# To wedge or not to wedge

Vic Vander Linden

(June 2023)

## Abstract

The AdS/CFT correspondence is an explicit realization of the holographic
principle relating a theory of gravity in a volume of space to a lower
dimensional quantum field theory on its boundary. By exploiting elements of
quantum error correction, qubit toy models of this correspondence have been
constructed for which the bulk logical operators are representable by
operators acting on the boundary [1]. Given a boundary subregion, wedges in
the volume space are used to enclose the bulk qubits for which logical
operators are reconstructable on that boundary subregion. In this thesis a
number of different wedges, such as the causal wedge, greedy entanglement
wedge and minimum entanglement wedge, are examined. More specifically, Monte-
Carlo simulations of boundary erasure are performed with various toy models to
study the differences between wedges and the effect on these wedge by the type
of the model, non-uniform boundaries and stacking of models. It has been found
that the minimum entanglement wedge is the best approximate for the true
geometric wedge. This is illustrated by an example toy model for which an
operator beyond the greedy entanglement wedge was also reconstructed. In
addition, by calculating the entropy of these subregions, the viability of a
mutual information wedge is rejected. Only for particular connected boundary
subregions was the inclusion of the central tensor by the geometric wedge
associated to a rise in mutual information.

###### Contents

  1. 1 Introduction
    1. 1.1 Qubits
    2. 1.2 Quantum Error Correction
    3. 1.3 Tensors
    4. 1.4 AdS/CFT Correspondence
  2. 2 Wedges and Operator Reconstructability
    1. 2.1 Causal Wedge
    2. 2.2 Entanglement Wedge
    3. 2.3 Minimum Entanglement Wedge
    4. 2.4 Operator Reconstructability
  3. 3 Mutual Information
  4. 4 Conclusion
  5. Acknowledgment

##  1 Introduction

In 1981, Richard Feynman jokingly called to close the doors in an attempt to
trap his attendees at the first Physics and Computation conference. He did so
as he revealed the only way to truly simulate physics might be through quantum
mechanics [2]. His ideas about an inherent probabilistic computer helped to
form and popularize a new branch of physics, quantum information.

In this thesis, a brief introduction to quantum information, quantum error
correction, tensors and tensor networks and the AdS/CFT correspondence will be
given. For this purpose, a literary study of the following works was conducted
[1] \- [19]. Thereafter, wedges and operator reconstructability for qubit toy
models of this AdS/CFT correspondence are investigated using Monte-Carlo
simulations of central qubit inclusion with boundary erasure, inspired by the
similar simulations performed by Pastawski et al. in [1]. These original
simulations are recreated in Section 2. In addition, further Monte-Carlo
simulations are carried out to investigate other properties of tensor networks
such as a different shape of the network and the effect of a non-uniform
boundary. Additionally, the effect of stacking tensor networks, as introduced
by [20], is investigated using the same simulations. Furthermore, an
additional wedge will be proposed, the minimal entanglement wedge. The Monte-
Carlo simulations using this wedge will depend on the state of the bulk
qubits. Its particular behaviour for single states will be explained using a
theorem constructed and proven in this thesis. This minimal entanglement wedge
will be shown to be useful in approximating the true geometric entanglement
wedge of toy tensor networks. Finally, the relationship between these wedges
and operator reconstructability of bulk qubits will be investigated using
newly constructed examples.

In the third chapter of this thesis the viability of a mutual information
wedge will be determined. The connection between inclusion in the geometric
wedge and the rise in mutual information will be investigated for both
connected and non connected boundary subregions. Additionally, the effect of
different bulk states, tensor networks and layer depth on the mutual
information will be examined. Furthermore, the computation of both the Von
Neumann- and Ryu-Takayanagi entropy will be compared for a growing boundary
subregion of one explicit example.

###  1.1 Qubits

The foundation of modern quantum information theory lies in the qubit. This
term, first coined by Schumacher in 1995 [3], represents a system in a
superposition of two possible energy levels which are denoted by $\ket{0}$ and
$\ket{1}$,

$\ket{\Psi}=\alpha\ket{0}+\beta\ket{1}=\alpha\begin{pmatrix}1\\\
0\end{pmatrix}+\beta\begin{pmatrix}0\\\ 1\end{pmatrix},$ (1)

where $\ket{\Psi}$ represents the state of the system and
$\alpha,\beta\in\mathbb{C}$ with $|\alpha|^{2}+|\beta|^{2}=1$.

These qubits are isomorphic to spin-$\frac{1}{2}$ systems, spanning a two
dimensional Hilbert space $\mathcal{H}_{2}$. In recent quantum computers, such
as Google’s 53 qubit and IBM’s 433 qubit quantum processor, qubits are
commonly approximated by letting current flow through a superconducting
material [4, 5]. Qubits have a property that regular bits do not possess,
entanglement. As an example, consider an Einstein–Podolsky–Rosen (EPR) pair or
Bell state of two qubits [6],

$\ket{\Psi}=\frac{\ket{00}+\ket{11}}{\sqrt{2}},$ (2)

where $\ket{00}$ and $\ket{11}$ are the tensor products
$\ket{0}\otimes\ket{0}$ and $\ket{1}\otimes\ket{1}$ respectively.

Suppose one wishes to describe only one of these qubits independently. This
can be done by tracing out the other qubit. This mathematical procedure is
conducted by performing a partial trace of the density matrix $\rho$. This
density matrix is defined for a quantum system as
$\rho=\sum_{j}{p_{j}\ket{\Psi_{j}}\bra{\Psi_{j}}}$, where $j$ sums over a
basis of pure states for the system with their probability $p_{j}$. As an
example the density matrix of the EPR pair is represented by,

$\rho=1\cdot\left(\frac{\ket{00}+\ket{11}}{\sqrt{2}}\right)\left(\frac{\bra{00}+\bra{11}}{\sqrt{2}}\right)=\begin{pmatrix}1/2&0&0&1/2\\\
0&0&0&0\\\ 0&0&0&0\\\ 1/2&0&0&1/2\\\ \end{pmatrix}.$ (3)

where the matrix is the result of the tensor product of the individual Hilbert
spaces. This density matrix represents a pure state because for the normalized
density matrix the trace of the square, $\operatorname{Tr}(\rho^{2})$, is
equal to one [7]. Tracing out a subsystem $A$ of a density matrix then
involves taking the partial trace of subsystem B defined as [8],

$\operatorname{Tr}_{B}(\rho_{AB})=\sum_{j}{(I_{A}\otimes\bra{j}_{B})\rho_{AB}(I_{A}\otimes\ket{j}_{B})}$
(4)

where $\set{\ket{j}_{B}}$ is any orthonormal basis for the Hilbert space
$\mathcal{H}_{B}$ of subsystem B and $I_{A}$ is the identity operator of the
remaining Hilbert space $\mathcal{H}_{A}$. As an example, tracing out one of
the qubits of the EPR pair, will yield for the remaining qubit the mixed state
with density matrix,

$\rho_{A}=\operatorname{Tr}_{B}(\rho_{AB})=\frac{1}{2}(\ket{0_{A}}\bra{0_{A}}+\ket{1_{A}}\bra{1_{A}})=\begin{pmatrix}1/2&0\\\
0&1/2\\\ \end{pmatrix}.$ (5)

This procedure will not result a pure linear superposition as defined in
Formula (1), but a mixed state made up of an ensemble of the states
$\set{\bra{0_{A}},\bra{1_{A}}}$ with a probability of $\frac{1}{2}$ each.
Since it is a mixed state, $\operatorname{Tr}(\rho^{2})$ will no longer be
equal to one.

In addition, calculating the Von Neumann entropy defined as,

$S_{A}=-\operatorname{Tr}(\rho_{A}\cdot\log(\rho_{A})),$ (6)

for a subsystem A, yields information about the degree of entanglement between
A and B. In this case the entropy is $\log(2)$, which is the largest possible
value for a two qubit system. In this thesis, as is often the case, will the
entropy of qubits be given in terms of $\log(2)$, so the entropy would be
$S_{A}$ = 1. This value highlights the fact that measuring the state of A will
also yield knowledge of the state of B, causing qubit B to collapse to that
measured energy level. In other words, the two qubits of an EPR pair are
maximally entangled. In this example it holds true that the entanglement
entropy of qubit A is equal to the entanglement entropy of qubit B. This
property can be generalized to any pure quantum state partitioned into $A$ and
$A^{c}$ and yields $S_{A}=S_{A^{c}}$. Lastly, the mutual information $I_{AB}$
between subsystems $A$ and $B$ is defined as $I_{AB}=S_{A}$ \+ $S_{B}$ \-
$S_{AB}$. Because the sum of entropy of the individual systems will always be
larger or equal to the entropy of the combination, the mutual information will
be larger or equal to zero.

