in dolev2022holography  to connecting
bulk interaction and (non-augmented) non-local computation is the need to
restrict to bulk regions of at least AdS scale. It would be interesting to
explore further if this limitation can be overcome, which would provide a
quantum information theoretic point of view on the emergence of sub-AdS local
interactions in AdS/CFT. To construct such a situation within our setting, we
would need to find geometries where our branes can come within an AdS distance
of each other while remaining the dominant saddle in the path integral, and
while keeping the connected (non-brane anchored) extremal surface minimal.
Towards keeping the connected brane geometry dominant, we could consider
variants of our geometries involving charged black hole and charged branes,
and we could consider branes with non-zero (and perhaps differing) tensions,
and we could adjust the black hole temperature, which here was implicitly
fixed. This would give a richer phase space for the phase transition from
connected to disconnected branes, potentially allowing the branes to come
within a sub-AdS distance of each other while remaining disconnected. To keep
the connected extremal surface minimal, we could consider adding a dilaton
field to the branes, which can be used to raise the generalized entropy
associated with minimal surfaces ending on the brane almheiri2020page .

ETW brane geometries and information processing

The work van2020spacetime  proposes entangled BCFTs as an alternative set of
degrees of freedom that, prepared in appropriate states, can approximate
holographic CFTs. An advantage to this alternative description is the finite
entanglement among subsystems. Combined with the perspective of bao2019beyond
on compressing holographic states, this allowed us to move from a description
of a quantum information processing protocol in a holographic CFT, involving
infinite dimensional systems, to a (approximate) description using only finite
dimensional systems.

Finding a finite dimensional description of these holographic protocols was
discussed in dolev2022holography  under an assumption the holographic CFT is
well approximated by a lattice system, and more commonly relating AdS/CFT to
quantum information processing protocols has been discussed in the context of
tensor network toy models. AdS/BCFT combined with compression results has a
set of advantageous features not reproduced in those settings: it gives a
concrete model for what bulk physics is being captured (the original geometry,
ended by an ETW brane), can be justified and studied directly from the
standpoint of the Euclidean path integral, and has top-down realizations.
Moving beyond non-local computation, this seems a promising approach to
relating other quantum information processing protocols to holography. Another
recent example of AdS/BCFT being used to model interesting information
theoretic manipulations of holographic CFTs is given in
antonini2022holographic ; antonini2023holographic .

Acknowledgements: We thank Patrick Hayden, David Pérez-Garcia, Shreya Vardhan,
Henry Lin, Stefano Antonini, and Raghu Mahajan for helpful discussions. AM is
supported by the Simons Foundation It from Qubit collaboration, a PDF
fellowship provided by Canada’s National Science and Engineering Research
council, and by Q-FARM. MX is supported by an NSF Graduate Research Fellowship
and the Simons Foundation.

##  Appendix A Coordinate systems

We summarize the coordinate systems used in this article. We describe our
coordinates and their relationships using the embedding space formalism. In
particular, our coordinates are parameterizations of the surface
$X^{A}X_{A}=X_{0}^{2}-X_{1}^{2}-X_{2}^{2}+X_{3}^{2}=\ell^{2}$, in the metric
$\text{diag}(1,-1,-1,1)$.

Lorentzian Poincare AdS2+1 is described by coordinates.

$\displaystyle X_{0}$
$\displaystyle=\frac{z}{2}\left(1+\frac{\ell^{2}+x^{2}-t^{2}}{z^{2}}\right)$
$\displaystyle X_{1}$
$\displaystyle=\frac{z}{2}\left(1-\frac{\ell^{2}-x^{2}+t^{2}}{z^{2}}\right)$
$\displaystyle X_{2}$ $\displaystyle=\ell\frac{x}{z}$ $\displaystyle X_{3}$
$\displaystyle=\ell\frac{t}{z}$ (61)

