Operator algebra, quantum entanglement, and emergent geometry

from matrix degrees of freedom

Vaibhav Gautama,b, Masanori Hanadaa, and Antal Jevickic

a School of Mathematical Sciences, Queen Mary University of London
Mile End Road, London, E1 4NS, United Kingdom
b Department of Mathematics, University of Surrey
Guildford, Surrey, GU2 7XH, United Kingdom
c Department of Physics, Brown University
182 Hope Street, Providence, RI 02912, United States
c Brown Theoretical Physics Center, Brown University
340 Brook Street, Providence, RI 02912, United States

Abstract

For matrix model and QFT, we discuss how dual gravitational geometry emerges
from matrix degrees of freedom (specifically, adjoint scalars in super Yang-
Mills theory) and how operator algebra that describes an arbitrary region of
the bulk geometry can be constructed. We pay attention to the subtle
difference between the notions of wave packets that describe low-energy
excitations: QFT wave packet associated with the spatial dimensions of QFT,
matrix wave packet associated with the emergent dimensions from matrix degrees
of freedom, and bulk wave packet which is a combination of QFT and matrix wave
packets. In QFT, there is an intriguing interplay between QFT wave packet and
matrix wave packet that connects quantum entanglement and emergent geometry.
We propose that the bulk wave packet is the physical object in QFT that
describes the emergent geometry from entanglement. This proposal sets a
unified view on two seemingly different mechanisms of holographic emergent
geometry: one based on matrix eigenvalues and the other based on quantum
entanglement. Further intuition comes from the similarity to a traversable
wormhole discussed as the dual description of the coupled SYK model by
Maldacena and Qi: the bulk can be seen as an eternal traversable wormhole
connecting boundary regions.

###### Contents

  1. 1 Introduction
  2. 2 Review of wave packet in matrix model
    1. 2.1 Gauge-invariant Hilbert space and extended Hilbert space
    2. 2.2 Matrix wave packet in $\mathbb{R}^{9N^{2}}$
    3. 2.3 Geometry from matrix wave packets
    4. 2.4 Gravity dual
  3. 3 Algebra of small excitations in matrix model
    1. 3.1 Small excitations around the ground state
    2. 3.2 Small excitation around black hole geometry
    3. 3.3 A remark on terminology
  4. 4 QFT wave packet and AdS5/CFT4 duality
    1. 4.1 Global matrix wave packet and D3-brane in AdS5/CFT4
    2. 4.2 QFT wave packet and bulk wave packet
    3. 4.3 Bulk wave packet and closed string in AdS5/CFT4
      1. 4.3.1 Radial coordinate in gravity vs QFT
      2. 4.3.2 Size of boundary region vs radial coordinate
      3. 4.3.3 Comment on ($2+1$)-dimensional theory
      4. 4.3.4 Bulk wave packet and quantum entanglement
      5. 4.3.5 Traversable wormhole and bulk wave packet
  5. 5 Conclusion and discussion
  6. A Polchinski’s puzzle
  7. B Computation of three-point function in Sec. 4.3.1

##  1 Introduction

In this paper, we discuss how holographic emergent geometry can be described
by matrix degrees of freedom of the dual QFT or matrix model, polishing basic
ideas provided in Refs. [1, 2, 3].

Matrix degrees of freedom in gauge theory play important roles in holographic
emergent geometry [4, 5, 1, 2, 3]. For example, a small black hole in AdS can
be described by a partially-deconfined state in gauge theory [4, 5], and
entanglement between the confined sector and deconfined sector explains the
Bekenstein-Hawking entropy as von Neumann entropy [3]. The use of wave packet
in the space of matrix degrees of freedom [1, 2] is crucial in resolving a
puzzling conceptual issue raised by Polchinski [6] and Susskind [7]. To read
the emergent geometry from matrices, one has to carefully distinguish two
different notions of “target space”. In the case of the D0-brane matrix model,
there is a space $\mathbb{R}^{9}$ in which string theory lives, and a space of
matrix coordinates $\mathbb{R}^{9N^{2}}$ (i.e., there are nine matrices and
$N^{2}$ real numbers are needed to specify each of them). Wave packets live in
$\mathbb{R}^{9N^{2}}$, and from wave packets, we can extract the information
of $\mathbb{R}^{9}$. The confusion in the past came from the ignorance of this
point.111It is also possible to make a wave packet in $\mathbb{R}^{9}$ as a
superposition of wave packets in $\mathbb{R}^{9N^{2}}$.

Recently, holographic emergent geometry was discussed in terms of von Neumann
algebra associated with effective field theory around semiclassical geometry
on the gravity side [8, 9, 10]. To make further progress, it is important to
understand how this approach and matrix degrees of freedom are related. In
this paper, we tackle this problem. We will study the matrix model first, then
generalize the idea to QFT.

As a byproduct, we find a simple connection between the two seemingly
different mechanisms of emergent geometry: one from matrix degrees of freedom
[11, 4, 1, 2, 3] and the other from quantum entanglement [12, 13]. This is
achieved by paying attention to subtle differences in the notions of wave
packets that describe low-energy excitations: QFT wave packet associated with
the spatial dimensions of QFT, matrix wave packet associated with the emergent
dimensions from matrix degrees of freedom, and bulk wave packet which is a
combination of QFT and matrix wave packets. The duality dictionary we propose
for AdS5/CFT4 correspondence is the following:

QFT |  | gravity
---|---|---
bulk wave packet | $\longleftrightarrow$ | closed string
$\Big{\downarrow}$condense | dual |  $\Big{\downarrow}$condense
global matrix wave packet | $\longleftrightarrow$ | D3-brane

The definition of bulk wave packet and global matrix wave packet will be given
in Sec. 4. To create a bulk wave packet sitting close to the center of the
spatial dimensions that emerge from matrix degrees of freedom, it is necessary
to entangle a large region on the boundary. Thus, we identify a physical
object in QFT that describes the emergent geometry from entanglement: a bulk
wave packet is associated with quantum entanglement on the boundary.

This paper is organized as follows. In Sec. 2, we review wave packets in the
matrix model. We will see how emergent geometry can be encoded into matrices.
In Sec. 3, the algebra of small excitations in the matrix model is discussed.
Generalization of these ideas to QFT is discussed in Sec. 4. We mainly
consider AdS5/CFT4 correspondence. In Sec. 4.1, the global matrix wave packet
and D3-brane are discussed. QFT wave packet and bulk wave packet are
introduced in Sec. 4.2, and their properties are discussed in Sec. 4.3 in the
context of holographic duality. Sec. 5 is devoted to conclusions and
discussions.

##  2 Review of wave packet in matrix model

In this section, we review the emergent geometry in the matrix model following
Refs. [1, 2, 3]. A more detailed presentation can be found in Sec. 2 of Ref.
[14].

There was a good old idea of emergent geometry based on the identification of
eigenvalues of matrices and locations of emergent space, for low-energy
dynamics of D-branes [15], supermembrane [16], and the Matrix Model of
M-theory [11]. However, a puzzling conceptual issue [6, 7] regarding the
application of this idea to gauge/gravity duality a la Maldacena [17]
distracted researchers from pursuing this direction. (We explain this issue
briefly in Appendix A.) This puzzle was resolved by noticing the importance of
the wave packet in the space of matrix degrees of freedom[1, 2], which we call
matrix wave packet in this paper.

In this section, we will consider a bosonic SU($N$) nine-matrix model
analogous to the BFSS matrix model. Each matrix has $N^{2}$ bosonic degrees of
freedom, and hence, the wave functions live on $\mathbb{R}^{9N^{2}}$. The
matrix wave packet is defined on $\mathbb{R}^{9N^{2}}$. From the matrix wave
packet, the information of nine-dimensional space on the gravity side can be
derived.

Based on the materials reviewed in this section, we will discuss the
description of an arbitrary region of emergent geometry from the matrix model
in Sec. 3. In Sec. 4, we discuss the generalization to QFT. There, it is
important to understand the relationship between two notions: matrix wave
packet and QFT wave packet. By combining matrix wave packet and QFT wave
packet, we will reach the notion of bulk wave packet that plays the
fundamental role in AdS5/CFT4 correspondence.

###  2.1 Gauge-invariant Hilbert space and extended Hilbert space

The matrix wave packet can be defined regardless of the details of the models,
although dual interpretation in weakly-coupled gravitational theory can exist
only for special kinds of models. Therefore, we consider the Hamiltonian
interpolating between the Gaussian matrix model and the bosonic part of the
BFSS matrix model as an example:222 This normalization corresponds to the
Lagrangian $L={\rm
Tr}\left(\frac{m}{2}(D_{t}X_{I})^{2}-\frac{m\omega^{2}}{2}X_{I}^{2}+\frac{g^{2}}{4}[X_{I},X_{J}]^{2}\right)$.
Another common normalization is $L=N{\rm
Tr}\left(\frac{m}{2}(D_{t}X^{\prime}_{I})^{2}-\frac{m\omega^{2}}{2}X_{I}^{\prime
2}+\frac{\lambda}{4}[X^{\prime}_{I},X^{\prime}_{J}]^{2}\right)$, where
$X^{\prime}_{I}=N^{-1/2}X_{I}$. The ’t Hooft scaling is
$\langle\mathrm{Tr}(X^{\prime}_{I_{1}}X^{\prime}_{I_{2}}\cdots
X^{\prime}_{I_{k}})\rangle\sim N$ and
$\langle\mathrm{Tr}(X_{I_{1}}X_{I_{2}}\cdots X_{I_{k}})\rangle\sim N^{1+k/2}$.

$\displaystyle\hat{H}={\rm
Tr}\left(\frac{1}{2m}\hat{P}_{I}^{2}+\frac{m\omega^{2}}{2}\hat{X}_{I}^{2}-\frac{g^{2}}{4}[\hat{X}_{I},\hat{X}_{J}]^{2}\right)\,.$
(1)

Here, $\hat{X}_{I}$ and $\hat{P}_{J}$ are operators describing Hermitian
matrices, and hence, $(\hat{X}_{ij})^{\dagger}=\hat{X}_{I,ji}$ and
$(\hat{P}_{ij})^{\dagger}=\hat{P}_{I,ji}$, where $i,j=1,2,\cdots,N$ and †
denotes Hermitian conjugation as operator acting on the Hilbert space. Trace
is over matrix indices (as $N\times N$ matrices), and $I,J$ run from 1 to 9.
The Gaussian matrix model is obtained at $g^{2}=0$, and in the strong ’t Hooft
coupling limit $\lambda=g^{2}N\to\infty$ or $\omega^{2}\to 0$ the bosonic part
of the BFSS matrix model is obtained. The canonical commutation relation is

$\displaystyle[\hat{X}_{I,ij},\hat{P}_{J,kl}]=i\delta_{IJ}\delta_{il}\delta_{jk}\,.$
(2)

To describe the matrix model through the quantum states, it is crucial to
discern between the gauge-invariant Hilbert space ${\cal H}_{\rm inv}$,
comprising solely of gauge-singlet states, and the extended Hilbert space
${\cal H}_{\rm ext}$ that encompasses non-singlet states as well. In ${\cal
H}_{\rm ext}$, we can employ the coordinate eigenstates $|X\rangle$ satisfying
$\hat{X}_{I,ij}|X\rangle=X_{I,ij}|X\rangle$ for all $9N^{2}$ combinations of
$(I,i,j)$ as the basis. Alternatively, we can utilize the momentum eigenstates
$|P\rangle$ as the basis. It is noteworthy that operators $\hat{X}_{I,ij}$ and
$\hat{P}_{I,ij}$ are defined within this extended Hilbert space because they
transform as adjoint representation under the SU($N$) transformation. In the
following, we will employ wave-packet states as an over-complete basis for
${\cal H}_{\rm ext}$.

The utilization of the extended Hilbert space finds natural justification when
employing the projector from $\mathcal{H}_{\rm ext}$ to $\mathcal{H}_{\rm
inv}$, which corresponds to integrating over the Polyakov loop [18, 1]. The
partition function at finite temperature can be expressed in two different yet
equivalent ways. The first expression is

$\displaystyle Z(T)$ $\displaystyle={\rm Tr}_{{\cal H}_{\rm
inv}}\left(e^{-\hat{H}/T}\right).$ (3)

Because of this expression, it is common to hear the phrase “physical states
are gauge-singlets.” The subsequent expression delves into the extended
Hilbert space that encompasses non-singlet modes:

$\displaystyle Z(T)$ $\displaystyle=\frac{1}{{\rm vol}G}\int_{G}dg{\rm
Tr}_{{\cal H}_{\rm ext}}\left(\hat{g}e^{-\hat{H}/T}\right)\equiv{\rm
Tr}_{{\cal H}_{\rm ext}}\left(\hat{{\cal P}}e^{-\hat{H}/T}\right)\,.$ (4)

Here, $G={\rm SU}(N)$ represents the gauge group, and $g$ and $\hat{g}$ are a
group element and its representation acting on the extended Hilbert space
${\cal H}_{\rm ext}$. The Haar measure is utilized for the integral. The
second expression is directly linked to the path integral involving the
temporal gauge field $A_{t}$, where $g\in G$ corresponds to the Polyakov loop.
In this second expression, it can be stated that “states connected by a gauge
transformation are identified,” meaning that $|\phi\rangle$ and
$\hat{g}|\phi\rangle$ are indistinguishable, analogous to what is observed in
the classical theory or in the path-integral formulation. $\hat{{\cal
P}}\equiv\frac{1}{{\rm vol}G}\int_{G}dg\hat{g}$ is a projection operator from
${\cal H}_{\rm ext}$ to ${\cal H}_{\rm inv}$ that satisfies $\hat{{\cal
P}}^{2}=\hat{{\cal P}}$. From a state $|\phi\rangle\in{\cal H}_{\rm ext}$,
which can be non-singlet, we can obtain a singlet $\hat{{\cal
P}}|\phi\rangle\in{\cal H}_{\rm inv}$.

###  2.2 Matrix wave packet in $\mathbb{R}^{9N^{2}}$

Low-energy states can be written as superpositions of wave packets in the
space of matrices that is $\mathbb{R}^{9N^{2}}$. We use the name matrix wave
packet to emphasize that this is a wave packet in $\mathbb{R}^{9N^{2}}$ and
not in $\mathbb{R}^{9}$. We consider a wave packet around $Y_{I}$ in
coordinate space and $Q_{I}$ in momentum space. Such a wave packet naturally
characterizes the geometry formed by D-branes and open strings [1, 2],
following the approach introduced in Ref. [15]. A straightforward method to
obtain a low-energy wave packet is to identify a state $|Y,Q\rangle$ that
fulfills constraints

$\displaystyle\langle Y,Q|\hat{X}_{I}|Y,Q\rangle=Y_{I}\,,\qquad\langle
Y,Q|\hat{P}_{I}|Y,Q\rangle=Q_{I}\,.$ (5)

The state minimizing the energy

$\displaystyle\langle Y,Q|\hat{H}|Y,Q\rangle$ (6)

is particularly interesting. It is important to note that such a $|Y,Q\rangle$
belongs to the extended Hilbert space ${\cal H}_{\rm ext}$, but not
necessarily to the gauge-invariant Hilbert space ${\cal H}_{\rm inv}$. It
undergoes transformation as

$\displaystyle|Y,Q\rangle\to|U^{-1}YU,U^{-1}QU\rangle$ (7)

through gauge transformation, as illustrated in Fig. 1. States that transform
into each other via gauge transformation are considered identical. Notably,
the shape of the wave packet remains invariant under gauge transformation. The
only “matrices” that can be diagonalized are $Y_{I}$ and $Q_{I}$.