By exploiting the entanglement between qubits it is possible to perform
computations that classical Turing Machines are not able to do efficiently.
Such an example is simulating inherent quantum systems, like the output of a
pseudo-random quantum circuit as shown by Google’s quantum processor [4].
Another computational task is the factoring of large numbers into two prime
numbers, a task commonly used in the security of online exchanges. For this
purpose an efficient quantum algorithm was advanced by P. Shor in 1995 [9].

###  1.2 Quantum Error Correction

The main limitation of quantum processors is their vulnerability to noise.
Such noise is usually qubit interacting with its environment. Examples of this
include noisy quantum gates or spontaneous emission of the qubit [10]. In
general, quantum errors can be thought of as the qubit transforming to a mixed
state entangled with its environment. Today, this noise is still significant
in quantum processors. For this reason quantum computing is still classified
to be in its noisy intermediate-scale quantum stage or NISQ. One might think
to solve this problem with the same error correction framework as for
classical computers. Unfortunately, quantum theory prevents this.
Specifically, the state of a qubit cannot be cloned. Furthermore, any attempt
to measure the state of a qubit will destroy the quantum information contained
in that qubit. Lastly, as opposed to discrete errors in regular qubits,
quantum errors are continuous [11]. As an example, in Formula (1) the
parameter $\alpha$ might change by an infinitesimal amount. Nevertheless,
quantum error correcting codes have been developed.

These errors are described using quantum noise operations $\varepsilon(\rho)$
that act on the density matrix $\rho$ of the system. All trace-preserving
quantum noise operations $\varepsilon(\rho)$ associated with a closed system,
may be approximated with error-operation elements acting on a single qubit,
$E_{i}$, which can then be expressed as a linear combination [11],

$E_{i}=e_{i0}I+e_{i1}X+e_{i2}Z+e_{i3}XZ,$ (7)

where $X$ and $Z$ are the Pauli operators acting on a single qubit. The
combination $XZ$ represents the operators acting in right to left order on the
qubit. These Pauli operators are,

$X=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\ Y=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\ Z=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$ (8)

where the operators $X$ and $Z$ will represent a bit flip and phase shift
respectively and $Y=iXZ$.
The constants $e_{i0},e_{i1},e_{i2},e_{i3}\in\mathbb{C}$ are sufficient to
describe any single operation element $E_{i}$. These will be able to describe
any arbitrary error acting on a single qubit. As an example, amplitude damping
might occur in a qubit interacting with its environment. This phenomenon can
be modeled with a constant $\gamma\in[0,1]$ existing in two operation elements
[11],

$E_{0}=\begin{pmatrix}1&0\\\ 0&\sqrt{1-\gamma}\end{pmatrix},\
E_{1}=\begin{pmatrix}0&\sqrt{\gamma}\\\ 0&0\end{pmatrix}$ (9)

acting on a density matrix $\rho$,

$\varepsilon(\rho)=E_{0}\rho E_{0}^{\dagger}+E_{1}\rho E_{1}^{\dagger}.$ (10)

It is crucial to note that two assumptions have already been made. Firstly, it
is assumed that the error operation is trace preserving, which corresponds to
a closed system. The operation described in Formula (10) is indeed trace-
preserving for all density matrices. However, measurements are an example of
operations which can be described by a set of single qubit operation elements
$E_{i}$ that are generally not trace-preserving. The second assumption made
with this description, is that correlated error operations will not occur on
multiple qubits at once. This is justified by the reasoning that if the chance
of an error occurring on one qubit is small $\mathcal{O}(\epsilon)$, the
chance of a correlated error acting on different qubits at the same time, is
even smaller $\mathcal{O}(\epsilon^{2})$.
The general framework of quantum error correction consists of performing
syndrome measurements to detect which error occurred without destroying the
quantum information, followed by the corresponding correction operation. The
combination of both of these is usually denoted by $R$. A powerful theorem in
quantum error correction states that when an error correcting code yields an
error correcting procedure $R$ which allows to recover from a trace-preserving
noise operation $\varepsilon$ with operation elements $\set{E_{i}}$, that $R$
will also be able correct the effects of any noise model $\mathcal{F}$ for
which its operation elements $\set{F_{i}}$ are linear combinations of
$\set{E_{i}}$ [11]. Consequently, since any operation element is always a
linear combination of $X$, $Z$ and $XZ=-iY$, an additional third assumption is
justified. The errors are assumed to be only the Pauli matrices X,Z and Y
denoted in Equations (8). With these assumptions, one can describe the quantum
error correcting codes using the stabilizer formalism.

#### Stabilizer Codes

The stabilizer formalism, eloquently formulated by D. Göttesman [10], has it
roots in group theory. Firstly, analogous to classical linear codes, a quantum
code C has three properties $[n,k,d]$, where $n$ will represent the number of
encoding or noisy qubits and $k$ the number of error corrected logical qubits.
A distance $d$ will be the smallest amount of noisy qubits an error needs to
act on, non-trivially, to change one logical state into another.
Suppose now one wishes to use $n$ noisy qubits to compute $k$ fault-tolerant
qubits. As previously explained, protecting against the basis of the Pauli
$X$, $Y$ and $Z$ matrices and the identity acting on every single qubit, will
be enough to also protect from linear combinations of those. This leads to the
introduction of the set,

$\mathcal{G}=\set{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm iZ},$ (11)

which will form a group, the Pauli group $\mathcal{G}$, under multiplication.
The tensor product of $n$ Pauli groups acting on $n$ qubits is denoted
$\mathcal{G}_{n}$. In this group each element will either commute or anti-
commute with each other. A stabilizer group representing an error correcting
code is defined as a subgroup $S$ of $\mathcal{G}_{n}$ that does not contain
the negative identity $-(I\otimes I...\otimes I)$. The stabilizer group
defines an action on the $2^{n}$ dimensional Hilbert space of noisy qubits
$\mathcal{H}^{\otimes n}_{2}$,

$S\times\mathcal{H}^{\otimes n}_{2}\to\mathcal{H}^{\otimes
n}_{2},(M,\ket{\psi})\to M\cdot\ket{\psi}.$ (12)

The subspace of the Hilbert space $\mathcal{H}^{\otimes n}_{2}$ made up of
states which stabilizes under $S$, defined as,

$P=\set{\ket{\psi}\in\mathcal{H}^{\otimes
n}_{2}}{M\ket{\psi}=\ket{\psi},\forall M\in S},$ (13)

will subsequently span the $2^{k}$ dimensional Hilbert space of fault-tolerant
qubits. These are also called logical qubits [10]. Lastly, it is possible to
write the stabilizer code $S$ more compactly using generators. S will then be
the smallest group that contains those generator elements.

#### The Five-Qubit Code

The five-qubit code is an example of a stabilizer code with properties
[5,1,3], meaning it will encode five noisy qubits into one logical qubit. The
distance three is the minimum amount of noisy qubits for which an error needs
to act on that will transform one logical state into another, which in this
case of one logical qubit is flipping the logical qubit.

$M_{1}$ | $X\otimes Z\otimes Z\otimes X\otimes I$
---|---
$M_{2}$ | $I\otimes X\otimes Z\otimes Z\otimes X$
$M_{3}$ | $X\otimes I\otimes X\otimes Z\otimes Z$
$M_{4}$ | $Z\otimes X\otimes I\otimes X\otimes Z$
Table 1: The generators of the stabilizer group of the five-qubit code [10].

The generators of the stabilizer group representing this code are given in
Table 1. By construction all generators in this table will leave the code
subspace $P$ unchanged. The five-qubit code constructs this code subspace $P$
of the logical qubit as [10],

$\overline{\ket{0}}=\frac{1}{4}(\ket{00000}+\ket{10010}+\ket{01001}+\ket{10100}+\ket{01010}-\ket{11011}-\ket{00110}-\ket{11000}$
$-\ket{11101}-\ket{00011}-\ket{11110}-\ket{01111}-\ket{10001}-\ket{01100}-\ket{10111}+\ket{00101})$,

$\overline{\ket{1}}=\frac{1}{4}(\ket{11111}+\ket{01101}+\ket{10110}+\ket{01011}+\ket{10101}-\ket{00100}-\ket{11001}-\ket{00111}$
$-\ket{00010}-\ket{11100}-\ket{00001}-\ket{10000}-\ket{01110}-\ket{10011}-\ket{01000}+\ket{11010})$,

where $\overline{\ket{0}}$ and $\overline{\ket{1}}$ represent the two states
of the logical qubit. The five-qubit code has an additional property that
makes it interesting. It is the qubit code with the smallest amount of noisy
qubits that is classified as perfect [10]. This means that all possible error
measurements correspond to a correctable error. These error measurements are
operators that either commute or anti-commute with all stabilizer elements.
For the five-qubit code these are all possible single qubit errors $X_{i}$,
$Y_{i}$ and $Z_{i}$. This means that, while the five-qubit code will protect
against all single qubit errors, it will not protect against errors acting on
multiple qubits at a time. As shown in Section 1.3, the holographic state
associated with this code is defined as
$\frac{\ket{0\overline{0}}+\ket{1\overline{1}}}{\sqrt{2}}$.