Lorentzian global AdS2+1 is described by

$\displaystyle X_{0}$ $\displaystyle=\ell\frac{\cos(t)}{\cos(r)}$
$\displaystyle X_{1}$ $\displaystyle=\ell\tan(r)\sin(\phi)$ $\displaystyle
X_{2}$ $\displaystyle=\ell\tan(r)\cos(\phi)$ $\displaystyle X_{3}$
$\displaystyle=\ell\frac{\sin(t)}{\cos(r)}$ (62)

The global, Lorentzian coordinates of the planar BTZ black hole are

$\displaystyle X_{0}$ $\displaystyle=\ell\cos(s)\sec(w)\cosh(X)$
$\displaystyle X_{1}$ $\displaystyle=\ell\cos(s)\sec(w)\sinh(X)$
$\displaystyle X_{2}$ $\displaystyle=\ell\tan(w)$ $\displaystyle X_{3}$
$\displaystyle=\ell\sin(s)\sec(w)$ (63)

One exterior region of the planar BTZ black hole is covered by the coordinates

$\displaystyle X_{0}$
$\displaystyle=\ell\sqrt{1+\frac{\rho^{2}}{\ell^{2}}}\cosh X$ $\displaystyle
X_{1}$ $\displaystyle=\ell\sqrt{1+\frac{\rho^{2}}{\ell^{2}}}\sinh X$
$\displaystyle X_{2}$ $\displaystyle=\rho\cosh t$ $\displaystyle X_{3}$
$\displaystyle=\rho\sinh t$ (64)

##  Appendix B Brane in Poincare AdS2+1

###  B.1 Single brane solution

(a)

(b)

Figure 17: (a) The Euclidean CFT path integral with boundary conditions set
along a disk at $x^{2}+\tau^{2}=R^{2}$. (b) The dual bulk geometry, with a
zero tension brane at $x^{2}+\tau^{2}+z^{2}=R^{2}$.

The simplest brane solution we study is dual to the Euclidean path integral
shown in figure LABEL:fig:puncturedplane. There, we are considering the
Euclidean CFT path integral on the plane with no operator insertions, and a
boundary condition set at $x^{2}+\tau^{2}=R^{2}$. We will choose this boundary
condition such that the boundary entropy is zero. The bulk dual then will be a
zero tension brane extending into the bulk and meeting the asymptotic boundary
at $x^{2}+\tau^{2}=R^{2}$.

The solution was studied already in rozali2020information , who found a
Euclidean Poincare AdS3 bulk metric

$\displaystyle ds^{2}=\frac{\ell^{2}}{z^{2}}(dx^{2}+d\tau^{2}+dz^{2})$ (65)

and brane trajectory defined by

$\displaystyle x^{2}+z^{2}+\tau^{2}=R^{2}.$ (66)

Variations of this solution, obtained by coordinate transformations, adding a
second circular boundary, or Wick rotating appear throughout this article. One
Lorentzian brane solution we will use is

$\displaystyle x^{2}+z^{2}-t^{2}=R^{2}.$ (67)

With the metric being Lorentzian Poincare AdS2+1. This is obtained by Wick
rotating the $\tau$ coordinate.

###  B.2 Minimal surfaces

Note that for $R=1$ these results can be extracted from rozali2020information
. We are just modifying their calculation to allow arbitrary $R$, and showing
the calculation directly in Lorentzian signature.

We start with minimal surfaces in the $t=0$ slice of Lorentzian Poincare AdS.
The brane is sitting at

$\displaystyle x^{2}+z^{2}=R^{2}.$ (68)

Consider an interval extending from the brane to $x=L$. The minimal surface
extending from this point is a portion of a circle, meeting the brane
orthogonally. In figure 18, we do some elementary geometry to determine the
radius of this circle and the angle it extends over.