Each wave packet smoothly extends in both coordinate space
$\mathbb{R}^{9N^{2}}$ and momentum space $\mathbb{R}^{9N^{2}}$. Along each of
the $9N^{2}$ dimensions, the size is at most of order $N^{0}$. Two wave
packets centered around $Y$ and $Y^{\prime}$ in $\mathbb{R}^{9N^{2}}$ exhibit
vanishingly small overlap if the centers of wave packets are well separated,
i.e., $\sqrt{{\rm Tr}(Y-Y^{\prime})^{2}}\gg 1$; see Fig. 2.

Figure 1:  Wave packets in coordinate space $\mathbb{R}^{9N^{2}}$ are
depicted. The black points represent the centers of wave packets ($Y$,
$Y^{(U)}$, and $Y^{(U^{\prime})}$), while the gray disks represent the wave
packets, specifically the regions where the wave functions are not vanishingly
small. Under gauge transformation, the center of the wave packet shifts as
$Y\to Y^{(U)}=U^{-1}YU$, while the shape of the wave packet remains unchanged.
The center of the wave packet characterizes the geometry comprising D-branes
and strings. This figure is adapted from Ref. [3].  Figure 2:  Each wave
packet smoothly extends in $\mathbb{R}^{9N^{2}}$. Along each of the $9N^{2}$
dimensions, the size of the wave packet is at most of the order $N^{0}$. Two
wave packets centered around $Y$ and $Y^{\prime}$ in $\mathbb{R}^{9N^{2}}$
exhibit vanishingly small overlap if the centers of the wave packets are well
separated, i.e., $\sqrt{{\rm Tr}(Y-Y^{\prime})^{2}}\gg 1$. This figure is
adapted from Ref. [3].

#### Gaussian matrix model and coherent state

We will examine the Gaussian matrix model as an illustrative example. We set
the coupling in the toy model Hamiltonian (1) to zero, while assuming $m$ and
$\omega$ to be 1:

$\displaystyle\hat{H}=\sum_{I}{\rm
Tr}\left(\frac{1}{2}\hat{P}_{I}^{2}+\frac{1}{2}\hat{X}_{I}^{2}\right)=\frac{1}{2}\sum_{I,\alpha}\left(\hat{P}_{I,\alpha}^{2}+\hat{X}_{I,\alpha}^{2}\right).$
(8)

Specifically, we examine $9N^{2}$ harmonic oscillators under the SU($N$) gauge
symmetry. The ground state simply corresponds to the tensor product of
$9N^{2}$ Fock vacua,

$\displaystyle|0\rangle\equiv\otimes_{I,\alpha}|0\rangle_{I,\alpha}.$ (9)

The ground state $|0\rangle$ is SU($N$)-invariant. The wave packet defined
above is the coherent state,

$\displaystyle|Y,Q\rangle=e^{-i\sum_{I}{\rm
Tr}(Y_{I}\hat{P}_{I}-Q_{I}\hat{X}_{I})}|0\rangle.$ (10)

The ground state $|0\rangle$ is a wave packet about $Y=0$ and $Q=0$.

#### Superselection at large $N$

The wave packet $|Y,Q\rangle$ defined above lacks SU($N$) invariance unless
$Y$ and $Q$ are proportional to the identity matrix. We can interpret it as a
superselection sector within the gauge-invariant Hilbert space as follows
[19]. We can obtain the gauge-invariant counterpart of $|Y,Q\rangle$ by
symmetrizing it over all possible SU($N$) transformations:

$\displaystyle\mathcal{N}^{-1/2}\int dU|U^{-1}YU,U^{-1}QU\rangle\,,$ (11)

where $\mathcal{N}$ is a normalization factor. In the large-$N$ limit, the
spread of the wave function along each direction $(I,\alpha)$ is of order
$N^{0}$ or less, while typical eigenvalues of $Y_{I}$ and $Q_{I}$ are of order
$\sqrt{N}$. Even for small SU($N$) transformations (specifically,
$U=e^{i\epsilon^{\alpha}\tau_{\alpha}}$, where
$\sum_{\alpha}(\epsilon^{\alpha})^{2}$ is small but of order $N^{0}$), the
overlap between $|Y,Q\rangle$ and $|U^{-1}YU,U^{-1}QU\rangle$ becomes
parametrically small. For coherent states, we can explicitly observe that the
overlap behaves as $e^{-\frac{1}{8}\sum_{I}{\rm Tr}(Y_{I}-U^{-1}Y_{I}U)^{2}}$
when $Q=0$. In typical situations of interest, ${\rm Tr}(Y^{2})$ grows with
$N$. Therefore, the overlap is essentially a delta function.

As observables, we focus on gauge-invariant operators that do not
significantly change the energy of the system, such as ${\rm
Tr}(\hat{X}_{I_{1}}\hat{X}_{I_{2}}\cdots\hat{X}_{I_{k}})$, where $k=O(N^{0})$.
Such operators cannot connect $|Y,Q\rangle$ and $|U^{-1}YU,U^{-1}QU\rangle$
unless $U$ is parametrically close to $G_{Y,Q}$, where $G_{Y,Q}\equiv\\{g\in
G|g^{-1}Yg=Y,g^{-1}Qg=Q\\}$ represents the stabilizer of $(Y,Q)$. Whether we
compute the expectation value of such an operator using $|Y,Q\rangle$ or its
gauge-invariant counterpart, we obtain the same value. In this sense,
different gauge choices correspond to different superselection sectors. Both
$|Y,Q\rangle$ and its gauge symmetrization (11) produce identical expectation
values, and thus, they cannot be distinguished.

###  2.3 Geometry from matrix wave packets

Matrices $Y_{I}$ are related to the geometry consisting of D-branes and open
strings through Witten’s interpretation [15]. Let us list a few features
justifying this identification [1]:

  * •

Reference [15] examined low-energy states, which should correspond to wave
packets. The only “matrices” associated with a wave packet that can be
diagonalized are $Y_{I}$ and $Q_{I}$.

  * •

Let us consider a diagonal configuration,

$\displaystyle(Y_{I})_{ij}=y_{I,i}\delta_{ij},\qquad\vec{y}_{i}=(y_{1,i},\cdots,y_{9,i})\in\mathbb{R}^{9}\,,$
$\displaystyle(Q_{I})_{ij}=q_{I,i}\delta_{ij},\qquad\vec{q}_{i}=(q_{1,i},\cdots,q_{9,i})\in\mathbb{R}^{9}\,.$
(12)

Around this background, fluctuations of the $(i,j)$ component have a mass
proportional to $|\vec{y}_{i}-\vec{y}_{j}|$. This framework allows for
interpreting $\vec{y}_{i}$ and $\vec{q}_{i}$ as the location and momentum of
the $i$-th D0-brane, while the off-diagonal entries describe the open strings
stretched between D-branes.

  * •

If the locations of D-branes coincide, the symmetry of the state is enhanced.
If $N_{1}$ D-branes sit at $\vec{y}_{1}=\cdots=\vec{y}_{N_{1}}$ (with momentum
$\vec{q}_{1}=\cdots=\vec{q}_{N_{1}}$), while another $N_{2}$ D-branes sit at
$\vec{y}_{N_{1}+1}=\cdots=\vec{y}_{N_{1}+N_{2}}$ (with momentum
$\vec{q}_{N_{1}+1}=\cdots=\vec{q}_{N_{1}+N_{2}}$), and so forth, the symmetry
becomes ${\rm U}(N_{1})\times{\rm U}(N_{2})\times\cdots$. This is the correct
symmetry enhancement.

#### Partial deconfinement and small black hole

Partially-deconfined states [4, 5, 18, 19, 20] can describe a small black
hole. In partially-deconfined states, an SU($M$) subgroup of SU($N$) is
deconfined, where $M$ depends on the energy. Specifically, we consider wave
packets of the form

$\displaystyle Y_{I}^{[i]}=\left(\begin{array}[]{cc}\tilde{Y}_{I}^{[i]}&0\\\
0&0\end{array}\right),\qquad
Q_{I}^{[i]}=\left(\begin{array}[]{cc}\tilde{Q}_{I}^{[i]}&0\\\
0&0\end{array}\right)\,$ (17)

where $i=1,2,\cdots$ labels different wave packets in this class, and their
linear combinations,

$\displaystyle\sum_{i}c_{i}\ket{Y^{[i]},Q^{[i]}}\,.$ (18)

We assume $\tilde{Y}_{I}^{[i]}$ and $\tilde{Q}_{I}^{[i]}$ are significantly
excited, e.g., ${\rm Tr}(\tilde{Y}_{I}^{[i]})^{2}\sim{\rm
Tr}(\tilde{Q}_{I}^{[i]})^{2}\sim M^{2}$, while the other blocks are set to
zero. Both $M$ D-branes and open strings between them are excited, forming an
extended object. The remaining $N-M$ D-branes are positioned at the origin
with no open strings excited between them or between those $N-M$ D-branes and
the other $M$ D-branes. It is important to note that all D-branes contribute
to the exterior geometry.

When $M\ll N$, the coarse-grained entropy of the partially-deconfined phase,
which should agree with the Bekenstein-Hawking entropy when there is a weakly-
curved gravity dual, can be obtained from a typical partially-deconfined state
as von Neumann entropy by tracing out the confined sector [3].

###  2.4 Gravity dual

The embedding of geometry into matrices discussed above did not assume the
details of the models. Whether weakly-coupled dual gravitational description
exists depend on the details. Supersymmetric matrix models, specifically the
BFSS model [16, 11, 21] and the BMN model [22] provide well-controlled gravity
duals that have been tested quantitatively via numerical simulations, starting
with Refs. [23, 24]. See Ref. [25] and Ref. [26] for recent investigation in
the context of type IIA superstring theory and M-theory, respectively.

Seen from matrix model, the appearance of the causal structure of the bulk
geometry, specifically the speed of light, is highly nontrivial. In this
context, Ref. [27] discussed that, if ‘D0-brane’ (wave packet in matrix model
that corresponds to D0-brane) moves faster than the speed of light in the bulk
geometry, tachyonic modes appear. This could mean either D0-brane dynamically
slows down or the interpretation as semiclassical geometry breaks down.

##  3 Algebra of small excitations in matrix model

In this section, we use the matrix wave packet reviewed in Sec. 2 to probe an
arbitrary region of the emergent geometry in the matrix model.

###  3.1 Small excitations around the ground state

Firstly, we consider small excitations around the ground state.333 For the
BFSS matrix model, we consider the threshold bound state [28, 29, 30].  The
ground state is a wave packet in $\mathbb{R}^{9N^{2}}$ localized around
$Y_{I}=0$ in the coordinate basis and $Q_{I}=0$ in the momentum basis. To
excite a small extended object, we can take a wave packet around $N\times N$
matrices $Y_{I}$ and $Q_{I}$ that are nonzero only in the lower-right $n\times
n$ block where $n$ is an order $N^{0}$ number (see Fig. 3):

$\displaystyle Y_{I}=\left(\begin{array}[]{cc}0&0\\\
0&\tilde{Y}_{I}\end{array}\right)\,,\qquad
Q_{I}=\left(\begin{array}[]{cc}0&0\\\ 0&\tilde{Q}_{I}\end{array}\right)\,,$
(23)

where

$\displaystyle\tilde{Y}_{I}=y_{I}\cdot\text{1}_{n}+\tilde{\tilde{Y}}_{I}\,,\qquad\tilde{Q}_{I}=q_{I}\cdot\text{1}_{n}+\tilde{\tilde{Q}}_{I}\,.$
(24)

Here, $y_{I}$ and $q_{I}$ are the location and momentum of the extended object
and they can be of order $\sqrt{N}$.
$\vec{y}=(y_{1},\cdots,y_{9})\in\mathbb{R}^{9}$ and
$\vec{q}=(q_{1},\cdots,q_{9})\in\mathbb{R}^{9}$ specify the location and
momentum of the object in the nine-dimensional target space of string theory.
$\tilde{\tilde{Y}}_{I}$ and $\tilde{\tilde{Q}}_{I}$ are traceless and take at
most order $N^{0}$ values. In the Gaussian limit, such a wave packet is a
coherent state (10) (the left panel in Fig. 3). We can make the gauge-
invariant version of this operator as

$\displaystyle\hat{\mathcal{O}}_{Y,Q}=\frac{1}{\mathrm{vol}(\mathrm{SU}(N))}\int_{\mathrm{SU}(N)}dg\
e^{-i\mathrm{Tr}((gY_{I}g^{-1})\hat{P}_{I}-(gQ_{I}g^{-1})\hat{X}_{I})}\,.$
(25)

Note that, by construction,
$\hat{\mathcal{O}}_{Y,Q}=\hat{\mathcal{O}}_{hYh^{-1},hQh^{-1}}$ for any
$h\in\mathrm{SU}(N)$.

When there is an interaction, the off-diagonal blocks (last $n$ rows and
columns on the right panel in Fig. 3 shown in pink) are also affected.
Specifically, as the eigenvalues of $\tilde{Y}_{I}$ become large, the width of
the quantum fluctuation of the off-diagonal entry shrinks. This can easily be
understood by noticing that, when $|\vec{y}|$ is large, off-diagonal blocks
behave as harmonic oscillators whose frequency is proportional to $|\vec{y}|$.
A conceptually simple way to obtain such a low-energy wave packet is to
minimize the energy $\bra{Y,Q}\hat{H}\ket{Y,Q}$ fixing the center of the wave
packet to $\bra{Y,Q}\hat{X}_{I,ij}\ket{Y,Q}=Y_{I,ij}$ and
$\bra{Y,Q}\hat{P}_{I,ij}\ket{Y,Q}=Q_{I,ij}$. Note that the back-reaction to
the $(N-n)\times(N-n)$ block can be neglected in the large-$N$ limit with
fixed $n$.

We can act two or more operators $\hat{\mathcal{O}}_{Y,Q}$,
$\hat{\mathcal{O}}_{Y^{\prime},Q^{\prime}}$, $\cdots$ (see Fig. 4). In the
large-$N$ limit, we can neglect the possibility of exciting the same color
degrees of freedom. If we act a finite number of such operators, say $k$ of
them, and send $N$ to infinity, then $k$ independent blocks are excited. For
example, for $k=2$, we can write it as

$\displaystyle\hat{\mathcal{O}}_{Y,Q}\hat{\mathcal{O}}_{Y^{\prime},Q^{\prime}}=\hat{\mathcal{O}}_{Y\oplus
Y^{\prime},Q\oplus Q^{\prime}}\,,$ (26)

where

$\displaystyle Y\oplus Y^{\prime}\equiv\left(\begin{array}[]{ccc}0&0&0\\\
0&\tilde{Y}_{I}&0\\\ 0&0&\tilde{Y}^{\prime}_{I}\end{array}\right)\,,\qquad
Q\oplus Q^{\prime}\equiv\left(\begin{array}[]{ccc}0&0&0\\\
0&\tilde{Q}_{I}&0\\\ 0&0&\tilde{Q}^{\prime}_{I}\end{array}\right)\,.$ (33)

Generalizations to generic $k$ are straightforward. Interaction between small
excitations is higher order in the $1/N$-expansion.

By collecting the operators $\hat{\mathcal{O}}_{Y,Q}$ whose centers
$\vec{y}=(y_{1},\cdots,y_{9})$ are in a region $R\subset\mathbb{R}^{9}$ and
taking product of them, we can construct operator algebra corresponding to
region $R$.

Figure 3: One object is excited (red). [Left] free theory. Off-diagonal blocks
are not affected. [Right] Interacting theory. Off-diagonal entries (pink) are
affected; specifically, though the center remains zero, width of quantum
fluctuation changes. Such effects are small when the excited block does not go
too far from the origin of $\mathbb{R}^{9}$.

Figure 4: Two objects are excited (red). [Left] free theory. Off-diagonal
blocks are not affected. [Right] Interacting theory. Off-diagonal entries
(pink) can be affected. (See the caption of Fig. 3.)