###  1.3 Tensors

Quantum error correcting codes can be represented as a tensor $T$ mapping a
collection of $k$ logical qubits to a bigger collection of $n$ noise-
susceptible qubits,

$T:\mathcal{H}^{\otimes k}_{2}\to\mathcal{H}^{\otimes
n}_{2},\quad\ket{q_{1},q_{2},....,q_{k}}\to\sum^{1}_{p_{1},p_{2},....,p_{n}=0}T_{p_{1},p_{2},....,p_{n}}^{q_{1},q_{2},....,q_{k}}\ket{p_{1},p_{2},...,p_{n}},$
(14)

where $\ket{q_{1},q_{2},....,q_{k}}$ represents a logical state and
$\ket{p_{1},p_{2},...,p_{n}}$ a state of noisy qubits. Note that this
definition can be generalized to $\nu$ level quantum bits, by changing the
number above the sum to $\nu-1$ [12]. As an example, consider the five-qubit
code, where the number of logical qubits is one and the number of noisy qubits
is five. In this case, Equation (14) simplifies to,

$T:\mathcal{H}_{2}\to\mathcal{H}^{\otimes
5}_{2},\quad\ket{q}\to\sum^{1}_{a,b,c,d,e=0}T_{abcde}^{q}\ket{abcde},$ (15)

where values $T_{abcde}^{q}$ can be derived from the logical qubit states of
the five-qubit code, noted in Section 1.2. This tensor will be the building
block used for the holographic states in this thesis.

#### Isometric Tensors

A linear mapping between any two Hilbert spaces is called an isometry when the
inner product between all vectors is conserved. For a tensor acting on finite
Hilbert spaces, this is equivalent to stating that $T^{{\dagger}}T$ is the
identity on the domain of $T$. Equivalently, it can also be stated as,

$\sum_{a,b,c,d,e}({T^{q^{\prime}}_{abcde}})^{{\dagger}}\cdot
T^{q}_{abcde}=\delta_{qq^{\prime}}.$ (16)

Isometric tensors have many interesting properties that will be exploited in
this thesis. Firstly, in order for a tensor
$T:\mathcal{H}^{A}\to\mathcal{H}^{B}$ to be possibly isometric, their
dimensions must satisfy dim($\mathcal{H}^{A}$) $\leq$ dim($\mathcal{H}^{B}$).
Secondly, isometric tensors allow for easy contractions between itself and its
hermitian adjoined. This is visible in Figure 1, where T and $T^{{\dagger}}$
are executed consecutively. Due to the isometric nature of this tensor, this
will be equivalent to the identity operator acting on $A$.

Figure 1: The Penrose graphical notation of an isometric tensor $T$
contracting the five outgoing legs with its hermitian $T^{{\dagger}}$ to
create the maximally entangled state between $A$ and $A^{\prime}$.

While no actual state has been defined in the Hilbert spaces $A$ and
$A^{\prime}$, the holographic state associated to this tensor can be found by
reshaping the contracted tensors to a density matrix. For isometric tensors
this density matrix, and consequently the subspace A, will always be maximally
entangled. However, when actually assigning a state to $A$ the entanglement
entropy might change. As an example, projecting all three legs of $A$ to be
the ground level $\ket{0}$, will create the state on the Hilbert spaces
$A\otimes A^{\prime}$ as $\ket{000000}$ for which the Von Neumann entropy
$S_{A}$ is zero.

Lastly, isometric tensors have a property that is at the core of this thesis.
It is the ability to push operators through. Consider an operator $O$ acting
on the domain of an isometric tensor $T$. This operator is able to be pushed
through and transformed to an operator $O^{\prime}$ acting on the image of the
tensor by performing $OT=TT^{{\dagger}}OT$ = $TO^{\prime}$, where
$O^{\prime}=T^{{\dagger}}OT$. This is a feature used by quantum error
correcting codes to push logical operators to operators acting on the noisy
qubits. As an example the logical $\overline{X}$ operator of the five-qubit
code can be pushed through the isometric five-qubit tensor and transformed
into $X_{1}X_{2}X_{3}X_{4}X_{5}$ [10].

#### Perfect Tensors

Perfect tensors are a subset of isometric tensors for which an additional
property holds. Any bipartition of the indices of a tensor
$T_{a_{1},a_{2},..a_{n}}$, regardless of them being an upper or lower index,
into $A$ and $A^{c}$ with dim($A$) $\leq$ dim($A^{c}$), will represent a
tensor $T$: $A\to A^{c}$ that is proportional to an isometric tensor. Using
the properties of isometric tensors, this allows to state that when a tensor
is split up into $A$ smaller or equal than its complement $A^{c}$, that
$TT^{{\dagger}}$ will be proportional the identity. For example, any
bipartition of the indices of the perfect five-qubit tensor, regardless of
them representing a logical state or not, into $A$ (incoming legs) and $A^{c}$
(outgoing legs) where dim($A$) is one, two or three, will have
$TT^{{\dagger}}$ be proportional to the identity. Using the fact that
$T^{{\dagger}}$ is just the transpose of $T$, this allows for the contraction
of subsequent five-qubit code tensors, into a tensor proportional to the
identity, as long as there are three or more connected legs. The holographic
states that can be associated with these tensors are absolutely maximally
entangled (AME) states [12]. This means that any bipartition of the state into
$A$ and $A^{c}$ with dim($A$) $\leq$ dim($A^{c}$) will result in a maximally
entangled state $A$. As an example, the addition of any leg of the five-qubit
tensor to the region $A$ with dim(A) $\leq$ dim($A^{c}$) will result in the
rise of the entanglement entropy $S_{A}$ with one. Furthermore, due to the
property $S_{A}$ = $S_{A^{c}}$, this also means that the addition of any leg
to region $A$ with dim(A) $\geq$ dim($A^{c}$) will decrease entanglement
entropy $S_{A}$ by one. Yet again, actually assigning a quantum state to the
Hilbert space $A$ might decrease the entanglement entropy. This will be
further explored in Section 3.

#### Tensor Networks

Using these perfect tensors, it is possible to create networks with certain
desirable mathematical properties. As an example, in Figure 2 a network is
created from five-qubit tensors represented by blue pentagons. In addition,
red dots represent the outgoing logical/bulk qubits and white dots at the
boundary the pure states of the noisy/boundary qubits. This is a so called
HaPPY code, named after the initials of the authors [1].

Figure 2: A two layer holographic pentagon code, where blue pentagons
represent the five-qubit tensor, red dots the outgoing logical qubits and
white dots at the boundary the pure states of the noisy qubits. Figure is
taken from [1].

The tensors, more precisely their underlying tessellations, are arranged such
that a hyperbolic space is created around a central qubit in the middle.
Moreover, the negative curvature of the tessellations of the disk around a
central tile together with its maximal symmetry constitutes it being an Anti-
de Sitter space or AdS [1]. Therefore can these tensor networks be considered
as toy models of AdS. The maximal distance between the central tensor and
boundary legs, defined as the amount of tensors one needs to pass through to
get there, is called the size $n$ of the code. It is also said that the code
has $n$ layers. Here the space is two dimensional. However, this formalism can
be extended to any dimension [1].

What is often studied with these tensor networks is the ability to reconstruct
operators acting on bulk qubits as operators acting on certain subregion $A$
of boundary qubits. This has a lot of similarities with determining the
equivalent noisy qubits operators of logical operators, as done in Section
1.2. For this purpose is the entanglement entropy, as defined in Formula (6)
of the subregion $A$, relevant. For an arbitrary region $A$ a cut $c$ through
the tensor network ending on the boundary $\partial A$ between $A$ and $A^{c}$
may be drawn. The length of this cut is defined as the amount of tensor legs
it passes through. It has been found that the entanglement entropy of a
boundary region $A$ will satisfy the bound [12],

$S_{A}\leq|\gamma_{A}|\log(\nu),$ (17)

where $|\gamma_{A}|$ is the length of the smallest cut $\gamma_{A}$ on the
boundary of $A$ and $\log(\nu)$ the natural log of the dimension of the
boundary states, which for qubits is $\nu=2$.

###  1.4 AdS/CFT Correspondence

An attentive reader might wonder why this whole set-up has any relevancy to
modern physics. For this the most curious objects in the universe, black
holes, are to be examined. In 1974 Stephen Hawking put forth the
Bekenstein–Hawking formula for the entropy of a black hole [13],

$S_{BH}=\frac{c^{3}A}{4G\hbar},$ (18)

where $A$ is the surface area of the black hole and $c$, $G$ and $\hbar$ are
the speed of light, gravitational constant and Planck’s constant divided by
$2\pi$, respectively. This essentially states that the entropy of a black hole
does not increase with its volume but with its boundary size, the surface area
$A$. The similarity between this observation and the bulk/boundary qubits of
last chapter, seems to suggest that the quantum states of the black hole are
encoded in its boundary states. It was indeed conjectured by Leonard Susskind
and Gerardus ’t Hooft that a theory of quantum gravity, not only black holes,
would have to be able to reduce a 3+1-dimensional space-time to a
2+1-dimensional description on its boundary [14, 15]. This is called the
holographic principle.