$R$$x=0$$x=L$$\alpha$$r_{H}$ Figure 18: The $t=0$ slice of Euclidean Poincare
AdS3. An ETW brane (gray) centered at $x=0$ sits at $x^{2}+z^{2}=R^{2}$. A
minimal surface (blue) anchored to $x=L$ is a portion of a semi-circle and
meets the brane orthogonally.

We find

$\displaystyle r_{H}$ $\displaystyle=\frac{L^{2}-R^{2}}{2L},$
$\displaystyle\tan\alpha$ $\displaystyle=\frac{2LR}{L^{2}-R^{2}}.$ (69)

Thus the geodesic is given parametrically by

$\displaystyle x(\theta)$ $\displaystyle=L-r_{H}+r_{H}\cos(\theta),$
$\displaystyle t(\theta)$ $\displaystyle=0,$ $\displaystyle z(\theta)$
$\displaystyle=r_{H}\sin(\theta).$ (70)

where $0\leq\theta\leq\pi-\alpha$.

Calculating the area of these curves, we find, for the area of a geodesic
starting at $x=x_{0},t=0$,

$\displaystyle
A[x_{0},0]=\ell\log\left(\frac{x_{0}^{2}-R^{2}}{R\delta}\right)$ (71)

Note that here we use a cut-off at $z=\delta$, which means
$\theta=\delta/r_{H}$.

Observing that the brane trajectory is invariant under boosts in the $(x,t)$
plane, we can obtain the area of geodesics starting at general coordinates
$x_{0},t_{0}$ by writing the area in terms of the invariant interval,

$\displaystyle
A[x_{0},t_{0}]=\ell\log\left(\frac{x_{0}^{2}-t_{0}^{2}-R^{2}}{R\delta}\right)$
(72)

We will make use of this below, by exploiting a local equivalence of the ETW
branes in the planar BTZ black hole to this Poincare AdS brane.

##  Appendix C Planar BTZ black hole

###  C.1 Brane trajectories

Consider the Euclidean path integral on a finite cylinder,

$\displaystyle ds^{2}=dX^{2}+d\phi^{2}\qquad\phi\in[0,2\pi)$ (73)

with boundary conditions set at $X=\pm X^{0}$. Choose the boundary conditions
to have zero boundary entropy, dual to a bulk solution with a zero tension
brane.

There are two possible bulk solutions corresponding to this Euclidean path
integral, which correspond to having either two separate branes ending on each
CFT edge, or a single brane that attaches the two edges together. Putting
boundary conditions at $X=\pm X^{0}$, a result of cooper2019black  (see
equation A.24 there) implies the disconnected solution is the minimal action
one for

$\displaystyle X^{0}\geq\frac{\pi}{2}.$ (74)

With this condition, the bulk metric dual to the path integral on the cylinder
is

$\displaystyle
ds^{2}=\left(\frac{\rho^{2}}{\ell^{2}}+1\right)dX^{2}+\frac{d\rho^{2}}{\frac{\rho^{2}}{\ell^{2}}+1}+\rho^{2}d\phi^{2},$
(75)

and the brane trajectory is just

$\displaystyle X=\pm X^{0}.$ (76)

Lorentzian solution

Assuming we are in the disconnected phase, Wick rotate $\phi\rightarrow it$ in
the metric 75 to obtain

$\displaystyle
ds^{2}=\left(\frac{\rho^{2}}{\ell^{2}}+1\right)dX^{2}+\frac{d\rho^{2}}{\frac{\rho^{2}}{\ell^{2}}+1}-\rho^{2}dt^{2}$
(77)

This covers one exterior region of the black hole. A global metric for this
black hole, covering the full spacetime, is

$\displaystyle
ds^{2}=\frac{\ell^{2}}{\cos^{2}(w)}\left(-ds^{2}+dw^{2}+\cos^{2}(s)dX^{2}\right)$
(78)

where $s\in[-\pi/2,\pi/2]$, $w\in[-\pi/2,\pi/2]$. From section A we have that
these two coordinate systems are related by