For practical applications, it would be convenient to consider a state
obtained by acting a gauge-invariant operator consisting of $n$ matrices,
e.g., $\mathrm{Tr}(\hat{X}_{I_{1}}\hat{X}_{I_{2}}\cdots\hat{X}_{I_{n}})$, to
the ground state. Such an operator excites an $n\times n$ block, because

$\displaystyle\mathrm{Tr}(\hat{X}_{I_{1}}\hat{X}_{I_{2}}\cdots\hat{X}_{I_{n}})=\hat{X}_{I_{1},12}\hat{X}_{I_{2},23}\cdots\hat{X}_{I_{n},n1}+\textrm{permutations}\,$
(34)

up to $1/N$-suppressed terms.

###  3.2 Small excitation around black hole geometry

Next, we consider an SU($M$)-partially-deconfined state. We can prepare either
a pure state or a mixed state. (It is conjectured that the 11d BH is described
by the partially-deconfined states in the matrix model [25].) If small
excitations are well separated from the partially-deconfined sector
(specifically, if $\vec{y}$ is larger than the eigenvalues of the
SU($M$)-deconfined block), we can repeat essentially the same logic. The only
difference is that we should take a superselection sector, e.g., the
deconfined sector sits at the upper-left $M\times M$ block as depicted in Fig.
5 [19], and define the operators that create excitations in the confined
sector. Fixing the superselection sector on the gauge theory side in this way
corresponds to fixing the semiclassical geometry on the gravity side [10].

Because a matrix wave packet has the information of locations and momenta of
small excitations, we can build both ‘algebra of states that stay outside BH’
(such excitations can be created in supersymmetric theories due to the flat
directions), and ‘algebra of states that fall into BH’. Late-time behaviors of
operators in the former and latter algebras must be drastically different. See
Ref. [31] for the relationship between the late-time behaviors and types of
algebras.

Figure 5: One object is excited (small red block) on top of a partially-
deconfined state. The large red block is the SU($M$)-deconfined sector. [Left]
free theory. Off-diagonal blocks are not affected. [Right] Interacting theory.
Off-diagonal entries (pink) can be affected. (See the caption of Fig. 3.)

###  3.3 A remark on terminology

A remark is in order concerning the nuance of the word matrix wave packet. As
we have seen, it is of crucial importance to consider wave packets in the
space of matrix degrees of freedom, which spreads in $\mathrm{R}^{9N^{2}}$ in
the case of the D0-brane matrix model. Naturally, we called such a wave packet
in $\mathrm{R}^{9N^{2}}$ matrix wave packet. In the rest of the paper, we will
mainly consider small excitations added to semi-classical geometry.
Specifically, we are interested in objects expressed by small blocks in
$X_{I}$ corresponding to the nonzero part of $Y_{I}$ and $Q_{I}$ (e.g., small
red blocks in Fig. 3, Fig. 4, and Fig. 5). Therefore, we will use the same
word matrix wave packet to mean these small blocks as well. Ultimately, the
difference is only in their nuances — whether our focus is on the entire
geometry or small excitations — because the full $N\times N$ matrices contain
all sub-blocks.

##  4 QFT wave packet and AdS5/CFT4 duality

In this section, we discuss emergent geometry in QFT, specifically 4d
$\mathcal{N}=4$ super Yang-Mills on S3. (Essentially the same picture holds on
$\mathbb{R}^{3}$. For simplicity, we use $\mathbb{R}^{3}$ for concrete
computations. The same expressions are valid for S3 near the boundary.) We
call the three-dimensional space of QFT as QFT space, in order to distinguish
it from the space that emerges from matrix degrees of freedom. This theory has
six scalars and their momentum conjugates. We denote them as
$\hat{X}_{I,ij,\vec{x}}$ and $\hat{P}_{I,ij,\vec{x}}$, where $I$ runs from 1
to 6, $\vec{x}$ is the coordinate in the QFT space, and $i,j=1,\cdots,N$. The
canonical commutation relation is

$\displaystyle[\hat{X}_{I,ij,\vec{x}},\hat{P}_{J,kl\vec{x}^{\prime}}]=i\delta_{IJ}\delta_{il}\delta_{jk}\delta^{3}(\vec{x}-\vec{x}^{\prime})\,.$
(35)

There are at least two kinds of wave packets: a wave packet in the QFT space
which we call QFT wave packet, and a wave packet in the space of matrix
degrees of freedom which we call matrix wave packet. A QFT wave packet can
specify the location in the three-dimensional space. A matrix wave packet can
specify the location in $\mathbb{R}^{6}$. AdS5 is described by four-
dimensional spacetime in QFT and the radial coordinate of $\mathbb{R}^{6}$
from matrices. S5 is described by the angular coordinates of $\mathbb{R}^{6}$.
We use the name bulk wave packet for an excitation which is a QFT wave packet
and matrix wave packet simultaneously. A location in AdS${}_{5}\times$S5 can
be associated with a bulk wave packet. We argue that a bulk wave packet
describes a closed string.

###  4.1 Global matrix wave packet and D3-brane in AdS5/CFT4

Let us start with a review of the matrix wave packet in QFT. This is a
straightforward generalization of Sec. 2. The partition function is the same
as (4), except that we need to consider all possible local gauge
transformations instead of $G$ that can be schematically written as
$\mathcal{G}=\otimes_{\vec{x}}G_{\vec{x}}$ where $G_{\vec{x}}$ is the gauge
group $G$ acting on a spatial point $\vec{x}$ in the QFT space. We can use a
lattice regularization to make sense of this expression. The canonical
partition function can be written as

$\displaystyle Z(T)$ $\displaystyle=\frac{1}{{\rm
vol}\mathcal{G}}\int_{\mathcal{G}}dg{\rm Tr}_{{\cal H}_{\rm
ext}}\left(\hat{g}e^{-\hat{H}/T}\right).$ (36)

As a concrete example, let us consider $(3+1)$-dimensional maximal super Yang-
Mills theory. The D3-brane geometry on
$\mathbb{R}^{6}=\mathbb{R}_{>0}\times{\rm S}^{5}$ can be generated from six
scalar fields. $\mathbb{R}_{>0}$ is interpreted as the radial coordinate of
AdS5 [17]. If we use a lattice regularization for the spatial dimensions, wave
functions are defined on $\mathbb{R}^{6\times N^{2}\times n_{\rm site}}$,
where $6$ is the number of scalar fields and $n_{\rm site}$ is the number of
spatial lattice sites. The center of a wave packet is specified by
$Y_{I}(\vec{x})$ and $Q_{I}(\vec{x})$. Note that we assume $\vec{x}$ runs
through the 3-dimensional space of QFT, and we assume they are continuous
functions in the continuum limit. (Strictly speaking, we have to take into
account the spatial directions of the gauge field as well, whose discretized
version lives on the links.) In the ground state,444This is the case for the
theory on S3. On $\mathbb{R}^{3}$, $Y_{I}=0$, $Q_{I}=0$ is one of the super-
selection sectors. $Y_{I}=0$ and $Q_{I}=0$, and hence, all D3-branes are
sitting at the origin of AdS5 and no open string is excited. This is exactly
the expected property of the BPS limit of the black three-brane. We can take
some D3-branes out of the BPS black hole by giving nontrivial values to
$Y_{I}$ and $Q_{I}$. This is essentially the Higgsing of scalar fields. See
Fig. 6.

A quantum state describing a D3-brane with a nontrivial shape can be written
as

$\displaystyle\int_{\vec{x}\in\mathrm{S}^{3}}[dX_{\vec{x}}]\
F[X_{\vec{x}}]\ket{X_{\vec{x}}}\,,$ (37)

where $F[X_{\vec{x}}]$ is localized around a certain value $Y_{I}(\vec{x})$ at
each $\vec{x}$. Specifically, we can observe excitations at different spatial
points in S3 simultaneously, corresponding to the location of the D3-brane
away from $\vec{0}\in\mathbb{R}^{6}$. We use a name global matrix wave packet
to denote such a matrix wave packet. Here, ‘global’ means it spreads globally
on S3. An important point for holographic duality is:

  * •

Global matrix wave packet describes D3-brane (black 3-brane).

More precisely, small sub-blocks of scalar matrices describe D3-branes (see
Sec. 3.3 for a remark on terminology). In this context, the values of scalars
specify the locations of the D3-branes in the bulk (specifically, S5 and the
radial coordinate of AdS5). This can be seen through the effective description
of Higgsed phase: the Dirac-Born-Infeld action can be obtained [17], and
further nontrivial evidence is obtained from the consistency of special
conformal transformation on the gravity side and gauge theory side [32]. The
same are expected for D$p$-brane and $(p+1)$-dimensional super Yang-Mills
theory for $p\neq 3$ as well [17, 33].

Figure 6: A schematic picture of a global matrix wave packet, e.g., a
D3-brane. It requires excitation at each spatial point $\vec{x}$. More
precisely, small sub-blocks of scalar matrices describe D3-branes (see Sec.
3.1.3 for a remark on terminology).

###  4.2 QFT wave packet and bulk wave packet

Next, let us introduce QFT wave packets and bulk wave packets. To simplify the
notation and see the essence, let us use one-component real scalar
$\hat{\phi}_{\vec{x}}$ and its momentum conjugate $\hat{\pi}_{\vec{x}}$. For
each $I$ and $\alpha$, $\hat{X}_{I,\vec{x}}^{\alpha}$ and
$\hat{P}_{I,\vec{x}}^{\alpha}$ behaves similar to $\hat{\phi}_{\vec{x}}$ and
$\hat{\pi}_{\vec{x}}$. We keep the spatial dimension $d$ general in this
subsection, though we will focus on $d=3$ later. The canonical commutation
relation is

$\displaystyle[\hat{\phi}_{\vec{x}},\hat{\pi}_{\vec{x}^{\prime}}]=i\delta^{d}(\vec{x}-\vec{x}^{\prime})\,.$
(38)

We can make smeared operators that can create wave packets in the QFT space,
which we call QFT wave packets. A simple example is the wave packets created
by Gaussian-smeared field variables

$\displaystyle\hat{\Phi}_{\vec{x}_{0},\vec{p}_{0}}$
$\displaystyle\equiv\frac{1}{(2\pi\sigma^{2})^{d/2}}\int
d^{d}\vec{x}e^{-\frac{(\vec{x}-\vec{x}_{0})^{2}}{2\sigma^{2}}-i\vec{p}_{0}\cdot(\vec{x}-\vec{x}_{0})}\hat{\phi}_{\vec{x}}\,,$
$\displaystyle\hat{\Pi}_{\vec{x}_{0},\vec{p}_{0}}$
$\displaystyle\equiv\frac{1}{(2\pi\sigma^{2})^{d/2}}\int
d^{d}\vec{x}e^{-\frac{(\vec{x}-\vec{x}_{0})^{2}}{2\sigma^{2}}-i\vec{p}_{0}\cdot(\vec{x}-\vec{x}_{0})}\hat{\pi}_{\vec{x}}\,.$
(39)

These operators can create wave packets in the QFT space, localized around
$\vec{x}_{0}$ in the coordinate space and $\vec{p}_{0}$ in the momentum space.
The commutation relation is

$\displaystyle[\hat{\Phi}_{\vec{x}_{0},\vec{p}_{0}},\hat{\Pi}_{\vec{x}^{\prime}_{0},\vec{p}^{\prime}_{0}}^{\dagger}]=[\hat{\Phi}_{\vec{x}_{0},\vec{p}_{0}},\hat{\Pi}_{\vec{x}^{\prime}_{0},-\vec{p}^{\prime}_{0}}]=\frac{i}{(4\pi\sigma^{2})^{d/2}}\cdot
e^{-\frac{(\vec{x}_{0}-\vec{x}^{\prime}_{0})^{2}}{4\sigma^{2}}-\frac{\sigma^{2}(\vec{p}_{0}-\vec{p}^{\prime}_{0})^{2}}{4}+i\frac{\vec{p}_{0}+\vec{p}^{\prime}_{0}}{2}\cdot(\vec{x}_{0}-\vec{x}^{\prime}_{0})}\,.$
(40)

A QFT wave packet including a bulk wave packet is written schematically as

$\displaystyle\int d^{3}\vec{x}f(\vec{x})\hat{\phi}_{\vec{x}}\ket{\rm
g.s.}\,.$ (41)

We can also use a composite operator $\hat{O}_{\vec{x}}$ made of
$\hat{\phi}_{\vec{x}}$, and we can act with multiple smeared fields on the
ground state. In (41), $\hat{\phi}_{\vec{x}}\ket{\rm g.s.}$ can be seen as a
small excitation around point $\vec{x}$. We take a linear combination of such
states. Therefore, unlike the global matrix wave packet, we observe
significant difference from the ground state only locally.

A QFT wave packet can be a matrix wave packet as well. We call such a wave
packet that is simultaneously a QFT wave packet and matrix wave packet as bulk
wave packet; see Fig. 7. We can immediately see that:

  * •

A global matrix wave packet is a condensation of bulk wave packets; see Fig.
8.

This simple observation sits at the heart of geometric interpretation provided
in Sec. 4.3.

Figure 7: A schematic picture of a bulk wave packet in QFT (specifically, 4d
$\mathcal{N}=4$ SYM). It is obtained by smearing the locations of finitely
many excitations.

Figure 8: Physical interpretation of bulk wave packet and global matrix wave
packet on the gravity side. Bulk wave packet describes short strings such as
graviton. A global matrix wave packet is a D-brane. Note that a D-brane is a
condensation of strings and a global matrix wave packet is a condensation of
bulk wave packets, consistent with this interpretation.

###  4.3 Bulk wave packet and closed string in AdS5/CFT4

Now we focus on the 4d $\mathcal{N}=4$ super Yang-Mills theory and AdS5/CFT4
duality. The meaning of bulk wave packet on the gravity side can be inferred
from the following:

  * •

Global matrix wave packet describes D3-brane. The values of scalars specify
the location of the D3-brane in the bulk (specifically, S5 and the radial
coordinate of AdS5) [17].

  * •

A global matrix wave packet is a condensation of bulk wave packets.

  * •

D3-brane (black 3-brane) is a condensation of strings.

Naturally, we interpret a bulk wave packet as a string or a few strings. With
this identification, because a D3-brane ($=$ global wave packet) is a
condensation of strings ($=$ bulk wave packet), and because the values of
scalars in the global matrix wave packet specify the location of D3-brane in
the bulk, we can naturally interpret that the values of scalars in a bulk wave
packet specify the location of the corresponding string in the bulk; see Fig.
8.

Below, we argue further that nontrivial aspects of other bulk reconstruction
programs naturally follow from this interpretation. Specifically:

  * •

The size of the region on the boundary where the bulk wave packet has support
is related to the radial coordinate of AdS.

  * •

Quantum entanglement plays an important role in bulk reconstruction.

We will first provide a consistency check of our interpretation of the radial
coordinate in Sec. 4.3.1 assuming the standard AdS/CFT dictionary. In Sec.
4.3.2, we give a qualitative argument for the relationship between the size of
the region on boundary and radial coordinate without assuming dual gravity
description. We comment on AdS4/CFT3 correspondence in Sec. 4.3.3. In Sec.
4.3.4, it is explained how the bulk wave packet is related to quantum
entanglement. In Sec. 4.3.5, the similarity to the case of traversable
wormhole is discussed. The entanglement associated with the bulk wave packet
is O($N^{0}$) correction to the leading O($N^{2}$) part captured by the Ryu-
Takayanagi formula [34] and its variants. The importance of the leading part
is suggested in Sec. 4.3.5.

As a disclaimer, we comment that, although the boundaries of the bulk regions
shown in Fig. 10, Fig. 11, and Fig. 12 look like Ryu-Takayanagi (RT) surface,
our argument does not confirm they are indeed the RT surface. Our discussions
are more qualitative.

####  4.3.1 Radial coordinate in gravity vs QFT

Suppose there is an excitation localized in the bulk geometry. Then, we argue
that, if we assume the standard AdS/CFT dictionary, the ‘radial coordinate’
determined from $X_{I}$ indeed corresponds to the radial coordinate of the
excitation on the gravity side.