The first concrete realization of the principle was proposed by J. Maldacena
in 1997. He conjectured the Anti-de Sitter/Conformal field theory
correspondence or AdS/CFT in short, which relates a $d+1$-dimensional gravity
theory to a $d$-dimensional quantum field theory on its boundary [16]. This
theory has been widely discussed, reaching 23.000 citations in 2023. Toy
models of this correspondence include the two layer pentagon code, shown in
Figure 2, and other HaPPY codes, such as the single qubit code and the
pentagon/hexagon code shown in Figure 3. In these models, the bulk/boundary
degrees of freedom will correspond to physical/logical degrees of freedom [1].
Within this correspondence a different entropy, the Ryu-Takayanagi or RT
entropy, determines the entanglement entropy of a connected boundary subsystem
$A$ as [17],

$S_{A}=\frac{|\gamma_{A}|}{4G},$ (19)

where $|\gamma_{A}|$ is the minimal area in Planck units ($l_{p}$) of a
$d-1$-dimensional surface $\gamma_{A}$ with its boundary on $A$ and $G$ the
gravitational constant. Similarly, will the entropy of subregions $A$ of
tensor networks in this AdS space be calculated analogously to this RT
formula, where $|\gamma_{A}|$ becomes the minimal cut through this tensor
network ending on $A$. The RT Formula is not only a generalization of the
black hole entropy, Formula (18), but it is also in agreement with this bound
in entropy of Formula (17). The AdS/CFT correspondence has already experienced
success in modeling complex gravitation systems such as black holes and
wormholes [18, 19].

Figure 3: Pentagon/hexagon code (left) and single qubit code (right). Figure
from [1].

##  2 Wedges and Operator Reconstructability

The purpose for which the introduced tools are utilized is operator
reconstructability in AdS/CFT. In this theory a $d+1$-dimensional gravity bulk
is modeled with $d$-dimensional quantum field theory on its boundary.
Consequently, operations in this gravitational bulk correspond to operators on
its boundary. This is usually done by pushing the operators outwards to the
boundary, analogously to the method introduced in Section 1.3. However, when
working with a fixed subregion $A$ of the boundary, bulk operations might not
allow to be modeled with only $A$.

For this purpose, a number of wedges will be introduced which correspond to
cuts within the bulk of tensor networks that encapsulates bulk qubits. These
include the causal wedge, the greedy entanglement wedge, the minimal
entanglement wedge and the geometric entanglement wedge. In the next section
the differences between these wedges will be examined. Additionally, it is
investigated how changes in the tensor network such as different types of
networks, non uniform boundaries and stacking of the networks, impact the
shape of the wedges.

The framework to study these changes will be Monte-Carlo simulations of the
boundary erasure and central qubit inclusion. Specifically, for every
iteration of the simulation a new erasure probability $p$ will be uniformly
chosen between zero and one. With this probability a boundary subregion $A$
will be established by performing a Bernoulli trial for every boundary leg
with the probability $p$ where success corresponds to the erasure of the
boundary qubit and failure to the inclusion of the boundary qubit into $A$.
Thereafter, the respective wedge will be constructed on the boundary of $A$.
If the wedge contains the central bulk qubit, defined as the bulk qubit of the
central tensor in the tensor network, the simulation will be deemed a success.
Finally, the success ratio of the simulation will be averaged over 30 even
intervals of $p$, corresponding to bins, and plotted in function of the middle
of those bins. This method was first introduced by [1]. The method allows for
the detection of threshold probabilities $p_{c}$ defined as the maximum
erasure probability where the chance of the central qubit not included in a
defined wedge, becomes exponentially small with increasing $n$ for all
$p<p_{c}$. The threshold probability will be the erasure probability for which
the plot will converge to a step function for $n$ going to infinity.

Figure 4: Example of the causal wedge (left) and the entanglement wedge
(right) indicating the operator reconstructability in the holographic pentagon
code of a bulk qubit, labeled input, and the boundary of respectively $A$ and
$A\cup C$. Figure is taken from [21].

###  2.1 Causal Wedge

The causal wedge received its name from its bulk gravitational interpretation.
It is defined in [22] for a boundary subregion $A$ as the intersection of the
bulk past and future of the domain of dependence of $A$. This domain of
dependence is the region of bulk points where their physics is fully
determined by the initial conditions of $A$. However, what is often
investigated in literature, is the reduced causal wedge where the wedge is
projected to the same time slice as $A$. Hence, the initial wedge shape in the
space-time diagram is reduced to a space slice visible in Figure 4 (left).
Additionally, in the setting of tensor networks, the construction of this
causal wedge will not rely on causal connections, but will rather rely on
extremizing a $d-1$-dimensional surface $\gamma_{A}$ in accordance with the RT
entropy depicted in Formula (19). This construction was described by Pastawski
et al. [1] and relies on performing a so called greedy algorithm on the
boundary.

This algorithm consists of first performing a trivial cut $c$ along the
boundary $A$ and then extending that cut each iteration. In an iteration all
tensors outside the cut that have at least half of their legs connected to
tensors within the cut will force the cut to move and include the tensor to
the region. The algorithm will end in the resulting greedy wedge
$\gamma^{*}_{A}$ when no new tensors are added in one iteration. Because the
tensors added to cut $c$ had at least half their legs intersecting with $c$,
the new cut will always have a smaller or equal size. Consequently,
$|\gamma^{*}_{A}|$ will be a local minimum of the cut sizes. As stated in [1],
that local minimum will be an absolute minimum, resulting in
$\gamma^{*}_{A}=\gamma_{A}$, when $A$ is a connected region on the boundary.
However, when the region is disconnected, the causal greedy algorithm may not
succeed in finding the true minimal geodesic or geometric wedge $\gamma_{A}$.
This is visible in Figure 4 (right), where the greedy algorithm applied to the
disconnected region $A$ and $C$ individually, highlighted by the black dotted
line, does not find the geometric wedge $\gamma_{A}$ which encapsulates the
blue region. The Monte-Carlo simulation for the holographic pentagon code of
the central qubit inclusion when performing the greedy algorithm for each
disconnected region resulting from the simulated erasure is visible in Figure
5.

Figure 5: A Monte-Carlo simulation of the central tensor being incorporated by
the greedy causal algorithm applied to the holographic pentagon code with $n$
layers. The size of the Monte-Carlo simulation is $10^{5}$ for the one and
three layer code and $250$ for the five and seven layer code.

In Figure 5 it is visible that there is no convergence of the inclusion curve
to a step function, but rather it converges to the zero function corresponding
to an erasure probability $p_{c}$ of zero. In other words, the erasure of an
infinitesimal small percentage of boundary qubits will, in the large $n$
limit, almost always result in the omission of the central qubit in the causal
wedge. It has been found in [1] that for the pentagon code the amount of
boundary qubits rises as
$\mathcal{O}\left(\left(\frac{3+\sqrt{5}}{2}\right)^{n}\right)$ with the
amount of layers $n$. Consequently, Figure 5 indicates that the number of
boundary qubits that are allowed to be erased while still maintaining central
qubit inclusion rises significantly slower than that.

This is in agreement with a clever example in [1], where it is stated that the
erasure of just four boundary qubits at certain places in a large layer
pentagon code will result in the omission of the central qubit. The behaviour
of the inclusion graph is a result of the disconnected regions, created due to
the erasure of just a couple of qubits, not being able to individually attain
the true minimal bulk geodesic $\gamma_{A}$. Moreover, simulations with the
single qubit code and the pentagon/hexagon code showed the same convergence
behaviour.

###  2.2 Entanglement Wedge

To be compatible with disconnected regions of the boundary, the greedy
entanglement wedge was introduced by Pastawski et al. [1]. Similar to the
causal wedge, is the greedy entanglement wedge constructed by applying the
greedy algorithm to the boundary. However, rather than applying the algorithm
to each disconnected boundary region individually, the algorithm will now be
applied to all regions simultaneously. As a result, disconnected regions are
now able to contribute together to advancing the cut. The greedy entanglement
wedge is visible in Figure 4 (right). Here, by applying the greedy
entanglement algorithm to boundary regions $A$ and $C$, the greedy
entanglement wedge $\gamma_{A}^{*}$ that encapsulates the blue region is
created. For this example this is also the geometric wedge $\gamma_{A}$. On
this figure, the distinction between the causal- and entanglement wedge is
clearly visible. The entanglement wedge includes the central tensors and the
individual causal wedges, noted by the black dotted line, fail to include it.

This difference is also visible in the Monte-Carlo simulation for the boundary
erasure/central qubit inclusion in the holographic pentagon network visible in
Figure 6. In this figure, it is visible that the graphs are descending
significantly later than the causal wedge. However, by looking at
probabilities where the graphs start to drop, the inclusion graph seems to
still be converging to the zero function. Therefore, the erasure of any
fraction of the boundary qubits will, in the big $n$ limit, almost always
result in the omission of the central qubit. In other words no threshold
probability is detected. This conclusion and simulation is in agreement with
the same Monte-Carlo simulation performed by Pastawski et al. [1].