$\displaystyle X_{0}/\ell$
$\displaystyle=\sqrt{1+\frac{\rho^{2}}{\ell^{2}}}\cosh
X=\cos(s)\sec(w)\cosh(X)$ $\displaystyle X_{1}/\ell$
$\displaystyle=\sqrt{1+\frac{\rho^{2}}{\ell^{2}}}\sinh
X=\cos(s)\sec(w)\sinh(X)$ $\displaystyle X_{2}/\ell$
$\displaystyle=\frac{\rho}{\ell}\cosh t=\tan(w)$ $\displaystyle X_{3}/\ell$
$\displaystyle=\frac{\rho}{\ell}\sinh t=\sin(s)\sec(w)$ (79)

Note that in particular the $X$ coordinates in the two spacetimes are
identified trivially.

###  C.2 Minimal surfaces

We’ll use the global coordinates 78 for the two sided planar BTZ black hole,

$\displaystyle
ds^{2}=\frac{\ell^{2}}{\cos^{2}(w)}\left(-ds^{2}+dw^{2}+\cos^{2}(s)dX^{2}\right).$
(80)

From the coordinate transformations in section A, this is related to
Lorentzian Poincare coordinates by the transformation

$\displaystyle t$ $\displaystyle=\ell\,e^{X}\tan(s),$ $\displaystyle x$
$\displaystyle=\ell\,e^{X}\sec(s)\sin(w),$ $\displaystyle z$
$\displaystyle=\ell\,e^{X}\sec(s)\cos(w).$ (81)

We would like to understand when the connected surface (the one that threads
through the black hole) is minimal. In global coordinates, the minimal surface
attached to $w=\pm\pi/2$, $s=T$, and at constant $X=0$ is

$\displaystyle s$ $\displaystyle=T,$ $\displaystyle w$
$\displaystyle=\arcsin(\tanh(\lambda)).$ (82)

The area, setting a cut-off $w\in[-\pi/2+\epsilon,\pi/2-\epsilon]$, is

$\displaystyle A_{c}[T]$
$\displaystyle=2\ell\log\left(\frac{2}{\epsilon}\right).$ (83)

Next, we look for the trajectory of the brane anchored geodesic. We find this
by starting with the solution in Poincare, and transforming to global
coordinates. In Poincare, the endpoint of our geodesic is

$\displaystyle x_{0}$ $\displaystyle=\pm\sec(T)$ $\displaystyle t_{0}$
$\displaystyle=\tan(T)$ (84)

and the brane radius $R$ is related to the position of the brane in the BTZ
black hole geometry by

$\displaystyle X_{0}=\ln R.$ (85)

We would like to use the area formula 72 to determine the area of these brane
attached surfaces, but to do so need to know what the $w=\pi/2-\epsilon$
cutoff translates to in terms of a $z=\delta$ cutoff. The point
$s=T,X=0,w=\pi/2-\epsilon$ translates to

$\displaystyle z=\delta=\epsilon\sec(T)$ (86)

Now using the area formula for the disconnected geodesic, we have

$\displaystyle
A_{d}[T]=2\ell\log\left(\frac{2\sinh|X_{0}|\cos(T)}{\epsilon}\right).$ (87)

To be in the connected phase, we need

$\displaystyle 2\ell\log\left(\frac{2\sinh|X_{0}|\cos
T}{\epsilon}\right)-2\ell\log\left(\frac{2}{\epsilon}\right)\geq 0$ (88)

which simplifies to

$\displaystyle\boxed{\sinh|X_{0}|\geq\sec T}.$ (89)

##  Appendix D Global AdS2+1

###  D.1 Euclidean brane trajectories

We are interested in finding branes ending the Euclidean global AdS spacetime

$\displaystyle
ds^{2}=\frac{\ell^{2}}{\cos^{2}(r)}(d\tau_{G}^{2}+dr^{2}+\sin^{2}(r)d\phi^{2}).$
(90)