We do this by relating certain correlation functions to the radial coordinate.
As a simple but instructive example, let us consider the Gaussian matrix
model. For the ground state, we have

$\displaystyle\bra{\rm g.s.}\hat{X}_{I,ij}\ket{\rm g.s.}=0$ (42)

and

$\displaystyle\bra{\rm g.s.}\mathrm{Tr}\hat{X}_{I}^{2}\ket{\rm g.s.}\sim
N^{2}\,.$ (43)

For a low-energy wave packet (coherent state) around $Y_{I,ij}\neq 0$, we have

$\displaystyle\bra{Y,Q}\hat{X}_{I,ij}\ket{Y,Q}=Y_{I,ij}$ (44)

and

$\displaystyle\bra{Y,Q}\mathrm{Tr}\hat{X}_{I}^{2}\ket{Y,Q}=\bra{\rm
g.s.}\mathrm{Tr}\hat{X}_{I}^{2}\ket{\rm g.s.}+\mathrm{Tr}Y_{I}^{2}\,.$ (45)

If we take a generic excitation that is gauge-invariant by symmetrizing the
coherent state as (25), the one-point function of $\hat{X}_{I,ij}$ becomes
zero. Therefore, it is more convenient to use $\mathrm{Tr}\hat{X}_{I}^{2}$ as

$\displaystyle\bra{Y,Q}\mathrm{Tr}\hat{X}_{I}^{2}\ket{Y,Q}-\bra{\rm
g.s.}\mathrm{Tr}\hat{X}_{I}^{2}\ket{\rm g.s.}=\mathrm{Tr}Y_{I}^{2}\,.$ (46)

This relation holds even when $\ket{Y,Q}$ is replaced with its
SU($N$)-symmetrized counterpart. Historically, people tried to use the
expectation value of $\mathrm{Tr}\hat{X}_{I}^{2}$ to estimate the radial
coordinate and got confused about the fact that this is of order $N^{2}$
already for the ground state. What we have seen above is we must subtract the
vacuum contribution that contains only quantum fluctuations. The same strategy
works for interacting theory as long as the back-reaction of small excitations
to the ground state is negligible. Note that the use of
$\mathrm{Tr}\hat{X}_{I}^{2}$ is not essential. Other functions of scalars
could also work.

We want to repeat a similar computation for QFT. For a product state of the
form $\prod_{\vec{x}}(\int
d\phi_{\vec{x}}f_{\vec{x}}(\phi_{\vec{x}})\ket{\phi_{\vec{x}}})$, we can
obtain the information about the location of the excitations in the emergent
space from the expectation value of $\mathrm{Tr}\hat{X}_{I}^{2}(\vec{x})$.
This is because, roughly speaking, we can treat each spatial point as a
different matrix model. The actual situation of interest can be more
complicated because there is entanglement along the QFT space. Still, we
expect at least a qualitative estimate of the location of the excitation from
this quantity because of the local nature of the QFT space.

To simplify the computation by using supergravity, we focus on chiral primary
operators. Let $\hat{O}_{r,\vec{x}}$ be a smeared operator that creates an
excitation localized at a bulk point $(r,\vec{x})$. Note that we do not care
exactly how such an operator is constructed: we only assume the existence of
such an operator and compute correlation functions using the standard AdS/CFT
dictionary. We consider a normalized state
$\mathcal{N}^{-1/2}\hat{O}_{r,\vec{x}}\ket{\rm g.s.}$, where
$\mathcal{N}=\bra{\rm
g.s.}\hat{O}_{r,\vec{x}}^{\dagger}\hat{O}_{r,\vec{x}}\ket{\rm g.s.}$. We use
the rank-2 chiral primary operator
$\mathrm{Tr}(X_{\\{I_{1}}X_{I_{2}\\}})\equiv\mathrm{Tr}(X_{I_{1}}X_{I_{2}})-\frac{\delta_{I_{1}I_{2}}}{6}\sum_{k}\mathrm{Tr}(X_{k}X_{k})$
instead of $\mathrm{Tr}\hat{X}_{I}^{2}$, and take $\hat{O}$ the rank-$k_{O}$
chiral primary operator. We should estimate555 The second term vanishes for
chiral primaries.

$\displaystyle\frac{\bra{\rm
g.s.}\hat{O}^{\dagger}_{r,\vec{x}}\mathrm{Tr}(X_{\\{I_{1}}X_{I_{2}\\}})(\vec{x}^{\prime})\hat{O}_{r,\vec{x}}\ket{\rm
g.s.}}{\bra{\rm g.s.}\hat{O}^{\dagger}_{r,\vec{x}}\hat{O}_{r,\vec{x}}\ket{\rm
g.s.}}-\bra{\rm
g.s.}\mathrm{Tr}(X_{\\{I_{1}}X_{I_{2}\\}})(\vec{x}^{\prime})\ket{\rm g.s.}\,.$
(47)

Equivalently, we study the connected part of the 3-point correlator

$\displaystyle\frac{\bra{\rm
g.s.}\hat{O}^{\dagger}_{r,\vec{x}}\mathrm{Tr}(X_{\\{I_{1}}X_{I_{2}\\}})(\vec{x}^{\prime})\hat{O}_{r,\vec{x}}\ket{\rm
g.s.}_{\rm conn}}{\bra{\rm
g.s.}\hat{O}^{\dagger}_{r,\vec{x}}\hat{O}_{r,\vec{x}}\ket{\rm g.s.}}\,.$ (48)

We can calculate it by using the Euclidean path integral at zero temperature,
inserting the operators at Euclidean time zero. An important point is that
$\mathrm{Tr}(X_{\\{I_{1}}X_{I_{2}\\}})$ is sitting at the boundary
$(r^{\prime}=\infty,\vec{x}^{\prime})$ from the bulk point of view; see Fig.
9.

To compute the three-point function, we insert a bulk three-point vertex that
connects $\hat{O}$ and $\hat{O}^{\dagger}$ to
$\mathrm{Tr}(X_{\\{I_{1}}X_{I_{2}\\}})$. To see the $r$-dependence, we can use
the bulk-to-boundary two-point correlation function for
$\mathrm{Tr}(X_{\\{I_{1}}X_{I_{2}\\}})$. Hence

$\displaystyle\frac{\bra{\rm
g.s.}\hat{O}^{\dagger}_{r,\vec{x}}\mathrm{Tr}(X_{\\{I_{1}}X_{I_{2}\\}})(\vec{x}^{\prime})\hat{O}_{r,\vec{x}}\ket{\rm
g.s.}_{\rm conn}}{\bra{\rm
g.s.}\hat{O}^{\dagger}_{r,\vec{x}}\hat{O}_{r,\vec{x}}\ket{\rm g.s.}}\sim
r^{2}\cdot\left(\frac{1}{1+r^{2}(\vec{x}-\vec{x}^{\prime})^{2}}\right)^{2}\,.$
(49)

More generally, by replacing $\mathrm{Tr}(X_{\\{I_{1}}X_{I_{2}\\}})$ with the
rank-$k$ chiral primary operator, we have

$\displaystyle\frac{\bra{\rm
g.s.}\hat{O}^{\dagger}_{r,\vec{x}}\mathrm{Tr}(X_{\\{I_{1}}\cdots
X_{I_{k}\\}})(\vec{x}^{\prime})\hat{O}_{r,\vec{x}}\ket{\rm g.s.}_{\rm
conn}}{\bra{\rm g.s.}\hat{O}^{\dagger}_{r,\vec{x}}\hat{O}_{r,\vec{x}}\ket{\rm
g.s.}}\sim
r^{k}\cdot\left(\frac{1}{1+r^{2}(\vec{x}-\vec{x}^{\prime})^{2}}\right)^{k}\,.$
(50)

This is $r^{k}$ at $\vec{x}-\vec{x}^{\prime}=0$ and decays quickly at
$|\vec{x}-\vec{x}^{\prime}|\gtrsim r^{-1}$. From this, we can see that the
excitation sits at $\textrm{radial\ coordinate}\sim r$ in the emergent
dimensions and localized at $|\vec{x}-\vec{x}^{\prime}|\lesssim r^{-1}$ in the
QFT space. Note that the same scaling can hold as long as $\hat{O}$ is a local
operator in the bulk, not just for chiral primaries. When $\hat{O}$ is
rank-$k_{O}$ chiral primary operator, a $k_{O}\times k_{O}$ block and $k_{O}$
eigenvalues can be excited; see (34). According to our proposal of the
emergent space from scalars, these $k_{O}$ eigenvalues should scale as $r$,
and hence, (49) and (50) should increase linearly with $k_{O}$. This is indeed
the case [35]. See Appendix B for details of the computations.

Typically, a recipe for the bulk reconstruction is provided by assuming the
AdS/CFT dictionary and reverse-engineering the CFT states from the ‘answer’ on
the gravity side. The argument above tells us where in the emergent dimensions
from matrices such CFT operators are sitting. Therefore, our analysis provides
a consistency check: if we assume the standard AdS/CFT dictionary, the ‘radial
coordinate’ determined from $X_{I}$ indeed corresponds to the radial
coordinate of the excitation on the gravity side.

Figure 9: Schematic picture of connected three-point function in (48). Because
$\hat{O}$ and $\hat{O}^{\dagger}$ are at the same point in the bulk, we only
need a three-point vertex and bulk-to-boundary two-point correlation function.

####  4.3.2 Size of boundary region vs radial coordinate

Widely believed pictures of bulk reconstruction suggest that the size of the
region on the boundary where the bulk wave packet has support is related to
the radial coordinate of AdS. The argument in Sec. 4.3.1 explains this
connection assuming that the previous results based on computations on the
gravity side are correct. In this section, we would like to understand this
relation without relying on the gravity dual. We consider a QFT wave packet,
which should be understood as a linear combination of bulk wave packets, and
show that the relation between the size of the wave packet in the QFT space
and the radial coordinate defined by scalars is consistent with the
holographic dictionary.

#### Intuition from lattice regularization

Let us use lattice regularization on a spatial lattice with lattice spacing
$a$. An excitation localized on only one point on the lattice is analogous to
a wave packet of size $a$ in the continuum theory. With $d$ spatial
dimensions, we can consider the square lattice with equal lattice spacing $a$
in all $d$ directions. The lattice Hamiltonian is defined by

$\displaystyle\hat{H}_{\rm
lattice}=a\hat{H}=\sum_{\vec{n}}\left(\frac{1}{2}\hat{\pi}_{{\rm
lattice},\vec{n}}^{2}+\frac{1}{2}\sum_{\mu=1}^{d}\left(\hat{\phi}_{{\rm
lattice},\vec{n}+\hat{\mu}}-\hat{\phi}_{{\rm
lattice},\vec{n}}\right)^{2}+V(\hat{\phi}_{{\rm lattice},\vec{n}})\right)\,.$
(51)

$\hat{\mu}$ is the unit vector along the $\mu$-th dimension of the spatial
lattice ($\mu=1,\cdots,d$). We have the canonical commutation relation

$\displaystyle[\hat{\phi}_{{\rm lattice},\vec{n}},\hat{\pi}_{{\rm
lattice},\vec{n}^{\prime}}]=i\delta_{\vec{n}\vec{n}^{\prime}}\,.$ (52)

Fields $\hat{\phi}_{\rm lattice}$ and $\hat{\pi}_{\rm lattice}$ are
dimensionless. To take the continuum limit, dimensions should be restored as

$\displaystyle\hat{\phi}_{\rm
lattice}=a^{(d-1)/2}\hat{\phi}\,,\qquad\hat{\pi}_{\rm
lattice}=a^{(d+1)/2}\hat{\pi}\,.$ (53)

Roughly speaking, $\phi$ in the lattice theory is average of $\phi(\vec{x})$
in the continuum theory over the unit lattice,
$a^{-d}\int_{|\vec{x}^{\prime}-\vec{x}|<a}d^{d}\vec{x}^{\prime}\phi(\vec{x}^{\prime})$.
Indeed, we can see that this $a$ corresponds to the width of the Gaussian wave
packet $\sigma$ in (40).

The lattice Hamiltonian $\hat{H}_{\rm lattice}$ can depend on lattice spacing
$a$ via couplings in $V(\hat{\phi}_{{\rm lattice},\vec{n}})$, i.e.,
dimensionful couplings such as mass $m^{2}$ is made dimensionless by
multiplying appropriate power of $a$, e.g., $m^{2}a^{2}$. In addition, there
can be a running of couplings associated with renormalization.

Suppose we could ignore the explicit $a$-dependence in $V(\hat{\phi}_{{\rm
lattice},\vec{n}})$, e.g., in a conformal theory such as 4d $\mathcal{N}=4$
SYM.666Strictly speaking, we need to include dynamical gauge fields and
fermions to the lattice action. Explicit $a$-dependence can be avoided by
using fine-tuning-free formulations of 4d SYM [36, 37].  Then, the location of
the center of lowest-energy wave packet made of a polynomial of
$\hat{\phi}_{{\rm lattice},\vec{n}}$ has to be of order $1$ in terms of
$\hat{\phi}_{{\rm lattice},\vec{n}}$ and $a^{-(d-1)/2}$ (which is $a^{-1}$ for
$d=3$) in terms of $\hat{\phi}$; see Fig. 10. This is the same as the scaling
we observed in Sec. 4.3.1, by identifying $a$ with $r^{-1}$.

Figure 10: Operators that has a support in a region of size $a^{3}$ in the QFT
space can create bulk wave packets at $r\gtrsim a^{-1}$ (equivalently,
$z\lesssim a$). Note that $r=\infty$ is the boundary.