The question may arise if the individual positions of the erased boundary
qubits have any significance in the inclusion of the central qubit. Figure 6
indicates this as well. For small layer networks the function gradually drops
to zero, indicating that for the same erasure probability there are some
configurations of $A$ which include the central qubit and some which do not.
For the pentagon code this can be illustrated by the example in Figure 7. For
this one layer pentagon network, a different boundary $A$ of the same size has
been created. On the left was for three tensors only one qubit erased. Due to
three qubits still remaining in the boundary $A$, the greedy entanglement
wedge will still advance past all these tensors, eventually including the
central qubit. This is due to the fact that $A$ has a very top heavy shape.
However, on the right an identical network with the same boundary size was
created where the central qubit is not included in the entanglement wedge.
Although these very asymmetric configurations of $A$ are still possible in
networks with more layers, they become exponentially unlikely as $n$
increases. Therefore, in Figure 6 the inclusion graphs drop more rapidly for
higher layer networks.

Figure 6: A Monte-Carlo simulation of the central tensor being incorporated by
the greedy entanglement algorithm applied to the holographic pentagon code
with $n$ layers. The size of the Monte-Carlo simulation is $10^{5}$ for the
one and three layer code and $250$ for the five and seven layer code.

Figure 7: An example of the shape of the boundary region $A$ having an impact
on the inclusion of the central qubit in the greedy entanglement wedge. On the
left the greedy entanglement wedge will advance to include the central qubit,
where on the right it will not.

#### Type of tensor networks

It can be investigated how changes in the type of tensor network impact a
possible threshold erasure probability $p_{c}$. For this purpose, the tensor
networks visible in Figure 3 were used. While these tensor networks are still
HaPPY codes, satisfying the requirements for the AdS/CFT correspondence, they
differ from the holographic pentagon by the use of hexagons. These hexagons
still are five-qubit tensors. However, the logical qubit leg, unlike
pentagons, do not represent bulk outgoing qubit but rather legs that remain in
the AdS-surface. As a result, the underlying structure of the tensor network
changes. These changes can be represented by the fraction
$f=\frac{N_{bulk}}{N_{boundary}}$, where $N_{bulk}$ and $N_{boundary}$ are the
total amount of bulk and boundary qubits respectively. The value of this
parameter was found by [1] for the different tensor networks. While the
fraction may differ based on which layer the boundary was cut off, in general
it will hold $f_{pent}<f_{pent/hex}<f_{singlequbit}$. In Figure 8, it can be
found how this fraction impacts the Monte-Carlo simulations of central qubit
inclusion and boundary erasure.

Figure 8: A Monte-Carlo simulation of the central tensor being incorporated by
the greedy entanglement algorithm applied to the pentagon/hexagon code (left)
and the single qubit code (right) with $n$ layers. The size of the Monte-Carlo
simulation is $10^{5}$ for the one and three layer code and $250$ for the five
and seven layer code.

The Monte-Carlo simulations indicate that, in contrast to the causal wedge,
the inclusion graphs do change with type of tensor network. Recall that for
the holographic pentagon code using the greedy entanglement there was no
threshold probability detected by the Monte-Carlo simulation in Figure 6. In
Figure 8, it is visible that the single qubit network does have an erasure
thresholds around the value $p_{c}\approx 0.50$. This means that about half of
boundary qubits may be erased before the central qubit will be excluded from
the greedy entanglement wedge. While for the pentagon/hexagon network it is
unsure if a threshold probability exists, it still shows better inclusion than
the pentagon network. The results indicate that decreasing the fraction
$\frac{N_{bulk}}{N_{boundary}}$ by changing tensor networks will improve the
threshold probability $p_{c}$ until a maximum value of 0.5.

One hypothesis to possibly justify this behaviour is that a lower fraction
allows the mutual information between the central qubit and boundary region to
be more spread out across the boundary. Therefore, some minimal value of
mutual information could be reached more quickly for a lower fraction.
Consequently, the operators acting on the central qubit could allow for a
faster reconstructability. This idea will be further investigated in Section
3.

#### Shape of the boundary

As suggested by [1], another way to decrease the fraction
$\frac{N_{bulk}}{N_{boundary}}$ is by performing a non-uniform cut off on the
boundary. This is executed by allowing boundary qubits to have different
distances to the central tensor. In Figure 3, such a non-uniform cut off is
visible of the pentagon/hexagon- and single qubit code. For these networks,
the cut off is executed by removing the tensors which are not connected to two
or more tensors of the previous layer. The Monte-Carlo simulations for these
networks are visible in Figure 9. These results are consistent with the
simulations performed by [1]. By comparing these inclusion graphs with the
equivalent uniform boundary graphs depicted in Figure 8, it is visible that a
change in boundary shape has a small effect on the inclusion graph. A
significant change in shape occurs for the three layer tensor networks, where
the graph seems to be more spread out with a non uniform boundary. Because the
cut off for the one layer networks remains the same, the three layer networks
are the smallest simulated tensor networks that are effected by the change in
boundary. A three layer network with a non-uniform cut off will have less
boundary legs and consequently will be more susceptible to asymmetric boundary
division, resulting in the stretching of the inclusion graph.

Figure 9: A Monte-Carlo simulation of the central tensor being incorporated by
the greedy entanglement algorithm applied to the pentagon/hexagon code (left)
and the single qubit code (right) with n layers with a non-uniform boundary.
The size of the Monte-Carlo simulation is $10^{5}$ for the one and three layer
code and $250$ for the five and seven layer code.

The inclusion graph of a single qubit network has the same threshold
probability, graphically determined as $p_{c}\approx 0.50$. For the
pentagon/hexagon code the plausibility of a threshold probability seems to
have increased with respect to the uniform boundary inclusion graph. This is
in agreement with [1], where it was proposed that through a bounding procedure
it was possible to find a lower bound on the non zero threshold probability of
the non-uniform pentagon/hexagon code. For the non-uniform single qubit
network, such a lower bound was indeed found by [1]. They have proven
analytically that for any erasure probability $p$ with $p<\frac{1}{12}$, the
probability to have guaranteed operator reconstructability is higher than
$1-\frac{1}{12}\left(12p\right)^{\lambda^{n}}$, with
$\lambda=\frac{1+\sqrt{5}}{2}$. This means that the chance of exclusion of the
central tensor becomes exponentially small for increasing layer depth $n$ for
any $p<\frac{1}{12}$. For this estimation a hierarchical recovery model was
used based on the propagation models of erasures on the boundary. This model
is weaker than the greedy entanglement wedge since it does not allow for lower
layer tensors to contribute to higher layer tensor inclusions. This means that
the threshold probability $p_{c}$ for inclusion in the greedy entanglement
wedge is at least larger or equal than $\frac{1}{12}$. However, this model
fails with the pentagon code since the recursion relationship between errors
of different layers does not allow for a scaling dimension $\lambda$ different
than one. This basically means that the error rate does not decrease or
increase towards the central qubit for this hierarchical model [1]. Therefore,
no lower bound on the threshold probability was able to be established. An
attempt was made to construct this same hierarchical model for the
pentagon/hexagon code, but no threshold probability was found.

#### Stacking of tensor networks

One last possible method to possibly increase the fraction
$\frac{N_{bulk}}{N_{boundary}}$ is to stack multiple tensor networks. This
method, first introduced by [20], consists of stacking multiple tensor
networks on top of each other and routing the corresponding bulk legs to a
single bulk leg using random all-to-all (AA) scrambling circuits, illustrated
in Figure 10. These circuits will be simulated by constructing tensors where
all the coefficients are randomly chosen from a standard normal distribution.
Through this stacking method, the fraction can be made arbitrarily low by
increasing stack size. There are two possible ways of defining a boundary
region for stacked tensor networks. The first method is to create a boundary
region $A$ for a single stack and copying that boundary region to all stacks.
This is the method introduced by [20] and visible in Figure 10. However, one
can also define new regions $A$ with the same erasure probability $p$ on each
stack of the network. For the first method the greedy entanglement wedge will
be identical for all stacks and the resulting inclusion graph will remain the
same, whereas with the second method the entanglement wedge might increase to
include some of the random all-to-all tensor resulting in a net positive in
central qubit inclusions. This is visible in the Monte-Carlo simulations of
Figure 11.

Figure 10: Stacking the pentagon code. On the left a single holographic
pentagon code and on the right $M$ stacked pentagon codes connected to random
all-to-all scrambling circuits. Bulk legs are highlighted in red, and the
boundary is noted with $A$. Figure is taken from [20]. Figure 11: A Monte-
Carlo simulation of the central tensor being incorporated by the greedy
entanglement algorithm applied to $M$ stacked holographic pentagon codes with
n layers, where the boundary subregion is redefined in each stack. The size of
the Monte-Carlo simulation is $10^{5}$ for the one and three layer code and
$250$ for the five layer code.