Heuristically, we would like solutions that have boundary conditions set along
roughly circular curves located antipodally on the cylinder. To construct a
precise brane geometry, consider the coordinate transformation to Poincare
AdS, given by

$\displaystyle\tan(r)$ $\displaystyle=\frac{\sqrt{x^{2}+\tau^{2}}}{z},$
$\displaystyle\tanh(\tau_{G})$
$\displaystyle=\frac{z^{2}+x^{2}+\tau^{2}-\ell^{2}}{z^{2}+x^{2}+\tau^{2}+\ell^{2}},$
$\displaystyle\tan(\phi)$ $\displaystyle=\frac{\tau}{x}.$ (91)

Or, inverting this,

$\displaystyle\tau$ $\displaystyle=\ell\sin(r)\sin(\phi)e^{\tau_{G}},$
$\displaystyle x$ $\displaystyle=\ell\sin(r)\cos(\phi)e^{\tau_{G}},$
$\displaystyle z$ $\displaystyle=\ell\cos(r)e^{\tau_{G}}.$ (92)

The resulting Poincare AdS metric is

$\displaystyle ds^{2}=\frac{\ell^{2}}{z^{2}}(dz^{2}+dx^{2}+d\tau^{2}).$ (93)

We will choose a very simple placement of the boundary conditions in Poincare
AdS, which will map to branes in global AdS with appropriate qualitative
features, and in particular will give the brane trajectory stated in the main
text as equation 44. In Poincare, we set boundary conditions at

$\displaystyle(x\pm\ell x_{0})^{2}+\tau^{2}=R^{2}$ (94)

There is a solution consisting of two separate hemispheres attached to each of
these edges, and a solution where the brane connects the two edges. We would
like to understand when the disconnected solution has minimal action. By doing
a conformal transformation, we can observe that this phase transition is the
same one as was studied in cooper2019black , and which we also exploited in
appendix C. In particular we can map the exterior of the two disks to the
finite cylinder. The disk radii and separation fix the cylinder height.

Conformal map to the cylinder

Figure 19: The stereographic mapping from the plane with two disks removed to
the sphere is conformal, and takes twice punctured plane to a band around the
sphere. A second stereographic map from the sphere to the dashed plane takes
the band to an annulus. The radial conformal map then maps the annulus to the
cylinder.

To map our plane with two disks removed to the cylinder, we will first map to
the stereographic sphere, then project back down into a rotated plane. In the
rotated plane, the resulting region is an annulus, which is conformal to a
cylinder with finite height by the usual exponential map. The basic idea is
shown in figure 19.

To go from the stereographic coordinates to Cartesian coordinates, we have the
map

$\displaystyle x$ $\displaystyle=\frac{2r\cos\phi}{1-\sin\phi\cos\theta},$
$\displaystyle\tau$
$\displaystyle=\frac{2r\sin\phi\sin\theta}{1-\sin\phi\cos\theta}.$ (95)

We can check that constant $\phi=\phi_{0}$ curves map to circles in the
$(x,\tau)$ plane. The circle parameters $(x_{0},R)$ are related to
$(r,\phi_{0})$ by

$\displaystyle R$ $\displaystyle=2r\tan\phi_{0},$ $\displaystyle x_{0}$
$\displaystyle=2r\sec\phi_{0}.$ (96)

Recall that to make the Wick rotation, we needed to set $x_{0}^{2}-R^{2}=1$.
Here this condition amounts to $r=1/2$. We can also invert the above (keeping
$r$ free), to find

$\displaystyle r$ $\displaystyle=\frac{1}{2}\sqrt{x_{0}^{2}-R^{2}},$
$\displaystyle\tan\phi_{0}$ $\displaystyle=\frac{R}{\sqrt{x_{0}^{2}-R^{2}}}.$
(97)