#### Gaussian wave packet in the weak-coupling limit

Let us see how the logic above works for the simplest explicit example: the
Gaussian-smeared operators (39) acting on the ground state of massless free
scalar theory on $\mathbb{R}^{3}$. The Hamiltonian is

$\displaystyle\hat{H}_{\rm free}$ $\displaystyle=\frac{1}{2}\int
d^{3}x\left(\hat{\pi}_{\vec{x}}^{2}+\sum_{j=1}^{3}(\partial_{j}\hat{\phi}_{\vec{x}})^{2}\right)\,.$
(54)

In the momentum basis, the Hamiltonian is a collection of harmonic
oscillators:

$\displaystyle\hat{H}_{\rm free}$ $\displaystyle=\frac{1}{2}\int
d^{3}\vec{k}\left(\hat{\tilde{\pi}}_{\vec{k}}\hat{\tilde{\pi}}_{-\vec{k}}+\vec{k}^{2}\cdot\hat{\tilde{\phi}}_{\vec{k}}\hat{\tilde{\phi}}_{-\vec{k}}\right)=\frac{1}{2}\int
d^{3}\vec{k}\left(\hat{\tilde{a}}^{\dagger}_{\vec{k}}\hat{\tilde{a}}_{\vec{k}}+\frac{1}{2}\right)\omega_{\vec{k}}\,.$
(55)

Here,

$\displaystyle\phi_{\vec{x}}=\frac{1}{(2\pi)^{3/2}}\int
d^{3}\vec{k}e^{ikx}\hat{\tilde{\phi}}_{\vec{k}}\,,\qquad\hat{\tilde{\phi}}_{\vec{k}}=\frac{1}{(2\pi)^{3/2}}\int
d^{3}\vec{x}e^{-ikx}\phi_{\vec{x}}\,,$
$\displaystyle\pi_{\vec{x}}=\frac{1}{(2\pi)^{3/2}}\int
d^{3}\vec{k}e^{ikx}\hat{\tilde{\pi}}_{\vec{k}}\,,\qquad\hat{\tilde{\pi}}_{\vec{k}}=\frac{1}{(2\pi)^{3/2}}\int
d^{3}\vec{x}e^{-ikx}\pi_{\vec{x}}\,,$ (56)
$\displaystyle\hat{\tilde{\phi}}_{\vec{k}}^{\dagger}=\hat{\tilde{\phi}}_{-\vec{k}}\,,\qquad\hat{\tilde{\pi}}_{\vec{k}}^{\dagger}=\hat{\tilde{\pi}}_{-\vec{k}}\,.$
(57)

Note that the momentum conjugate to $\hat{\phi}_{\vec{k}}$ is
$(\hat{\pi}_{\vec{k}})^{\dagger}=\hat{\pi}_{-\vec{k}}$, and the canonical
commutation relation is

$\displaystyle[\hat{\tilde{\phi}}_{\vec{k}},\hat{\tilde{\pi}}_{-\vec{k}^{\prime}}]=i\delta^{3}(\vec{k}-\vec{k}^{\prime})\,.$
(58)

The annihilation operator $\hat{\tilde{a}}_{\vec{k}}$ and creation operator
$\hat{\tilde{a}}^{\dagger}_{\vec{k}}$ are defined as

$\displaystyle\hat{\tilde{a}}_{\vec{k}}=\frac{\sqrt{\omega_{\vec{k}}}\hat{\tilde{\phi}}_{\vec{k}}+\frac{i}{\sqrt{\omega_{\vec{k}}}}\hat{\tilde{\pi}}_{\vec{k}}}{\sqrt{2}}\,,\qquad\hat{\tilde{a}}_{\vec{k}}^{\dagger}=\frac{\sqrt{\omega_{\vec{k}}}\hat{\tilde{\phi}}_{-\vec{k}}-\frac{i}{\sqrt{\omega_{\vec{k}}}}\hat{\tilde{\pi}}_{-\vec{k}}}{\sqrt{2}}\,,$
(59)

where $\omega_{\vec{k}}=\sqrt{\vec{k}^{2}}$. They satisfy

$\displaystyle[\hat{\tilde{a}}_{\vec{k}},\hat{\tilde{a}}_{\vec{k}^{\prime}}^{\dagger}]=\delta^{3}(\vec{k}-\vec{k}^{\prime})\,.$
(60)

The ground state is the Fock vacuum $\ket{0}$ that is annihilated by all
annihilation operators:

$\displaystyle\hat{\tilde{a}}_{\vec{k}}\ket{0}=0\,.$ (61)

In the momentum basis, the Gaussian-smeared operators (39) are written as

$\displaystyle\hat{\Phi}_{\vec{x}_{0},\vec{p}_{0}}$
$\displaystyle=\frac{1}{(2\pi)^{3/2}}\int
d^{3}\vec{k}e^{-\frac{\sigma^{2}}{2}(\vec{k}-\vec{p}_{0})^{2}+i\vec{k}\cdot\vec{x}_{0}}\hat{\tilde{\phi}}_{\vec{k}}$
$\displaystyle=\frac{1}{(2\pi)^{3/2}}\int
d^{3}\vec{k}e^{-\frac{\sigma^{2}}{2}(\vec{k}-\vec{p}_{0})^{2}+i\vec{k}\cdot\vec{x}_{0}}\cdot\frac{\hat{\tilde{a}}_{\vec{k}}+\hat{\tilde{a}}^{\dagger}_{-\vec{k}}}{\sqrt{2\omega_{\vec{k}}}}\,,$
(62)

etc. Acting $\hat{\Phi}_{\vec{x}_{0},\vec{p}_{0}}$ on the Fock vacuum, we
obtain

$\displaystyle\ket{\Psi}\equiv\hat{\Phi}_{\vec{x}_{0},\vec{p}_{0}}\ket{0}=\frac{1}{(2\pi)^{3/2}}\int
d^{3}\vec{k}e^{-\frac{\sigma^{2}}{2}(\vec{k}-\vec{p}_{0})^{2}+i\vec{k}\cdot\vec{x}_{0}}\cdot\frac{\hat{\tilde{a}}^{\dagger}_{-\vec{k}}}{\sqrt{2\omega_{\vec{k}}}}\ket{0}\,.$
(63)

Following the idea presented in Sec. 4.3.1, let us compute

$\displaystyle\frac{\bra{\Psi}\hat{\phi}_{\vec{x}}^{2}\ket{\Psi}}{\bra{\Psi}\ket{\Psi}}-\bra{0}\hat{\phi}_{\vec{x}}^{2}\ket{0}\,.$
(64)

The ground-state contribution is

$\displaystyle\bra{0}\hat{\phi}_{\vec{x}}^{2}\ket{0}$
$\displaystyle=\frac{1}{(2\pi)^{3}}\int d^{3}\vec{k}\int
d^{3}\vec{k}^{\prime}e^{i(\vec{k}+\vec{k}^{\prime})\cdot\vec{x}}\bra{0}\hat{\tilde{\phi}}_{\vec{k}}\hat{\tilde{\phi}}_{\vec{k}^{\prime}}\ket{0}$
$\displaystyle=\frac{1}{(2\pi)^{3}}\int d^{3}\vec{k}\int
d^{3}\vec{k}^{\prime}e^{i(\vec{k}+\vec{k}^{\prime})\cdot\vec{x}}\bra{0}\frac{\hat{\tilde{a}}_{\vec{k}}}{\sqrt{2\omega_{\vec{k}}}}\cdot\frac{\hat{\tilde{a}}_{-\vec{k}^{\prime}}^{\dagger}}{\sqrt{2\omega_{\vec{k}^{\prime}}}}\ket{0}$
$\displaystyle=\frac{1}{(2\pi)^{3}}\int d^{3}\vec{k}\int
d^{3}\vec{k}^{\prime}e^{i(\vec{k}+\vec{k}^{\prime})\cdot\vec{x}}\cdot\frac{\delta^{3}(\vec{k}+\vec{k}^{\prime})}{2\sqrt{\omega_{\vec{k}}\omega_{\vec{k}^{\prime}}}}$
$\displaystyle=\frac{1}{(2\pi)^{3}}\int\frac{d^{3}\vec{k}}{2k}\,,$ (65)

which is ultraviolet divergent. This term cancels with the counterpart in the
first term of (64). The normalization factor $\bra{\Psi}\ket{\Psi}$ is

$\displaystyle\bra{\Psi}\ket{\Psi}$ $\displaystyle=\frac{1}{(2\pi)^{3}}\int
d^{3}\vec{k}\frac{e^{-\sigma^{2}(\vec{k}-\vec{p}_{0})^{2}}}{2\omega_{k}}\,.$
(66)

For $\vec{p}_{0}=\vec{0}$, we have

$\displaystyle\bra{\Psi}\ket{\Psi}_{\vec{p}_{0}=\vec{0}}$
$\displaystyle=\frac{1}{(2\pi)^{3}}\int_{0}^{\infty}4\pi
k^{2}dk\frac{e^{-\sigma^{2}k^{2}}}{2k}=\frac{1}{8\pi^{2}\sigma^{2}}\,.$ (67)

Finally,

$\displaystyle\bra{\Psi}\hat{\phi}_{\vec{x}}^{2}\ket{\Psi}$
$\displaystyle=\frac{1}{(2\pi)^{6}}\int
d^{3}\vec{k}e^{-\frac{\sigma^{2}}{2}(\vec{k}-\vec{p}_{0})^{2}+i\vec{k}\cdot\vec{x}_{0}}\int
d^{3}\vec{k}^{\prime}e^{-\frac{\sigma^{2}}{2}(\vec{k}^{\prime}-\vec{p}_{0})^{2}+i\vec{k}^{\prime}\cdot\vec{x}_{0}}$
$\displaystyle\qquad\int d^{3}\vec{l}\int
d^{3}\vec{l}^{\prime}e^{i(\vec{l}+\vec{l}^{\prime})\cdot\vec{x}}\bra{0}\frac{\hat{\tilde{a}}_{-\vec{k}^{\prime}}}{\sqrt{2\omega_{\vec{k}^{\prime}}}}\cdot\frac{\hat{\tilde{a}}_{\vec{l}}+\hat{\tilde{a}}_{-\vec{l}}^{\dagger}}{\sqrt{2\omega_{\vec{l}}}}\cdot\frac{\hat{\tilde{a}}_{\vec{l}^{\prime}}+\hat{\tilde{a}}_{-\vec{l}^{\prime}}^{\dagger}}{\sqrt{2\omega_{\vec{l}^{\prime}}}}\cdot\frac{\hat{\tilde{a}}^{\dagger}_{-\vec{k}}}{\sqrt{2\omega_{\vec{k}}}}\ket{0}$
$\displaystyle=\frac{1}{(2\pi)^{6}}\int
d^{3}\vec{k}e^{-\frac{\sigma^{2}}{2}(\vec{k}-\vec{p}_{0})^{2}+i\vec{k}\cdot\vec{x}_{0}}\int
d^{3}\vec{k}^{\prime}e^{-\frac{\sigma^{2}}{2}(\vec{k}^{\prime}-\vec{p}_{0})^{2}+i\vec{k}^{\prime}\cdot\vec{x}_{0}}$
$\displaystyle\qquad\int d^{3}\vec{l}\int
d^{3}\vec{l}^{\prime}\frac{e^{i(\vec{l}+\vec{l}^{\prime})\cdot\vec{x}}}{4\sqrt{\omega_{\vec{k}}\omega_{\vec{k}^{\prime}}\omega_{\vec{l}}\omega_{\vec{l}^{\prime}}}}$
$\displaystyle\qquad\times\left\\{\delta^{3}(\vec{l}+\vec{l}^{\prime})\delta^{3}(\vec{k}+\vec{k}^{\prime})+\delta^{3}(\vec{k}+\vec{l}^{\prime})\delta^{3}(\vec{k}^{\prime}+\vec{l})+\delta^{3}(\vec{k}+\vec{l})\delta^{3}(\vec{k}^{\prime}+\vec{l}^{\prime})\right\\}\,.$
(68)

From
$\delta^{3}(\vec{l}+\vec{l}^{\prime})\delta^{3}(\vec{k}+\vec{k}^{\prime})$, we
obtain

$\displaystyle\frac{1}{(2\pi)^{6}}\int
d^{3}\vec{k}e^{-\sigma^{2}(\vec{k}-\vec{p}_{0})^{2}}\int
d^{3}\vec{l}\frac{1}{4\omega_{\vec{k}}\omega_{\vec{l}}}$
$\displaystyle=\bra{\Psi}\ket{\Psi}\cdot\frac{1}{(2\pi)^{3}}\int\frac{d^{3}\vec{l}}{2l}\,,$
(69)

which cancels with $\bra{0}\hat{\phi}_{\vec{x}}^{2}\ket{0}$. From the other
terms, we obtain

$\displaystyle\frac{1}{(2\pi)^{6}}\int
d^{3}\vec{k}e^{-\frac{\sigma^{2}}{2}(\vec{k}-\vec{p}_{0})^{2}-i\vec{k}\cdot(\vec{x}-\vec{x}_{0})}\int
d^{3}\vec{k}^{\prime}e^{-\frac{\sigma^{2}}{2}(\vec{k}^{\prime}-\vec{p}_{0})^{2}-i\vec{k}^{\prime}\cdot(\vec{x}-\vec{x}_{0})}\cdot\frac{1}{4\omega_{\vec{k}}\omega_{\vec{k}^{\prime}}}$
(70)

This takes maximum at $\vec{x}=\vec{x}_{0}$. Furthermore, if we assume
$\vec{p}_{0}=\vec{0}$ for simplicity, we obtain

$\displaystyle\frac{1}{(2\pi)^{6}}\left(\int
d^{3}\vec{k}\frac{e^{-\frac{\sigma^{2}}{2}\vec{k}^{2}-i\vec{k}\cdot(\vec{x}-\vec{x}_{0})}}{2\omega_{\vec{k}}}\right)^{2}=\frac{1}{(2\pi)^{6}}\times\left(\frac{2\pi}{\sigma^{2}}e^{-\frac{(\vec{x}-\vec{x}_{0})}{2\sigma^{2}}}\right)^{2}=\frac{e^{-\frac{(\vec{x}-\vec{x}_{0})}{\sigma^{2}}}}{4\pi^{2}\sigma^{2}}\cdot\bra{\Psi}\ket{\Psi}\,.$
(71)

Therefore, for $\vec{p}_{0}=\vec{0}$, (64) becomes

$\displaystyle\frac{\bra{\Psi}\hat{\phi}_{\vec{x}}^{2}\ket{\Psi}}{\bra{\Psi}\ket{\Psi}}-\bra{0}\hat{\phi}_{\vec{x}}^{2}\ket{0}=\frac{e^{-\frac{(\vec{x}-\vec{x}_{0})}{\sigma^{2}}}}{4\pi^{2}\sigma^{2}}\,.$
(72)

Identifying $\sigma$, $a$ and $r^{-1}$, we see qualitatively the same scaling
as before.

####  4.3.3 Comment on ($2+1$)-dimensional theory

Repeating the arguments in Sec. 4.3.2 for $d=2$, we obtain ‘radial coordinate’
$a^{-(d-1)/2}=a^{-1/2}$ for a QFT wave packet of size $a$. This scaling is
natural in the following sense.

In the ABJM theory [38] with $\mathrm{U}(N)\times\mathrm{U}(N)$ gauge group,
there are four bi-fundamental scalars $C_{\alpha=1,\cdots,4}$. The potential
term of the scalars is proportional to

$\displaystyle-\frac{1}{6}\mathrm{Tr}(C_{\alpha}\bar{C}^{\alpha})^{3}-\frac{1}{6}\mathrm{Tr}(\bar{C}^{\alpha}C_{\alpha})^{3}-\frac{2}{3}\mathrm{Tr}(C_{\alpha}\bar{C}^{\gamma}C_{\beta}\bar{C}^{\alpha}C_{\gamma}\bar{C}^{\beta})+\mathrm{Tr}(C_{\alpha}\bar{C}^{\alpha}C_{\beta}\bar{C}^{\gamma}C_{\gamma}\bar{C}^{\beta})\,.$
(73)

Note that a product $C_{\alpha}\bar{C}^{\beta}$ transforms in the adjoint
representation of the first U($N$), analogous to scalars in 4d SYM. The
scaling is $C_{\alpha}\sim a^{-1/2}$ means $C_{\alpha}\bar{C}^{\beta}\sim
a^{-1}$, which leads to natural energy scale. For example, assuming $C_{1}$ is
diagonal, i.e., $C_{1}=D\equiv\mathrm{diag}(d_{1},\cdots,d_{N})$, the scalar
potential is proportional to

$\displaystyle\sum_{\alpha=2}^{4}\mathrm{Tr}\left([D\bar{D},C_{\alpha}][D\bar{D},\bar{C}^{\alpha}]\right)\,$
(74)

up to terms of higher order in $C_{\alpha=2,3,4}$. Therefore, $D\sim a^{-1/2}$
means the mass of off-diagonal entries of order $a^{-1}$, as in the case of 4d
SYM.

####  4.3.4 Bulk wave packet and quantum entanglement

Let us consider the union of two regions on the boundary A and B, as shown in
the left panel of Fig. 11. We can also consider a union of two disconnected
regions of the boundary, as shown in Fig. 12. Either way, region c is
analogous to a traversable wormhole as we will discuss in Sec. 4.3.5. Below,
we study the role of the entanglement of boundary regions in the creation of
bulk local excitations. The entanglement discussed here is the $O(N^{0})$
corrections on top of the $O(N^{2})$ entanglement of the ground state. The
role of the ground-state contribution is explained in Sec. 4.3.5.