In Figure 11 Monte-Carlo simulations performed for boundary erasure/central
qubit inclusion with the greedy entanglement wedge of six stacked holographic
pentagon codes are visible as the light colors. The inclusion graph has moved
from its one stack equivalent to the left by about 0.3 erasure probability. As
mentioned this happens due to boundary region $A$ being different for all
stacks. This results in some of the random all-to-all (AA) tensors being
included in the entanglement wedge. This in turn can allow to entanglement
wedge to be pushed further on other stacks. Note as well that only half the
legs of the central AA tensor have to be included for the central qubit to be
included and the simulation to be deemed successful. However, still no
threshold probability is detected for this pentagon code. Lastly, the
distribution of the AA tensors has no impact on the success rate. As an
example, identical simulations with the coefficients uniformly drawn between
zero and one, yielded the same inclusion graph.

While through this method of stacking, the fraction
$\frac{N_{bulk}}{N_{boundary}}$ can be made arbitrarily low, it is visible in
the shape of the inclusion graph that there still does not seem to be
convergence to a threshold probability. This shows that central qubit recovery
is not only affected by $\frac{N_{bulk}}{N_{boundary}}$ but also by the
inherent structure of the tensor network. Changing the fraction may move the
inclusion graphs of the same networks, but the inherent geometry of the tensor
network still determines if there is convergence or not. This is also visible
when attempting to do the same lower bound estimation for this stacked
pentagon code. By stacking the tensor networks, one is essentially just adding
a new ground layer to the tensor network. However, the scaling dimension of
the propagation of errors of an hierarchical model of this stacked pentagon
tensor network will still remain one, regardless of the fact that at the end a
new layer has been added.

###  2.3 Minimum Entanglement Wedge

The final wedge that is investigated in this thesis is the minimum
entanglement wedge. This newly introduced notion of entanglement wedge loses
its ensured operator reconstructability but will be a better approximate for
the entanglement entropy. Since the greedy algorithm applied to $A$ finds a
local minimum in wedge size $|\gamma^{*}_{A}|$, the greedy algorithm performed
on the compliment $A^{c}$ will do the same for $\gamma^{*}_{A^{c}}$. However,
this geodesic $\gamma^{*}_{A^{c}}$ will also have its boundary on A.
Consequently, the geodesic $\gamma^{*}_{A^{c}}$ is an additional candidate for
the true geometric entanglement wedge $\gamma_{A}$. From this idea the minimum
entanglement wedge can be created as the minimum in geodesic size,

$\gamma_{A}^{min}=\min\\{{\gamma_{A}^{*},\gamma_{A^{c}}^{*}}\\}$ (20)

where $\gamma_{A}^{*}$ and $\gamma_{A^{c}}^{*}$ are the greedy algorithm
performed on $A$ and $A^{c}$ respectively.
For any boundary subregion $A$, the length of the cut of this minimum
entanglement wedge $\gamma_{A}^{min}$ will be smaller or equal to the cut of
the greedy entanglement wedge, and thus be a better candidate to the minimum
cut of the true geometric wedge. This allows for a better approximation of the
RT entropy in Formula (19). However, as further explained in Section 2.4, the
wedge does not have the guaranteed reconstructability on $A$ for operators
acting on bulk qubits included in the wedge.

#### Classical RT Entropy

Before performing this algorithm, a clear idea of wedge size has to be
defined. In this thesis the wedge size will be defined as the number of tensor
legs the geodesic representing that wedge crosses. This wedge size will be
used to calculate the entropy of subregions analogously to the classic RT
formula depicted in Formula (19). However, for error correcting tensor
networks the holographic Von Neumann entropy for a boundary $A$ can be
approximated with [23, 24],

$S(\rho_{A})=\operatorname{Tr}(\rho\mathcal{L}_{A})+S_{bulk}(\rho_{\epsilon_{A}}),$
(21)

where $\mathcal{L}_{A}$ is $\frac{Area(\gamma_{A})}{4G}$ with the Newtonian
constant $G$ and $\rho_{\epsilon_{A}}$ is the density matrix of all bulk
qubits included in the entanglement wedge.

As is visible, Formula (19) and Formula (21) will no longer agree.
Consequently their tensor network equivalent will not either. However,
numerically it has been found that, similar to the example of Section 1.3,
projecting the bulk qubit of the five qubit tensor to $\ket{0}$ or $\ket{1}$
changes the entropy. More precisely, when tracing out any three qubits of the
tensor, the entanglement entropy will not increase to three anymore, but
rather remain at two. Because all other interactions remain the same, this can
be translated to removing all contributions of bulk qubits to the Von Neumann
entropy. In other words the second term of Formula (21) vanishes and all that
remains is the wedge size, which corresponds to using the classic RT Formula
(19). The entropy of these systems will be further investigated in Section 3.
Consequently, for the algorithm of the minimum wedge, the size of the wedge
will be considered as the true size, the amount of tensor legs the geodesic
crosses. This means that a geodesic which has its boundary on $A$ will have
the same size when looking from the compliment boundary $A^{c}$.

Figure 12: A Monte-Carlo simulation of the central tensor being incorporated
by the minimum entanglement wedge with classical size applied to the
holographic pentagon code with $n$ layers. The size of the Monte-Carlo
simulation is $10^{5}$ for the one and three layer code and $250$ for the five
and seven layer code.

The Monte-Carlo simulation of central qubit inclusion/boundary erasure using
the minimum entanglement wedge without bulk leg contributions is visible in
Figure 12 for the holographic pentagon code. Here, the inclusion graph
approaches a true step function around the value $p=0.50$. Consequently, the
threshold probability $p_{c}$ is 0.5. This was defined as the maximum erasure
probability where the chance of the central qubit not included in a defined
wedge becomes exponentially small with increasing $n$ for all $p<p_{c}$. This
is a property of not only the minimum entanglement wedge acting on the
pentagon code, but for the pentagon/hexagon- and single qubit code as well,
visible in Figure 13. In an effort to explain this behaviour Theorem (1) was
introduced.

######  Theorem 1.

The threshold probability $p_{c}$ for any wedge $\gamma_{A}$ is 0.50
regardless of the structure of a tensor network as long as the probability of
$\gamma_{A}\neq\gamma_{A^{c}}$ becomes exponentially small for an increasing
amount of layers.

###### Proof.

To proof by contradiction, assume $p_{c}>0.5$. Consequently, there exists an
erasure probability $p_{c}>p>0.5$, where the probability of exclusion of the
central qubit included in the wedge, becomes exponentially small with
increasing number of layers $n$. For all regions $A$ originating from this
probability $p$ will, due to the assumption, $\gamma_{A}$ almost always be
equal to $\gamma_{A^{c}}$. As a result a bulk region enclosed by $\gamma_{A}$
and $A^{c}$, originating from an erasure probability $1-p$ has been found for
which the probability of exclusion of the central qubit between $A^{c}$ and
$\gamma_{A^{c}}$ not become exponentially small with $n$. Consequently,
because $A$ was chosen arbitrarily, there exists an erasure probability
$1-p<0.5<p_{c}$ for which the definition of $p_{c}$ does not hold.

Conversely, assume that $p_{c}<0.5$. For almost every region $A$ created by a
erasure probability $p_{c}<p<0.5$ that includes the central tensor, there will
exist a region enclosed by $\gamma_{A}$ and $A^{c}$ which will not include the
central qubit. Consequently, an erasure probability $1-p<0.5<p_{c}$ was found
where the chance of exclusion of the central qubit does not become
exponentially small with increasing $n$, in contradiction with the definition
of the threshold probability. ∎

Figure 13: A Monte-Carlo simulation of the central tensor being incorporated
by the minimum entanglement wedge with classical size of the pentagon/hexagon
code (left) and the single qubit code (right) with n layers. The size of the
Monte-Carlo simulation is $10^{5}$ for the one and three layer code and $250$
for the five layer code.

This theorem applied to the minimum entanglement wedge states that if the
probability $\gamma_{A}^{min}\neq\gamma_{A^{c}}^{min}$ becomes exponentially
small for an increasing amount of layers, the threshold probability $p_{c}$
will be 0.50. The minimum entanglement wedge is not the same geodesic for a
region and its compliment when the sizes of the greedy entanglement wedges are
equal $|\gamma_{A}^{*}|=|\gamma_{A^{c}}^{*}|$, but the wedges itself are not
$\gamma_{A}^{*}\neq\gamma_{A^{c}}^{*}$. In this case, in accordance to the
geometric wedge, the minimum entanglement wedge of $A$ is $\gamma_{A^{c}}^{*}$
and of $A^{c}$ is $\gamma_{A}^{*}$ . While a definite proof of
$\gamma_{A}^{min}\neq\gamma_{A^{c}}^{min}$ becoming exponentially small for
increasing $n$ has not been found, an experimental case study, visible in
Figure 20, suggests that it might be true. In this study, similar to the
Monte-Carlo simulations, boundary regions $A$ were created corresponding to
different erasure probabilities. For these regions the greedy entanglement
algorithm was applied to both $A$ and $A^{c}$. Subsequently different greedy
entanglement wedges with the same size, corresponding to different minimal
entanglement wedges, were detected. In Figure 20, this estimated case
probability was plotted in 30 bins of erasure probabilities. It is visible
that for an arbitrary erasure probability $p$, the probability that
$\gamma_{A}^{min}$ is not equal to $\gamma_{A^{c}}^{min}$ becomes really small
for increasing layer depth $n$.