Next, we will map to a second plane, this one tilted 90 degrees compared to
the first. We show the setting again in figure 19. The new plane coordinates
$\bar{x},\bar{\tau}$ are given in terms of the spherical coordinates by

$\displaystyle\bar{x}$
$\displaystyle=\frac{2r}{1-\cos\phi}\sin\phi\cos\theta,$
$\displaystyle\bar{\tau}$
$\displaystyle=\frac{2r}{1-\cos\phi}\sin\phi\sin\theta.$ (98)

This takes the two caps of the sphere to a disk and an punctured plane, so
that the remaining CFT lives on an annulus. The inner and outer radii are

$\displaystyle r_{+}$
$\displaystyle=\frac{2r\sin\phi_{0}}{1-\cos\phi_{0}}=\frac{R\sqrt{x_{0}^{2}-R^{2}}}{x_{0}-\sqrt{x_{0}^{2}-R^{2}}},$
$\displaystyle r_{-}$
$\displaystyle=\frac{2r\sin\phi_{0}}{1+\cos\phi_{0}}=\frac{R\sqrt{x_{0}^{2}-R^{2}}}{x_{0}+\sqrt{x_{0}^{2}-R^{2}}}.$
(99)

We can also invert D.1 to obtain

$\displaystyle\tan\theta$ $\displaystyle=\frac{\bar{\tau}}{\bar{x}}$
$\displaystyle\sin\phi$
$\displaystyle=\frac{4r\sqrt{\bar{x}^{2}+\bar{\tau}^{2}}}{\bar{x}^{2}+\bar{\tau}^{2}+4r^{2}}$
(100)

The coordinate change from the initial $(x,\tau)$ plane to the
$(\bar{x},\bar{\tau})$ plane is

$\displaystyle\bar{x}$
$\displaystyle=2r\frac{x^{2}+\tau^{2}-4r^{2}}{(x-2r)^{2}+\tau^{2}}$
$\displaystyle\bar{\tau}$
$\displaystyle=8r^{2}\frac{\tau}{(x-2r)^{2}+\tau^{2}}$ (101)

We can check explicitly that this is conformal, and maps circles centered
around the origin in one set of coordinates to circles offset in the $x$
direction in the other coordinates.

Finally, we go to the cylinder. Using radial coordinates $(r,\phi)$ in the
plane, the map to the cylinder is given by $r=e^{X}$. The height of the
cylinder is therefore

$\displaystyle
H=X_{+}-X_{-}=\ln\left(\frac{r_{+}}{r_{-}}\right)=\ln\left(\frac{x_{0}+\sqrt{x_{0}^{2}-R^{2}}}{x_{0}-\sqrt{x_{0}^{2}-R^{2}}}\right).$
(102)

Comparison to the connected solution

Recall from section C equation 74 that the finite cylinder is in the
disconnected phase when

$\displaystyle H\geq\pi$ (103)

which using 102 leads to

$\displaystyle 1\leq x_{0}\leq\frac{e^{\pi}+1}{e^{\pi}-1}.$ (104)

Note the first inequality is automatically satisfied by our previous condition
for Wick rotation, $x_{0}^{2}-R^{2}=1$. In the next section, we will relate
the shift parameter $x_{0}$ to the angular opening of the brane when viewed in
global AdS. In particular, we find $x_{0}=\sec\Delta\phi$ for $\Delta\phi$ the
angular radius of the region. The condition on $\Delta\phi$ then is
$\Delta\phi\lesssim 24^{\circ}$.