In these figures, the vertical direction is the emergent space from scalars. A
wave packet that has support in one of the boundary regions A or B can
describe only an excitation in bulk regions a or b. In the right panel, we
showed a schematic picture of such smearing functions that can create wave
packets in the emergent bulk geometry. However, a wave packet that has support
in the union of two boundary regions can describe an excitation in a bulk
region c as well; see Fig. 11. Such a wave packet is created entangling the
excitations in region A and region B. To understand this intuitively, imagine
a pure state made of a QFT wave packet obtained by using a smearing function
depicted by the blue line in the right panel of Fig. 11. We can write it
trivially as

$\displaystyle\left(\int_{{\rm A}\cup{\rm
B}}d^{3}\vec{x}f(\vec{x})\hat{\phi}_{\vec{x}}\right)\ket{\rm
g.s.}=\left(\int_{\rm
A}d^{3}\vec{x}f(\vec{x})\hat{\phi}_{\vec{x}}\right)\ket{\rm
g.s.}+\left(\int_{\rm
B}d^{3}\vec{x}f(\vec{x})\hat{\phi}_{\vec{x}}\right)\ket{\rm g.s.}\,.$ (75)

(Here, $f(\vec{x})$ is a smearing function. We could use a generic composite
operator instead of $\hat{\phi}_{\vec{x}}$.) The excitation is either in
boundary region A or boundary region B; this is an entangled state. As
essentially the same case, let us construct an entangled state from one cat.
Suppose that a cat wants to go into a box and there are two boxes, $A$ and
$B$. Then,

$\displaystyle\frac{\ket{\rm cat}_{\rm A}\otimes\ket{\rm no\ cat}_{\rm
B}+\ket{\rm no\ cat}_{\rm A}\otimes\ket{\rm cat}_{\rm B}}{\sqrt{2}}$ (76)

is an entangled state made of only one cat. The similarity between this state
and the right-hand side of (75) must be obvious. It would be instructive to
see the difference between such an entangled state and more common entangled
states made of two cats, e.g.,

$\displaystyle\frac{\ket{\rm white\ cat}_{\rm A}\otimes\ket{\rm black\
cat}_{\rm B}+\ket{\rm black\ cat}_{\rm A}\otimes\ket{\rm white\ cat}_{\rm
B}}{\sqrt{2}}\,.$ (77)

The meaning of the entanglement of the form (75), (76) becomes clearer after
seeing the similarity to the traverse wormhole, which is discussed in Sec.
4.3.5.

Figure 11: [Left] bulk region c is generated due to the entanglement between
two boundary regions A and B. [Right] A cartoon picture of smearing functions.
The vertical axis is $f(\vec{x})$ in (75). To create a bulk wave packet in a,
b, or c, $f(\vec{x})$ should have support in A, B, or A$\cup$B like the line
with the corresponding color. The important point is that such a figure could
be drawn without referring to gravity dual, identifying the ‘radial
coordinate’ with the value of scalar field, or more specifically, the location
of the bulk wave packet.

Figure 12: Region c is generated due to the entanglement between two disjoint
regions $A$ and $B$ in the boundary. Region c is made of low-frequency modes
($f(\vec{x})$ with large support). Region d requires high-frequency modes
($f(\vec{x})$ with small support) and hence cannot be generated.

####  4.3.5 Traversable wormhole and bulk wave packet

The duality between a two-sided black hole and thermofield double state [12]
is arguably the clearest example of the emergence of spacetime (in this case,
wormhole) from entanglement [13]. We suggest that this case can also be
understood in terms of bulk wave packets, in the same way as in Sec. 4.3.4.

Let us consider a traversable wormhole. A well-known example is generated by
the coupled SYK model [39]. (Different from a famous work that related
traversable wormhole and quantum teleportation [40], we discuss an eternal
traversable wormhole that is static and dual to the ground state.) To make the
point even simpler, let us consider the simplest toy model: the coupled
harmonic oscillator [41, 42]. The Hamiltonian is

$\displaystyle\hat{H}=\frac{1}{2}\hat{p}_{\rm A}^{2}+\frac{1}{2}\hat{x}_{\rm
A}^{2}+\frac{1}{2}\hat{p}_{\rm B}^{2}+\frac{1}{2}\hat{x}_{\rm
B}^{2}-c\hat{x}_{\rm A}\hat{x}_{\rm B}\,,$ (78)

where $c$ is a small coupling. By using new variables
$\hat{X}_{\pm}=\frac{\hat{x}_{\rm A}\pm\hat{x}_{\rm B}}{\sqrt{2}}$ and
$\hat{P}_{\pm}=\frac{\hat{p}_{\rm A}\pm\hat{p}_{\rm B}}{\sqrt{2}}$, we can
write the Hamiltonian as a sum of two harmonic oscillators whose frequencies
are shifted from the original value:

$\displaystyle\hat{H}=\frac{1}{2}\hat{P}_{+}^{2}+\frac{\omega_{+}^{2}}{2}\hat{X}_{+}^{2}+\frac{1}{2}\hat{P}_{-}^{2}+\frac{\omega_{-}^{2}}{2}\hat{X}_{-}^{2}\,,$
(79)

where

$\displaystyle\omega_{\pm}^{2}=1\mp c\,.$ (80)

The ground state is the Fock vacuum in terms of these new harmonic oscillators
$\hat{X}_{\pm}$. In terms of the original oscillators $\hat{x}_{\rm A}$ and
$\hat{x}_{\rm B}$, a thermofield double state is realized; see Refs. [41, 42]
for details.777 Strictly speaking, we need to use slightly different
oscillators whose frequencies are slightly different from 1 to make the
thermofield-double structure manifest [42]. This fact does not affect the
following discussions.  The same holds for the coupled SYK model and the
ground state is interpreted as two copies of AdS connected by traversable
wormhole [39].

Creation and annihilation operators $\hat{A}^{\dagger}_{\pm}$, $\hat{A}_{\pm}$
in the $\pm$-basis and those in the ${\rm A,B}$-basis $\hat{a}^{\dagger}_{\rm
A,B}$, $\hat{a}_{\rm A,B}$ are related as follows:

$\displaystyle\hat{a}_{\rm A}^{\dagger}=\frac{\hat{x}_{\rm A}-i\hat{p}_{\rm
A}}{\sqrt{2}}\,,\qquad\hat{a}_{\rm B}^{\dagger}=\frac{\hat{x}_{\rm
B}-i\hat{p}_{\rm B}}{\sqrt{2}}\,,$ (81)

$\displaystyle\hat{X}_{\pm}=\frac{\hat{a}_{\rm A}+\hat{a}_{\rm
A}^{\dagger}\pm\hat{a}_{\rm B}\pm\hat{a}_{\rm
B}^{\dagger}}{2}\,,\qquad\hat{P}_{\pm}=\frac{\hat{a}_{\rm A}-\hat{a}_{\rm
A}^{\dagger}\pm\hat{a}_{\rm B}\mp\hat{a}_{\rm B}^{\dagger}}{2i}\,,$ (82)

$\displaystyle\hat{A}_{\pm}^{\dagger}=\frac{\sqrt{\omega_{\pm}}\hat{X}_{\pm}-i\frac{\hat{P}_{\pm}}{\sqrt{\omega_{\pm}}}}{\sqrt{2}}\simeq\hat{a}^{\dagger}_{\pm}\mp\frac{c\hat{a}_{\pm}}{4}\,,$
(83)

where

$\displaystyle\hat{a}_{\pm}\equiv\frac{\hat{a}_{\rm A}\pm\hat{a}_{\rm
B}}{\sqrt{2}}\,,\qquad\hat{a}^{\dagger}_{\pm}\equiv\frac{\hat{a}_{\rm
A}^{\dagger}\pm\hat{a}_{\rm B}^{\dagger}}{\sqrt{2}}\,.$ (84)

Therefore, the ground state $\ket{0}_{+}\ket{0}_{-}$ annihilated by
$\hat{A}_{\pm}$ is the same as $\ket{0}_{\rm A}\ket{0}_{\rm B}$ up to a small
difference of $O(c)$. Also, up to $O(c)$ terms,
$\ket{1}_{+}\ket{0}_{-}\simeq\frac{\ket{1}_{\rm A}\ket{0}_{\rm B}+\ket{0}_{\rm
A}\ket{1}_{\rm B}}{\sqrt{2}}$ and
$\ket{0}_{+}\ket{1}_{-}\simeq\frac{\ket{1}_{\rm A}\ket{0}_{\rm B}-\ket{0}_{\rm
A}\ket{1}_{\rm B}}{\sqrt{2}}$.

These expressions remind us of the low-energy wave functions of a particle in
the double-well potential: $\ket{1}_{+}\ket{0}_{-}$ and
$\ket{0}_{+}\ket{1}_{-}$ resembles symmetric ground state and anti-symmetric
excited state. The states $\ket{1}_{+}\ket{0}_{-}$ and
$\ket{0}_{+}\ket{1}_{-}$ have the energy levels $E_{\pm}=\omega_{\pm}=(1\mp
c)^{1/2}$ up to the ground-state energy, and hence, the difference is $\Delta
E\equiv E_{-}-E_{+}\simeq c$. Because of this energy difference, there is a
‘tunneling’ between the states localized in A or B. Namely, time evolution of
$\frac{\ket{1}_{+}\ket{0}_{-}+\ket{0}_{+}\ket{1}_{-}}{\sqrt{2}}\simeq\ket{1}_{\rm
A}\ket{0}_{\rm B}$ is

$\displaystyle\frac{e^{\frac{i}{2}t\Delta
E}\ket{1}_{+}\ket{0}_{-}}{\sqrt{2}}+\frac{e^{-\frac{i}{2}t\Delta
E}\ket{0}_{+}\ket{1}_{-}}{\sqrt{2}}$ $\displaystyle\qquad\simeq\cos\left(\pi
t/\tau_{0}\right)\ket{1}_{\rm A}\ket{0}_{\rm B}+i\sin\left(\pi
t/\tau_{0}\right)\ket{0}_{\rm A}\ket{1}_{\rm B}$ (85)

up to an overall phase, where $\tau_{0}\equiv\frac{2\pi}{\Delta E}$, and
hence, it goes back and force between $\ket{1}_{\rm A}\ket{0}_{\rm B}$ and
$\ket{0}_{\rm A}\ket{1}_{\rm B}$ with period $\tau_{0}$. At
$\frac{t}{\tau_{0}}=\frac{1}{4},\frac{3}{4},\frac{5}{4},\cdots$, we have the
states in the wormhole moving from A to B or from B to A:

$\displaystyle\frac{\ket{1}_{\rm A}\ket{0}_{\rm B}\pm i\ket{0}_{\rm
A}\ket{1}_{\rm B}}{\sqrt{2}}\,.$ (86)

These states have entanglement analogous to (75). The excitation can go into a
wormhole due to the entanglement between regions A and B just as the example
in Sec. 4.3.4. To quantify this statement, we can employ the time evolution of
entanglement entropy. At time $t$, the reduced density matrix obtained by
tracing out B is

$\displaystyle\hat{\rho}_{A}=\mathrm{Tr}_{\rm
B}\left(\ket{\psi(t)}\bra{\psi(t)}\right)\simeq\cos^{2}\left(\pi
t/\tau_{0}\right)\ket{1}_{\rm A}\bra{1}_{\rm A}+\sin^{2}\left(\pi
t/\tau_{0}\right)\ket{0}_{\rm A}\bra{0}_{\rm A}\,.$ (87)

Hence, the entanglement entropy evolves as

$\displaystyle-\mathrm{Tr}_{A}\left(\hat{\rho}_{A}\log\hat{\rho}_{A}\right)\simeq-\cos^{2}\left(\pi
t/\tau_{0}\right)\cdot\log\left(\cos^{2}\left(\pi
t/\tau_{0}\right)\right)-\sin^{2}\left(\pi
t/\tau_{0}\right)\cdot\log\left(\sin^{2}\left(\pi t/\tau_{0}\right)\right)\,.$
(88)

Indeed, entanglement is large when the excitation is going through the
wormhole; see Fig. 13. Such a wormhole is analogous to a link between two
regions A and B. In this specific example, the interaction term
$-c\hat{x}_{\rm A}\hat{x}_{\rm B}$ can come from $\frac{c}{2}(\hat{x}_{\rm
A}-\hat{x}_{\rm B})^{2}$, and hence, it is analogous to the spatial derivative
in the kinetic term of QFT.

This mechanism is rather general and is applicable to various theories with
discrete spectrum. For small coupling $c$,
$\frac{\ket{E_{n}}_{A}\ket{E_{0}}_{B}\pm\ket{E_{0}}_{A}\ket{E_{n}}_{B}}{\sqrt{2}}$,
where $\ket{E_{0}}$ and $\ket{E_{n}}$ are the ground state and the $n$-th
excited state of the original system, can be close to energy eigenstates of
the coupled system. Note that symmetric and antisymmetric combinations appear
because of the symmetry under the exchange of system A and system B. By using
these states, we can repeat the same argument presented above.888When the QFT
side is gauge theory, whether the singlet constraint should be imposed or not
is a nontrivial issue (see e. g. [43, 44]). This discussion is not sensitive
to the singlet constraint.

Note that the interaction term is crucial for the ‘wormhole’ to be
traversable: If there is no interaction, there is no energy splitting ($\Delta
E=0$), and hence, there is no ‘tunneling’. In the coupled SYK model or coupled
SU($N$) Yang-Mills, there is a large entanglement between two sides in the
ground state. The entanglement discussed above is an O($N^{0}$) correction to
the ground state contribution coming from the small excitation. The ground
state contribution assisted by ‘interaction’ (including the spatial
derivative) prepares the space that the small excitation (bulk wave packet)
can go through. An important open problem is if holographic QFTs admit such
nonlocal interactions at low energy that can explain the entanglement wedge
reconstruction [45, 46, 47, 48] (see also refs. [49, 50, 51]).

Region c in Fig. 11 and Fig. 12 is analogous to traversable wormhole. Region c
is a bridge connecting region a and region b, and an excitation in c is
created by entangling the boundary regions A and B in the same way as the
above example. In the philosophy of ER$=$EPR conjecture [52], we could say the
bulk geometry is (traversable) ER bridge connecting EPR pair (in this case,
two boundary regions).

The values of $x_{\rm A}$ and $x_{\rm B}$ are analogous to those of the scalar
fields $X_{I}$ in 4d SYM. When the excitation is localized in region A (resp.,
B), the increment of the expectation value of $\hat{x}_{\rm A}^{2}$ (resp.,
$\hat{x}_{\rm B}^{2}$) from the ground state is 1. For the state
$\frac{\ket{1}_{\rm A}\ket{0}_{\rm B}\pm i\ket{0}_{\rm A}\ket{1}_{\rm
B}}{\sqrt{2}}$, because the excitation spreads to $A\cup B$, the increment
from the ground state is $\frac{1}{2}$ for both $\hat{x}_{\rm A}^{2}$ and
$\hat{x}_{\rm B}^{2}$. In this sense, “radial coordinate” is smaller. This is
the same as the situation in QFT: as a bulk wave packet spreads in the QFT
space, scalars take smaller values.

Figure 13:  Time evolution of entanglement entropy
$-\mathrm{Tr}_{A}\left(\hat{\rho}_{A}\log\hat{\rho}_{A}\right)$, with the
approximation (88).

##  5 Conclusion and discussion

In this paper, we discussed how holographic emergent geometry can be described
by matrix degrees of freedom of the dual QFT or matrix model, making basic
ideas provided in Refs. [1, 2, 3] more concrete. Specifically, we considered
the matrix model and 4d $\mathcal{N}=4$ SYM. Historically, such a direction
analogous to the BFSS proposal [11] was not pursued much because of a puzzling
conceptual issue raised by Polchinski [6] and Susskind [7]. The key to
resolving this conceptual issue was the use of wave packet in the space of
matrix degrees of freedom [1, 2], which we called matrix wave packet (see Sec.
2).

In Sec. 3, we discussed how operators describing small excitations on semi-
classical gravitational geometry can be constructed. Our construction uses
only the language of the matrix model and does not assume anything about
duality. The duality dictionary plays a role only when we try to interpret it
in terms of gravity.