Figure 14: The estimated probabilities of the minimum entanglement wedge
applied to the holographic pentagon code being different for a Monte-Carlo
simulated boundary subregion $A$ and its complement $A^{c}$. The size of the
Monte-Carlo simulation is $10^{5}$ for the one and three layer code and $250$
for the five and seven layer code.

#### Error Corrected Entropy

It is also possible to investigate the minimum entanglement wedge for the
holographic state. By definition and numerical study of the five-qubit tensor,
this is equivalent to utilizing the holographic state
$\frac{\ket{0\overline{0}}+\ket{1\overline{1}}}{\sqrt{2}}$ in all tensors.
With these states the entanglement entropy will rise with one whenever a bulk
qubit is included in the wedge. Consequently, the entropy of the boundary
subregion will also rise according to Formula (21). For the algorithm acting
on the subregion $A$ it means that for $A$, one will be added to the size of
entanglement wedge $|\gamma^{*}_{A}|$ whenever a bulk qubit is included and
for $A^{c}$ one will be added to the size of $|\gamma^{*}_{A^{c}}|$ for all
bulk qubits encapsulated between $A$ and $\gamma^{*}_{A^{c}}$. As a result,
$|\gamma^{min}_{A}|$ and $|\gamma^{min}_{A^{c}}|$ is no longer guaranteed to
be equal.

Figure 15: A Monte-Carlo simulation of the central tensor being incorporated
by the minimum entanglement wedge with corrected size applied to the
holographic pentagon code with $n$ layers. The size of the Monte-Carlo
simulation is $10^{5}$ for the one and three layer code and $250$ for the five
and seven layer code.

In Figure 15 the central qubit inclusion for simulated boundary erasure with
different erasure probabilities is visible for the minimum entanglement wedge
that adds one to the wedge size for every bulk qubit included. Comparing this
graph with the Monte-Carlo simulation of the entanglement wedge, visible in
Figure 6, shows that the inclusion graph in the minimum entanglement wedge has
shifted a bit to the left of that of the greedy entanglement wedge. This was
expected since the minimum entanglement wedge will either encapsulate more or
an equal amount of tensors than the greedy entanglement wedge. However, this
shift in probability is only about 0.05. This means that the minimum
entanglement wedge will include the central bulk qubit for a fraction of the
boundary states that is only slightly higher than that the greedy entanglement
wedge. The inclusion graph still does not seem to converge to a step function.
In Figure 16, the Monte-Carlo simulations for the single qubit- and
pentagon/hexagon code are also visible. In this figure it is still visible
that the pentagon/hexagon network does seems to converge to a step function
with an erasure probability of about $p_{c}\approx 0.35$. The threshold
probability of the single qubit code seems to advance from its entanglement
wedge equivalent $p_{c}\approx 0.50$ to about $p_{c}\approx 0.60$. This
indicates that the minimum entanglement wedge is moving past the greedy
entanglement wedge. Lastly, there seem to be significant differences when
comparing the inclusion graphs of the classical RT formula and those of the
error corrected entropy. This happens since a geodesic which has its boundary
on $A$ will no longer have the same size when looking from the compliment
boundary $A^{c}$. Consequently, the proof of Theorem (1) is no longer
applicable.

Figure 16: A Monte-Carlo simulation of the central tensor being incorporated
by the minimum entanglement algorithm with corrected size applied to the
pentagon/hexagon code (left) and the single qubit code (right) with n layers.
The size of the Monte-Carlo simulation is $10^{5}$ for the one and three layer
code and $250$ for the five and seven layer code.

#### Geometric wedge

An example of the minimal entanglement wedge reaching the true geometric wedge
is visible in Figure 17. For this single qubit network with a non-uniform
boundary the minimum entanglement wedge $\gamma_{A}^{min}$ reaches the true
geometric entanglement wedge $\gamma_{A}$ as the blue geodesic. This blue
geodesic is the greedy algorithm applied to the complement boundary $B$ and
has the size $|\gamma_{B}^{*}|=9$. The greedy entanglement wedge of $A$
highlighted in red has a size $|\gamma_{A}^{*}|=12$. Consequently, the greedy
entanglement algorithm falls short in finding the true geometric wedge. These
wedge sizes have been calculated according to the classic RT formula, as the
amount of legs the geodesic crosses. However, the example still holds for any
state of the bulk qubit. For this example, an operator is reconstructed beyond
the greedy entanglement algorithm in Section 2.4.

Figure 17: An example of a non-uniform boundary two layer single qubit network
where the minimum entanglement wedge $\gamma_{A}^{min}$ reaches the true
geometric entanglement wedge $\gamma_{A}$ as the blue geodesic and the greedy
entanglement wedge of $A$ highlighted in red falls short.

The minimum entanglement wedge reaches the true geometric wedge for the one
layer pentagon-, single qubit- and pentagon/hexagon code for all boundaries.
However, in general, may the true geometric geodesic exist between the tensors
enclosed by $\gamma_{A}^{*}$ and those enclosed by $\gamma_{A^{c}}^{*}$. In
that case $\gamma_{A}^{min}$ will not find the true geodesic entanglement
wedge.

###  2.4 Operator Reconstructability

In this section, the explicit reconstructability of operators acting on bulk
qubits is investigated. Firstly, recall that the construction of the causal-
and entanglement wedge was to ensure reconstructability of all operators.
Since the tensor networks are made up of perfect tensors and the greedy
algorithm will only advance past a perfect tensor when half or more tensor
legs are included in the wedge, any operator may be reconstructed using the
framework of perfect tensors introduced in Section 1.3. As an example, in
Figure 18 an arbitrary operator $O$ acting on the central bulk qubit of the
one layer pentagon code with the boundary region visible in Figure 7 (left),
is pushed through to the boundary subregion $A$. For this, all other boundary
qubits and bulk qubits are considered as additional input legs to the five-
qubit perfect tensor. Nonetheless, the entanglement algorithm ensures that at
least half of the tensor legs are output. This ensures that the tensor will
remain proportional to an isometric tensor. Consequently, the operator
$\mathcal{O}$ = $O\otimes I\otimes I$ can be pushed through the tensor as
$\mathcal{O}T=TT^{{\dagger}}\mathcal{O}T$ = $TO^{\prime}$, where
$O^{\prime}=T^{{\dagger}}\mathcal{O}T$. For the subsequent layer, the same
framework can be applied to $O^{\prime}$. Yet again identity operators are
appended $O^{\prime}$ until it is an operator working on all non-region legs.
Only now the operator must be pushed through $T\otimes T\otimes T$. This poses
no problem since the greedy algorithm ensured that for each of the individual
tensors T at least half of the legs are output. Therefore, T will still be
proportional to an isometric tensor and the operator can be pushed through
using the fact that ($T\otimes T\otimes T$)($T^{{\dagger}}\otimes
T^{{\dagger}}\otimes T^{{\dagger}}$) is proportional to the identity. In the
end this procedure will result in an operator acting on solely the boundary
subregion. In this example the resulting operator will be $O^{\prime\prime}$ =
($T^{{\dagger}}\otimes T^{{\dagger}}\otimes
T^{{\dagger}}$)(($T^{{\dagger}}(O\otimes I\otimes I)T)\otimes I_{3}\otimes
I_{3}$) ($T\otimes T\otimes T$).
This procedure is not limited to the central bulk qubit and will work on all
bulk qubits included in the greedy entanglement wedge at the same time. Since
$I$, $X$, $Z$ and $Y$ from a basis for operators, these individual operators
are able to be written as the tensor product of individual operators acting on
all bulk qubits. As a result can these operators be pushed through by simply
applying the operators on the bulk qubits and pushing them through at the
correct moment. For example, in Figure 18, operators acting on bulk qubits
that are not the central qubit may be pushed through together with
$O^{\prime}$.

Figure 18: An example of an arbitrary operator $O$ acting on the central bulk
qubit of the one layer pentagon code with the boundary visible in Figure 7
(left), being pushed through as the operator $O^{\prime\prime}$ acting on that
boundary.