###  D.2 Disconnected branes in global AdS2+1

The disconnected solution is

$\displaystyle(x\pm\ell x_{0})^{2}+\tau^{2}+z^{2}=R^{2}$ (105)

Now we use the coordinate transformation D.1 again to express this solution in
global coordinates, finding

$\displaystyle\frac{1}{2}(e^{\tau_{G}}+e^{-\tau_{G}}(x_{0}^{2}-R^{2}))=\mp
x_{0}\sin(r)\cos(\phi)$ (106)

To obtain Lorentzian global AdS, we will Wick rotate $\tau_{G}\rightarrow
it_{G}$. To ensure this brane solution is well defined after the Wick
rotation, we need to set $x_{0}^{2}-R^{2}=1$, so that the surface remains
real. Doing so and Wick rotating, we obtain

$\displaystyle\cos(t_{G})=\mp x_{0}\sin(r)\cos(\phi)$ (107)

We can identify the two choices of sign with a shift in the angular
coordinate, so that we have two branes in the geometry, with

$\displaystyle\cos(t_{G})$ $\displaystyle=x_{0}\sin(r)\cos(\phi)$
$\displaystyle\cos(t_{G})$ $\displaystyle=x_{0}\sin(r)\cos(\phi-\pi)$ (108)

Finally, we can identify $x_{0}$ with $1/\cos(\Delta\phi)$ by looking at this
equation restricted to $r=\pi/2$, $t=0$.

###  D.3 Minimal surfaces

Next we need to find minimal surfaces in the global geometry. We consider an
interval ending at $t_{G}$, and $\phi=0,\pi$, and look for the minimal surface
that encloses it.

The transformation from global to Poincare, in Lorentzian signature, is

$\displaystyle t$
$\displaystyle=\frac{\ell\sin(t_{G})}{\cos(t_{G})-\sin(\phi)\sin(r)}$
$\displaystyle x$
$\displaystyle=\frac{\ell\cos(\phi)\sin(r)}{\cos(t_{G})-\sin(\phi)\sin(r)}$
$\displaystyle z$
$\displaystyle=\frac{\ell\cos(r)}{\cos(t_{G})-\sin(\phi)\sin(r)}$ (109)

Under this coordinate transformation, we get to the global AdS metric 43 with
the brane

$\displaystyle\cos(t_{G})=\frac{\ell^{2}+R^{2}}{\ell^{2}-R^{2}}\sin(\phi)\sin(r).$
(110)

Notice that $R$ is related to $\Delta\phi$ differently than when we
transformed from Euclidean Poincare to Euclidean global coordinates, in
particular

$\displaystyle\frac{\ell^{2}+R^{2}}{\ell^{2}-R^{2}}=\sec(\Delta\phi).$ (111)

We are interested in minimal surfaces anchored to $\phi=\pm\pi/2$, $t_{G}=s$.
This means $t=\tan(s)$, $x=\pm\sec(s)$, so that, using equation 72 for areas
of geodesics in Poincare, the brane anchored surface has area

$\displaystyle A_{b}=\ell\log\left(\frac{\ell^{2}-R^{2}}{R\delta}\right)$
(112)

where the cutoff is at $z=\delta$. To compare this to the connected solution,
we will need to translate this to a $r$ cut-off. Using D.3 with $\phi=0$, we
find

$\displaystyle\delta=\frac{\epsilon}{\cos(t_{G})+1}$ (113)

so that the disconnected minimal surface has area

$\displaystyle
A_{b}=2\ell\log\left(\frac{\cos(t)+1}{\epsilon\tan\Delta\phi}\right)$ (114)

where we used 111 to replace $R$ in favor of $\Delta\phi$. The minimal surface
extending directly though the bulk at constant $t$, $\phi$, has area

$\displaystyle A_{0}=2\ell\log\left(\frac{2}{\epsilon}\right).$ (115)

Requiring $A_{0}\leq 2A_{b}$ we have

$\displaystyle\boxed{\cos(t)\geq 2\tan\Delta\phi-1}$ (116)

Recall that in our protocol, we placed the cut at the top of the diamonds
$V_{i}$, which will be at time $t=\pi/2-\Delta\phi$. Inserting this into the
condition above, we find that we need $\Delta\phi\leq 39^{\circ}$ for the
connected surface to be minimal.

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