We discussed generalization to AdS5/CFT4 duality in Sec. 4. A crucial addition
to the case of the matrix model was that we must consider wave packets along
the QFT space as well. Of particular importance were the global matrix wave
packet and bulk wave packet, which describe the D3-brane and closed string,
respectively. The bulk wave packet is a wave packet both along the QFT space
and the space of matrix degrees of freedom. We found a simple interplay
between the emergent space from matrix degrees of freedom and entanglement on
the QFT-space that connects two seemingly different mechanisms of holographic
emergent geometry, i.e., one based on matrix eigenvalues and the other based
on quantum entanglement.999This combination reminds us of a tensor network
approach to holography [53].

We did not discuss how to define the ‘center’ of a bulk wave packet in the QFT
space that would be identified with the location of the bulk excitation. It is
a nontrivial issue when the support consists of multiple disconnected pieces
(see e.g., Fig. 12). If the entanglement wedge reconstruction proposal is
correct and if we use the location of the excitation on the gravity side to
define the ‘center’, then such a ‘center’ may be outside the bulk wave packet.
Finding a precise definition of the ‘center’ in a purely QFT language is an
important open problem. Note also that direct and explicit computations on the
QFT side are highly nontrivial. The demonstration in Sec. 4.3.1 assumed the
dual gravity description, or more precisely, a bulk reconstruction such as the
HKLL [54] that uses the dual gravity description for the derivation. The
demonstrations in Sec. 4.3.2 are essentially the dimensional analyses.

Our proposal does not immediately tell us whether the entanglement wedge
reconstruction proposal is correct or wrong.101010 See Refs. [55, 56] for
criticism against the entanglement wedge reconstruction.  If the entanglement
wedge reconstruction is valid, then the dual gravity interpretation of wave
packets would depend on the boundary region under consideration.

For the global-AdS reconstruction [54], whole the boundary is used. Still, the
argument in Sec. 4.3.4 can be applied with minor changes. An obvious change is
that (75) cannot hold exactly; instead, we need to consider wave packets that
decay outside certain regions (A$\cup$B, A, and B in this case). Accordingly,
we could slightly modify (76), for example, as

$\displaystyle\frac{\ket{\rm cat\mathchar 39\relax s\ body}_{\rm
A}\otimes\ket{\rm cat\mathchar 39\relax s\ tail}_{\rm B}+\ket{\rm cat\mathchar
39\relax s\ tail}_{\rm A}\otimes\ket{\rm cat\mathchar 39\relax s\ body}_{\rm
B}}{\sqrt{2}}\,.$ (89)

This is still an entangled state made of one cat; see Fig. 14.

Figure 14:  Schematic picture of entangled states (76) [top] and (89)
[bottom].

4d $\mathcal{N}=4$ SYM has six scalars and hence not just the radial
coordinate of AdS5 but also S5 can be described as emergent space from scalar
fields. The relationship between the size of the boundary region $a$ and the
‘radial coordinate’ $a^{-1}$ can, at least qualitatively, explain the
emergence of the AdS metric. Other models could still describe the radial
coordinate of AdS via the same relation although more intricate structures
such as $\mathcal{M}$ of AdS$\times\mathcal{M}$ may not be captured.

A potentially interesting application of our proposal is in the physics of
small black holes in AdS${}_{5}\times$S5. This is an ideal setup to study
black hole information problem because the evaporation of very small black
holes can be studied in the isolated system [57]. A small black hole can be
described by the partially-deconfined states in the CFT [4, 19] (see also Fig.
5). The emission of graviton can be visualized as the emission of bulk wave
packets. The size of the deconfined sector decreases as the black hole
shrinks. This picture provides us with an intuitive understanding of important
properties such as negative specific heat [58, 4] and Page curve [3]. An
important open problem is how the event horizon can be understood in terms of
matrix model or QFT.111111An intuitive picture could be as follows. For a
small block to be emitted from the deconfined sector (black hole), the off-
diagonal entries of $Y$ and $Q$ connecting the deconfined sector and small
block must vanish. Such a process is entropically suppressed; as the
deconfined sector becomes larger, the emission is more suppressed. This could
explain the formation of the event horizon. When a small block falls into the
deconfined sector, off-diagonal entries are excited. The dual interpretation
could be the condensation of open strings.

Conceptually the simplest test and potentially the most powerful application
are in quantum simulations. If we can simulate the matrix model or SYM on a
quantum computer, we can simply monitor the time evolution of the scalar field
and identify which degrees of freedom describe which object.

Acknowledgement

The authors would like to thank Tarek Anous, Masazumi Honda, Dan Kabat, Henry
Lin, Hong Liu, So Matsuura, Cheng Peng, Hidehiko Shimada, Evgeny Sobko, and
Brian Swingle for discussions and comments. V. G. thanks STFC for the Doctoral
Training Programme funding (ST/W507854-2021 Maths DTP). M. H. thanks his STFC
consolidated grant ST/X000656/1. A. J. was supported by the U.S. Department of
Energy under contract DE-SC0010010. M. H. and A. J. thank the participants of
the APCTP Focus Program “Entanglement, Large $N$, and Black Hole 2024” for
questions, comments, and discussions during and after M. H.’s presentation.

##  Appendix A Polchinski’s puzzle

Let us explain a seemingly puzzling conceptual issue on the emergent geometry
from matrix degrees of freedom [6, 7] by using a matrix model with the
Hamiltonian (1) as an example. The following argument uses only generic
properties of large-$N$ gauge theory. Therefore, essentially the same puzzle
existed for AdS5/CFT4 as well; see e.g. [59]. Note also that, though we will
focus on the ’t Hooft large-$N$ limit, essentially the same results are
obtained quite generally including the parameter region of the BFSS matrix
model that describes M-theory.

The simple relations that created huge, decades-long confusion are

$\displaystyle\bra{\Phi}\mathrm{Tr}\hat{X}_{I}^{2}\ket{\Phi}\sim N^{2}$ (90)

and

$\displaystyle\bra{\Phi}\mathrm{Tr}[\hat{X}_{I},\hat{X}_{J}]^{2}\ket{\Phi}\sim
N^{3}\,.$ (91)

These are consequences of ’t Hooft scaling and hence hold quite generally for
any theory and any state $\ket{\Phi}$ whose energy is of order $N^{2}$ or
smaller. In particular, these relations hold for the ground state as well.

Let us imagine we performed a measurement in the coordinate basis and obtained
a generic $X_{I,ij}$. In retrospect, we should have known that such a
$X_{I,ij}$ does not necessarily carry the information of emergent geometry —
we must take a very special point, i.e., the center of wave packet — but
suppose that such matrices $X_{I,ij}$ could describe geometry and see what
kinds of ‘physics’ would follow. Note that such an assumption could look
natural, given an apparent analogy to the master field. Then, such typical
matrices satisfy

$\displaystyle\mathrm{Tr}X_{I}^{2}\sim\bra{\Phi}\mathrm{Tr}\hat{X}_{I}^{2}\ket{\Phi}\sim
N^{2}\,,$ (92)

and hence, eigenvalues of each $X_{I}$ are typically of order $\sqrt{N}$,
which is as large as the radius of the bulk region where the weakly-curved
gravity picture is valid. This is the case even for the ground state. It looks
as though the D-branes are fluctuating a lot, and the wave function is not
localized and the resolution for the ‘location’ is so bad that it cannot
capture the geometry on the gravity side. Furthermore, if we diagonalize
$X_{1}$, then

$\displaystyle\mathrm{Tr}[X_{1},X_{J}]^{2}\sim\bra{\Phi}\mathrm{Tr}[\hat{X}_{1},\hat{X}_{J}]^{2}\ket{\Phi}\sim
N^{3}$ (93)

for $J\neq 1$, which means

$\displaystyle\sum_{i\neq j}(X_{1,ii}-X_{1,jj})^{2}|X_{J,ij}|^{2}\sim
N^{3}\,.$ (94)

Because $(X_{1,ii}-X_{1,jj})^{2}$ is typically of order $N$, each $|X_{J,ij}|$
is of order 1. Given that $\mathrm{Tr}X_{J}^{2}\sim N^{2}$, a significant
fraction of $\mathrm{Tr}X_{J}^{2}$ comes from the off-diagonal entries of
$X_{J}$. In other words, when $X_{1}$ is diagonalized, other $X_{J}$’s are not
close to diagonal. For this reason, it looks as though the basic assumption of
the geometric picture — $X_{I}$’s can be taken close to simultaneously
diagonal — does not hold.

Because this pathological feature appears even for the ground state, it was
clear that the quantum fluctuation must be removed in order to obtain the
right geometric information [6]. Ref. [1] pointed out that the geometric
information is obtained from the center of the wave packet..

##  Appendix B Computation of three-point function in Sec. 4.3.1

In this appendix, we use $z=r^{-1}$ frequently. As $\hat{O}$, we take chiral
primaries that consist of $k_{O}$ scalar fields. We use $\hat{O}_{r,\vec{x}}$
and $\hat{O}_{z,\vec{x}}$ to mean the same operator. To regularize the short-
distance divergence in the bulk, we can use Gaussian wave packet in the
bulk:121212 Note also that the examples discussed in Sec. 4.3.2 correspond to
such a wave packet on the gravity side with a finite value of $\sigma_{0}$.

$\displaystyle\hat{O}^{\rm(smear)}_{z,\vec{x}}=\frac{1}{4\pi^{2}\sigma^{4}(z)}\int
d^{3}\vec{x}^{\prime}dz^{\prime}e^{-\frac{(z-z^{\prime})^{2}+(\vec{x}-\vec{x}^{\prime})^{2}}{2\sigma^{2}(z)}}\hat{O}_{z^{\prime},\vec{x}^{\prime}}\,.$
(95)

We take $\sigma$ to vary with the radial coordinate $z$ as

$\displaystyle\sigma(z)=\sigma_{0}\times z\,,$ (96)

where $\sigma_{0}$ is a constant, so that the spatial volume of the wave
packet is fixed. We take the variance small enough such that $\sigma_{0}\ll
1$. The two-point function of $\hat{O}^{\rm(smear)}_{z,\vec{x}}$ at the same
point is

$\displaystyle\frac{1}{(4\pi^{2}\sigma^{4})^{2}}\int
d^{3}\vec{x}_{1}dz_{1}e^{-\frac{(z-z_{1})^{2}+(\vec{x}-\vec{x}_{1})^{2}}{2\sigma^{2}}}\int
d^{3}\vec{x}_{2}dz_{2}e^{-\frac{(z-z_{2})^{2}+(\vec{x}-\vec{x}_{2})^{2}}{2\sigma^{2}}}\langle\textrm{g.s.}|\hat{O}^{\dagger}_{z_{1},\vec{x}_{1}}\hat{O}_{z_{2},\vec{x}_{2}}|\textrm{g.s.}\rangle\,.$
(97)

The bulk-to-bulk propagator in AdS5 for a massive scalar field $\hat{O}$ of
mass squared $m^{2}=k_{O}(k_{O}-4)$ is given by, up to an overall factor,
131313Note that the one-point function vanishes for chiral primaries. [60, 61]

$\displaystyle\langle\textrm{g.s.}|\hat{O}^{\dagger}_{r_{1},\vec{x}_{1}}\hat{O}_{r_{2},\vec{x}_{2}}|\textrm{g.s.}\rangle=\xi^{k_{O}}F\left(\frac{k_{O}}{2},\frac{k_{O}+1}{2};k_{O}-1;\xi^{2}\right)\,,$
(98)

where $F$ is the hypergeometric function and

$\displaystyle\xi=\frac{2z_{1}z_{2}}{z_{1}^{2}+z_{2}^{2}+(\vec{x}_{1}-\vec{x}_{2})^{2}}\,.$
(99)

To regularize the denominator of (48), we replace $\hat{O}$ with
$\hat{O}^{\rm(smear)}$. We take $\sigma_{0}$ parametrically small such that
only $z_{1},z_{2}\simeq z$ and $\vec{x}_{1},\vec{x}_{2}\simeq\vec{x}$
contribute. Then, $\xi$ can be approximated as

$\displaystyle\xi\simeq\left(1+\frac{(z_{1}-z_{2})^{2}+(\vec{x}_{1}-\vec{x}_{2})^{2}}{2z^{2}}\right)^{-1}\,.$
(100)

Then, $F\left(\frac{k_{O}}{2},\frac{k_{O}+1}{2};k_{O}-1;\xi^{2}\right)$
diverges as
$(1-\xi^{2})^{-3/2}\sim\left(\frac{z^{2}}{(z_{1}-z_{2})^{2}+(\vec{x}_{1}-\vec{x}_{2})^{2}}\right)^{3/2}$.
Therefore, (97) becomes

$\displaystyle\frac{1}{(4\pi^{2}\sigma^{4})^{2}}\int
d^{3}\vec{x}_{1}dr_{1}e^{-\frac{(z-z_{1})^{2}+(\vec{x}-\vec{x}_{1})^{2}}{2\sigma^{2}}}\int
d^{3}\vec{x}_{2}dr_{2}e^{-\frac{(z-z_{2})^{2}+(\vec{x}-\vec{x}_{2})^{2}}{2\sigma^{2}}}\left(\frac{z^{2}}{(z_{1}-z_{2})^{2}+(\vec{x}_{1}-\vec{x}_{2})^{2}}\right)^{3/2}$
$\displaystyle=\frac{1}{16\pi^{2}\sigma^{4}}\int d^{3}\vec{\delta}d\epsilon
e^{-\frac{\epsilon^{2}+\vec{\delta}^{2}}{4\sigma^{2}}}\left(\frac{z^{2}}{\epsilon^{2}+\vec{\delta}^{2}}\right)^{3/2}\,,$
(101)

where $\epsilon=z_{1}-z_{2}$ and $\vec{\delta}=\vec{x}_{1}-\vec{x}_{2}$, and
hence, by using $\rho=\sqrt{\epsilon^{2}+\vec{\delta}^{2}}$,

$\displaystyle\frac{1}{16\pi^{2}\sigma^{4}}\int_{0}^{\infty}2\pi^{2}\rho^{3}d\rho
e^{-\frac{\rho^{2}}{4\sigma^{2}}}\frac{z^{3}}{\rho^{3}}=\frac{\sqrt{\pi}z^{3}}{4\sigma^{3}}=\frac{\sqrt{\pi}}{4\sigma_{0}^{3}}\,.$
(102)

We use the same regularization for the three-point function. Then, the
numerator of (48) becomes the product of three-point vertex, two bulk-to-bulk
propagators (98), and bulk-to-boundary propagator, integrated over the
location of the three-point vertex which we denote by
$\tau_{3},r_{3},\vec{x}_{3}$. Because the dominant contribution comes from
$\tau_{3}\sim 0$, $r_{3}\sim r$ and $\vec{x}_{3}\sim\vec{x}$, the bulk-to-
boundary propagator for the rank-$k$ chiral primary operator is, up to a
normalization factor [61] and negligible corrections,

$\displaystyle
K(r,\vec{x};\vec{x}^{\prime})=\left(\frac{r^{-1}}{r^{-2}+(\vec{x}-\vec{x}^{\prime})^{2}}\right)^{k}\,.$
(103)

This sets the $r$-dependence of the three-point function. Note that this
scaling is valid for generic local operators in the bulk. The divergence comes
from the product of two bulk-to-bulk propagators integrated over the location
of three-point vertex,

$\displaystyle\frac{1}{(4\pi^{2}\sigma^{4})^{2}}\int
d^{3}\vec{x}_{1}dz_{1}e^{-\frac{(z-z_{1})^{2}+(\vec{x}-\vec{x}_{1})^{2}}{2\sigma^{2}}}\int
d^{3}\vec{x}_{2}dz_{2}e^{-\frac{(z-z_{2})^{2}+(\vec{x}-\vec{x}_{2})^{2}}{2\sigma^{2}}}$
$\displaystyle\qquad\int\frac{d^{3}\vec{x}_{3}dz_{3}d\tau_{3}}{z^{5}}\left(\frac{z^{2}}{z_{13}^{2}+\vec{x}_{13}^{2}+\tau_{3}^{2}}\right)^{3/2}\left(\frac{z^{2}}{z_{23}^{2}+\vec{x}_{23}^{2}+\tau_{3}^{2}}\right)^{3/2}$
(104)

where $z_{13}=z_{1}-z_{3}$, etc. To evaluate the second line, let us introduce
$\tau_{1}$ and $\tau_{2}$, which are sent to zero later, and write the
integral by using a 5-component vector $\boldsymbol{x}=(\tau,z,\vec{x})$ as

$\displaystyle z\int
d^{3}\vec{x}_{3}dz_{3}d\tau_{3}\left(\frac{1}{z_{13}^{2}+\vec{x}_{13}^{2}+\tau_{13}^{2}}\right)^{3/2}\left(\frac{1}{z_{23}^{2}+\vec{x}_{23}^{2}+\tau_{23}^{2}}\right)^{3/2}=z\int\frac{d^{5}\boldsymbol{x}_{3}}{|\boldsymbol{x}_{13}|^{3}|\boldsymbol{x}_{23}|^{3}}\,.$
(105)

This integral depends only on the distance $|\boldsymbol{x}_{12}|$ and, by the
dimensional counting, we obtain $\frac{z}{|\boldsymbol{x}_{12}|}$ up to a
multiplicative factor. Plugging it back to (104), we obtain
$\frac{z}{\sigma}=\frac{1}{\sigma_{0}}$. Combining this factor, (102), and
(103), we obtain

$\displaystyle\frac{\bra{\rm
g.s.}\hat{O}^{\rm(smeared)\dagger}_{r,\vec{x}}\mathrm{Tr}(X_{\\{I_{1}}\cdots
X_{I_{k}\\}})(\vec{x}^{\prime})\hat{O}^{\rm(smeared)}_{r,\vec{x}}\ket{\rm
g.s.}_{\rm conn}}{\bra{\rm
g.s.}\hat{O}^{\rm(smeared)\dagger}_{r,\vec{x}}\hat{O}^{\rm(smeared)}_{r,\vec{x}}\ket{\rm
g.s.}}\sim\sigma_{0}^{2}\times\left(\frac{r^{-1}}{r^{-2}+(\vec{x}-\vec{x}^{\prime})^{2}}\right)^{k}\,.$
(106)

This gives us the expected scaling explained in Sec. 4.3.1. That the result
depends on the size of the wave packet through the overall factor
$\sigma_{0}^{2}$ may look strange at first sight, but that it vanishes as the
wave packet $\hat{O}^{\rm(smeared)}\ket{\rm g.s.}$ shrinks is actually
natural. The three-point function can be interpreted as the overlap of two
states
$\mathrm{Tr}(\hat{X}_{\\{I_{1}}\cdots\hat{X}_{I_{k}\\}})(\vec{x}^{\prime})\hat{O}^{\rm(smeared)}\ket{\rm
g.s.}$ and $\hat{O}^{\rm(smeared)}\ket{\rm g.s.}$. While the latter is
localized in the bulk by assumption, we expect that the former is not because
there is no reason to be so. (Note that $\hat{O}^{\rm(smeared)}\ket{\rm g.s.}$
is not necessarily an eigenstate of $\mathrm{Tr}(X_{\\{I_{1}}\cdots
X_{I_{k}\\}})(\vec{x}^{\prime})$.) Hence the overlap should become smaller as
the wave packet shrinks.

The normalization of a three-point vertex can depend on the normalization of
two-point functions. When the two-point function does not have a $k$-dependent
overall factor, the three-point vertex for rank-$k_{1}$, $k_{2}$ and $k_{3}$
chiral primary is proportional to $\sqrt{k_{1}k_{2}k_{3}}$ [35]. For
$k_{1}=k_{2}=k_{O}$, this is proportional to $k_{O}$. This factor $k_{O}$
appears in the three-point function normalized as (48).

## References

  * [1] M. Hanada, “Bulk geometry in gauge/gravity duality and color degrees of freedom,” Phys. Rev. D 103 no. 10, (2021) 106007, arXiv:2102.08982 [hep-th].
  * [2] M. Hanada, “Large-N limit as a second quantization,” PoS CORFU2021 (2022) 260, arXiv:2103.15873 [hep-th].
  * [3] V. Gautam, M. Hanada, A. Jevicki, and C. Peng, “Matrix entanglement,” JHEP 01 (2023) 003, arXiv:2204.06472 [hep-th].
  * [4] M. Hanada and J. Maltz, “A proposal of the gauge theory description of the small Schwarzschild black hole in AdS${}_{5}\times$S5,” JHEP 02 (2017) 012, arXiv:1608.03276 [hep-th].
  * [5] M. Hanada, G. Ishiki, and H. Watanabe, “Partial Deconfinement,” JHEP 03 (2019) 145, arXiv:1812.05494 [hep-th]. [Erratum: JHEP 10, 029 (2019)].
  * [6] J. Polchinski, “M theory and the light cone,” Prog. Theor. Phys. Suppl. 134 (1999) 158–170, arXiv:hep-th/9903165.
  * [7] L. Susskind, “Holography in the flat space limit,” AIP Conf. Proc. 493 no. 1, (1999) 98–112, arXiv:hep-th/9901079.
  * [8] E. Witten, “APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory,” Rev. Mod. Phys. 90 no. 4, (2018) 045003, arXiv:1803.04993 [hep-th].
  * [9] S. Leutheusser and H. Liu, “Causal connectability between quantum systems and the black hole interior in holographic duality,” Phys. Rev. D 108 no. 8, (2023) 086019, arXiv:2110.05497 [hep-th].
  * [10] S. Leutheusser and H. Liu, “Subalgebra-subregion duality: emergence of space and time in holography,” arXiv:2212.13266 [hep-th].
  * [11] T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, “M theory as a matrix model: A Conjecture,” Phys. Rev. D 55 (1997) 5112–5128, arXiv:hep-th/9610043.
  * [12] J. M. Maldacena, “Eternal black holes in anti-de Sitter,” JHEP 04 (2003) 021, arXiv:hep-th/0106112.
  * [13] M. Van Raamsdonk, “Building up spacetime with quantum entanglement,” Gen. Rel. Grav. 42 (2010) 2323–2329, arXiv:1005.3035 [hep-th].
  * [14] M. Hanada and H. Watanabe, “On Thermal Transition in QCD,” PTEP 2024 no. 4, (2024) 043B02, arXiv:2310.07533 [hep-th].
  * [15] E. Witten, “Bound states of strings and p-branes,” Nucl. Phys. B 460 (1996) 335–350, arXiv:hep-th/9510135.
  * [16] B. de Wit, J. Hoppe, and H. Nicolai, “On the Quantum Mechanics of Supermembranes,” Nucl. Phys. B 305 (1988) 545.
  * [17] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998) 231–252, arXiv:hep-th/9711200.
  * [18] M. Hanada, H. Shimada, and N. Wintergerst, “Color confinement and Bose-Einstein condensation,” JHEP 08 (2021) 039, arXiv:2001.10459 [hep-th].
  * [19] M. Hanada, A. Jevicki, C. Peng, and N. Wintergerst, “Anatomy of Deconfinement,” JHEP 12 (2019) 167, arXiv:1909.09118 [hep-th].
  * [20] D. Berenstein, “Submatrix deconfinement and small black holes in AdS,” JHEP 09 (2018) 054, arXiv:1806.05729 [hep-th].
  * [21] N. Itzhaki, J. M. Maldacena, J. Sonnenschein, and S. Yankielowicz, “Supergravity and the large N limit of theories with sixteen supercharges,” Phys. Rev. D 58 (1998) 046004, arXiv:hep-th/9802042.
  * [22] D. E. Berenstein, J. M. Maldacena, and H. S. Nastase, “Strings in flat space and pp waves from N=4 superYang-Mills,” JHEP 04 (2002) 013, arXiv:hep-th/0202021.
  * [23] K. N. Anagnostopoulos, M. Hanada, J. Nishimura, and S. Takeuchi, “Monte Carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature,” Phys. Rev. Lett. 100 (2008) 021601, arXiv:0707.4454 [hep-th].
  * [24] S. Catterall and T. Wiseman, “Black hole thermodynamics from simulations of lattice Yang-Mills theory,” Phys. Rev. D 78 (2008) 041502, arXiv:0803.4273 [hep-th].
  * [25] Monte Carlo String/M-theory (MCSMC), MCSMC Collaboration, G. Bergner, N. Bodendorfer, M. Hanada, S. Pateloudis, E. Rinaldi, A. Schäfer, P. Vranas, and H. Watanabe, “Confinement/deconfinement transition in the D0-brane matrix model — A signature of M-theory?,” JHEP 05 (2022) 096, arXiv:2110.01312 [hep-th].
  * [26] Monte Carlo String/M-theory (MCSMC) Collaboration, S. Pateloudis, G. Bergner, M. Hanada, E. Rinaldi, A. Schäfer, P. Vranas, H. Watanabe, and N. Bodendorfer, “Precision test of gauge/gravity duality in D0-brane matrix model at low temperature,” JHEP 03 (2023) 071, arXiv:2210.04881 [hep-th].
  * [27] D. N. Kabat and G. Lifschytz, “Gauge theory origins of supergravity causal structure,” JHEP 05 (1999) 005, arXiv:hep-th/9902073.
  * [28] S. Sethi and M. Stern, “D-brane bound states redux,” Commun. Math. Phys. 194 (1998) 675–705, arXiv:hep-th/9705046.
  * [29] P. Yi, “Witten index and threshold bound states of D-branes,” Nucl. Phys. B 505 (1997) 307–318, arXiv:hep-th/9704098.
  * [30] J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe, and S.-T. Yau, “Asymptotic form of zero energy wave functions in supersymmetric matrix models,” Nucl. Phys. B 567 (2000) 231–248, arXiv:hep-th/9904182.
  * [31] E. Gesteau and L. Santilli, “Explicit large $N$ von Neumann algebras from matrix models,” arXiv:2402.10262 [hep-th].
  * [32] A. Jevicki, Y. Kazama, and T. Yoneya, “Quantum metamorphosis of conformal transformation in D3-brane Yang-Mills theory,” Phys. Rev. Lett. 81 (1998) 5072–5075, arXiv:hep-th/9808039.
  * [33] A. Jevicki, Y. Kazama, and T. Yoneya, “Generalized conformal symmetry in D-brane matrix models,” Phys. Rev. D 59 (1999) 066001, arXiv:hep-th/9810146.
  * [34] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96 (2006) 181602, arXiv:hep-th/0603001.
  * [35] S. Lee, S. Minwalla, M. Rangamani, and N. Seiberg, “Three point functions of chiral operators in D = 4, N=4 SYM at large N,” Adv. Theor. Math. Phys. 2 (1998) 697–718, arXiv:hep-th/9806074.
  * [36] M. Hanada, S. Matsuura, and F. Sugino, “Two-dimensional lattice for four-dimensional N=4 supersymmetric Yang-Mills,” Prog. Theor. Phys. 126 (2011) 597–611, arXiv:1004.5513 [hep-lat].
  * [37] M. Hanada, “A proposal of a fine tuning free formulation of 4d N = 4 super Yang-Mills,” JHEP 11 (2010) 112, arXiv:1009.0901 [hep-lat].
  * [38] O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, “N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals,” JHEP 10 (2008) 091, arXiv:0806.1218 [hep-th].
  * [39] J. Maldacena and X.-L. Qi, “Eternal traversable wormhole,” arXiv:1804.00491 [hep-th].
  * [40] P. Gao, D. L. Jafferis, and A. C. Wall, “Traversable Wormholes via a Double Trace Deformation,” JHEP 12 (2017) 151, arXiv:1608.05687 [hep-th].
  * [41] M. Srednicki, “Entropy and area,” Phys. Rev. Lett. 71 (1993) 666–669, arXiv:hep-th/9303048.
  * [42] F. Alet, M. Hanada, A. Jevicki, and C. Peng, “Entanglement and Confinement in Coupled Quantum Systems,” JHEP 02 (2021) 034, arXiv:2001.03158 [hep-th].
  * [43] J. Maldacena and A. Milekhin, “To gauge or not to gauge?,” JHEP 04 (2018) 084, arXiv:1802.00428 [hep-th].
  * [44] A. Jevicki, D. Mukherjee, and J. Yoon, “Emergent factorization of Hilbert space at large $N$ and black hole,” arXiv:2404.07862 [hep-th].
  * [45] B. Czech, J. L. Karczmarek, F. Nogueira, and M. Van Raamsdonk, “The Gravity Dual of a Density Matrix,” Class. Quant. Grav. 29 (2012) 155009, arXiv:1204.1330 [hep-th].
  * [46] A. C. Wall, “Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy,” Class. Quant. Grav. 31 no. 22, (2014) 225007, arXiv:1211.3494 [hep-th].
  * [47] M. Headrick, V. E. Hubeny, A. Lawrence, and M. Rangamani, “Causality & holographic entanglement entropy,” JHEP 12 (2014) 162, arXiv:1408.6300 [hep-th].
  * [48] D. L. Jafferis, A. Lewkowycz, J. Maldacena, and S. J. Suh, “Relative entropy equals bulk relative entropy,” JHEP 06 (2016) 004, arXiv:1512.06431 [hep-th].
  * [49] A. Almheiri, X. Dong, and D. Harlow, “Bulk Locality and Quantum Error Correction in AdS/CFT,” JHEP 04 (2015) 163, arXiv:1411.7041 [hep-th].
  * [50] X. Dong, D. Harlow, and A. C. Wall, “Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,” Phys. Rev. Lett. 117 no. 2, (2016) 021601, arXiv:1601.05416 [hep-th].
  * [51] J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle, and M. Walter, “Entanglement Wedge Reconstruction via Universal Recovery Channels,” Phys. Rev. X 9 no. 3, (2019) 031011, arXiv:1704.05839 [hep-th].
  * [52] J. Maldacena and L. Susskind, “Cool horizons for entangled black holes,” Fortsch. Phys. 61 (2013) 781–811, arXiv:1306.0533 [hep-th].
  * [53] B. Swingle, “Entanglement Renormalization and Holography,” Phys. Rev. D 86 (2012) 065007, arXiv:0905.1317 [cond-mat.str-el].
  * [54] A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, “Holographic representation of local bulk operators,” Phys. Rev. D 74 (2006) 066009, arXiv:hep-th/0606141.
  * [55] S. Terashima, “Bulk locality in the AdS/CFT correspondence,” Phys. Rev. D 104 no. 8, (2021) 086014, arXiv:2005.05962 [hep-th].
  * [56] S. Sugishita and S. Terashima, “Rindler bulk reconstruction and subregion duality in AdS/CFT,” JHEP 11 (2022) 041, arXiv:2207.06455 [hep-th].
  * [57] G. T. Horowitz, “Comments on black holes in string theory,” Class. Quant. Grav. 17 (2000) 1107–1116, arXiv:hep-th/9910082.
  * [58] E. Berkowitz, M. Hanada, and J. Maltz, “Chaos in Matrix Models and Black Hole Evaporation,” Phys. Rev. D 94 no. 12, (2016) 126009, arXiv:1602.01473 [hep-th].
  * [59] I. Heemskerk, D. Marolf, J. Polchinski, and J. Sully, “Bulk and Transhorizon Measurements in AdS/CFT,” JHEP 10 (2012) 165, arXiv:1201.3664 [hep-th].
  * [60] C. Burgess and C. Lütken, “Propagators and effective potentials in anti-de sitter space,” Physics Letters B 153 no. 3, (1985) 137–141.
  * [61] D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, “Correlation functions in the CFT(d) / AdS(d+1) correspondence,” Nucl. Phys. B 546 (1999) 96–118, arXiv:hep-th/9804058.