#### Minimum entanglement wedge

As previously mentioned, the minimum entanglement wedge loses this assured
operator reconstructability. If the minimum entanglement wedge of $A$ is equal
the greedy entanglement wedge of $A$, there is no problem and all operators of
included bulk qubits are reconstructable on $A$. The troubles comes when the
minimum entanglement wedge is equal to $\gamma^{*}_{A^{c}}$. Because this
geodesic generally does not come from $\gamma^{*}_{A}$, it is not assured that
the formalism similar to Figure 18 is able to be applied. As an example, one
is unable to use this same greedy formalism to send operators acting on the
central qubit of Figure 17 to the boundary $A$. More precisely, this algorithm
will fail by the third time an operator is send through as the amount of input
legs becomes bigger then the output legs.

#### Geometric wedge

While bulk qubits that are outside the greedy entanglement wedge are not
guaranteed reconstructability on the boundary subregion, by exploiting
symmetrical elements of the tensor network, some operators acting on bulk
qubits beyond the greedy entanglement wedge can be reconstructed. While this
has also been shown by [1], a more explicit example can be found by examining
the network in Figure 17. In this two layer single qubit code with a non-
uniform boundary the greedy algorithm does not advance past the central qubit,
but the area enclosed by true geometric geodesic, found by applying the
minimum entanglement wedge, and the boundary subregion $A$ does enclose this
central qubit. Conveniently, operators acting on this central qubit exists
that are able to be reconstructed on $A$. To show this claim, inherent
properties of the five-qubit tensor as a quantum error correcting code are
exploited. Recall that this five-qubit code is a stabilizer code with its
generators given in Table 1. These generators bring forth the stabilizer group
$S$ of the five-qubit code under which the code subspace $P$ of logical qubits
remains invariant. In other words, applying any element of the stabilizer
group to the states which represent the logical qubits will simply rearrange
the order of elements, but keep the states invariant.

Consequently, as the code subspace remains invariant, the five-qubit tensor
that encodes this code subspace to the correct logical qubit will also remain
invariant under applying elements of the stabilizer. This is visible in Figure
19, where the five-qubit tensor remains invariant under the generator $M_{1}$
applied to its input legs. However, the legs of this tensors may be switched
from input leg to output leg without a problem. This has been verified by
explicitly determining the contracted tensor. Now, for the tensor network in
Figure 17 there has been no explicit assigning of the tensor legs to
corresponding qubits, apart from the bulk qubit in the central tensor. By
cleverly choosing the boundary leg-qubit correspondence, replacing the tensors
by tensors with operators according to Figure 19 and changing the tensor legs
as input and output accordingly, the outer layer of Figure 20 can be
constructed. As is visible in the figure, this arrangement will have operators
acting only on boundary qubits of the subregion $A$. These operators get
pushed trough to be Z operators that cancel with each other respectively and X
operators acting on the central tensor legs. These X operators will then
construct the logical $\overline{X}$ operator of the five-qubit code, which is
pushed through to the central bulk qubit. This argument can be repeated in the
opposite direction which allows the bulk X operator to be recreated on A.

Figure 19: An example of how the five-qubit tensor remains invariant under the
operator $M_{1}$ that is part of the stabilizer group of the five-qubit,
visible in Table 1.  Figure 20: An example of how the X operator acting on the
central qubit is recreated on a boundary $A$ for a tensor network where the
central qubit is included in the geometric entanglement wedge but not the
greedy entanglement wedge.

The Z operator acting on the central bulk qubit can be reconstruced in much
the same way. This can be done replacing all the X operators by Z operators in
Figure 20 and vice versa. While the X operators will cancel each other, the
bulk Z operator will be reconstructed by Z operators on the boundary subregion
$A$. However, it must be pointed out that for this to work with the generator
$M_{1}$ the tensor legs/code qubit correspondence must be changed.

While the tensor legs/code qubit correspondence was chosen according to our
wishes, it has been done in such a way that the symmetries of the tensor
network remain assuredly. It is unclear if allowing rationally invariant
tensors to have the same legs corresponding to different code qubits changes
something fundamentally of the tensor networks. As an example, it can be
argued that this stabilizer method is limited if one chooses the legs that are
connected to the central tensor to correspond the logical output of the
connected hexagons. Because all stabilizer elements applied to this hexagon
will always leave that leg unchanged, no non-trivial operator could be pushed
through by this method. Consequently, while one might push operators to other
legs, all legs connected the central tensor will remain the same. Thus, the
central bulk qubit would remain unchanged as well. However, this limitation
can be lifted when working with the six-qubit perfect state. For this state,
stabilizers that have non trivial operators on the logical leg exists. These
are written down by Pastawski et al. in appendix A [1]. By using these
stabilizer elements, one is able to push logical operators through by the same
method, regardless if the legs that are connected to the central tensor
correspond to the logical output of the connected hexagons. This is an example
that seems to indicate that the tensor legs/code qubit correspondence has no
immediate influence on the global properties of the tensor network.

##  3 Mutual Information

In this section the mutual information between a boundary subregion $A$ and
the central qubit will be investigated. It will be examined if the inclusion
of the central bulk qubit into the geometric wedge has any correlation with a
rise in mutual information between the bulk qubit and boundary A. For this,
the mutual information is given by,

$I_{A,bulk}=S_{A}+S_{bulk}-S_{A,bulk}.$ (22)

Because the sum of entropy of the individual systems will always be larger or
equal to the entropy of the combination, the mutual information will be larger
or equal to zero. The entropy for the toy tensor network models will be
calculated using cut sizes, analogously to the corresponding holographic
Formulas (19, 21). However, one can also determine the entropy by calculating
the Von Neumann entropy explicitly by tracing out the regions. A more in depth
examination between these two different calculations has been performed by M.
Grandjean [25].

#### Classical RT Entropy

When explicitly working with wedge size, one must define which algorithm was
utilized. In this first section, no bulk contributions to the wedge size was
allowed. As proven in [1], this wedge size will correspond to its holographic
equivalent, the RT Formula (19), for connected boundary subregions. Because of
this, the mutual information will be investigated for a connected boundary
subregion that is enlarged by one qubit at a time until the whole boundary
region is reached. For one layer tensor networks, the minimum entanglement
algorithm will be used to find the true geometric wedge. This allows in turn
to determine the classical RT entropy. Since the bulk qubits are not chosen to
contribute to wedge sizes, $S_{bulk}$ is always zero. The entropy of the
boundary subregion together with the mutual information between the central
bulk qubit and the moment of inclusion of that bulk qubit in geometric wedge,
is plotted in Figure 21 for the one layer pentagon- and single qubit code.

Figure 21: The classic RT entropy of an increasing connected boundary region
$A$ in function of the size of that boundary. In addition, the mutual
information between central qubit and $A$ and the moment of inclusion in
geometric wedge of the central bulk qubit. On the left the one layer
holographic pentagon code and on the right the one layer single qubit code.

In Figure 21 it is visible that the mutual information remains zero for all
boundary sizes. While this seems troublesome, it can be explained through the
fact that the bulk qubits are not accounted for in the wedge size. In
calculating the entropy $S_{A,bulk}$ the bulk central qubit was allowed to
contribute to the greedy entanglement of $A$. However, since the minimum
entanglement wedge was taken, the moment that the central bulk pushed the
greedy geodesic $|\gamma_{A}^{*}|$ to include the central tensor, the
compliment geodesic $|\gamma_{A^{c}}^{*}|$ had the smaller size and kept the
entropy equal to $S_{A}$. The entropy of both systems are also symmetric and
has the property that a full boundary has an entropy of zero. This first
property is an artifact of the starting point of the boundary and the inherent
symmetry of the tensor network. The connected boundary region was started at
an outer leg of a tensor and first moved to include that whole tensor.
However, when starting from a middle tensor leg this symmetry fades.

The behaviour of these entropy graphs can also be explained using the
holographic state of the five-qubit code. The holographic state of the five-
qubit tensors was given by
$\frac{\ket{0\overline{0}}+\ket{1\overline{1}}}{\sqrt{2}}$. where the logical
states where defined in Section 1.2. Not increasing the size of the wedge for
the bulk qubits encapsulated, is proven to be numerically equivalent to
assigning the collapsed states $\ket{0}$ or $\ket{1}$ to the bulk qubits.
Consequently, the state of the entire system will change from the holographic
state to be either $\ket{0\overline{0}}$ or $\ket{1\overline{1}}$. However,
this projection forces the bulk qubit to lose the entanglement entropy as
tracing out the noisy qubits will just yield $\ket{0}$ or $\ket{1}$
respectively. These have no entanglement entropy as they are pure states.
Consequently, $I_{A,bulk}$ remains zero and the entropy graph ends on zero as
well.

In Figure 22, approximations of the RT formula are visible. In the left most
figure random boundary qubits are gradually appended to the boundary. This is
in contrast with the connected region of Figure 21. As a result the wedge size
will only give an upper bound to the entropy according to Formula (17). It is
visible on this graph that the upper bound will rise to 8, significantly
higher than the connected region’s peak, until about half the qubits are
included in the boundary after which the upper bound on the entropy will
decline until zero.
On the right graph of Figure 22, the RT entropy of an increasing boundary
region for a two layer pentagon code is shown. However, since the pentagon
code has two layers, it is no longer guaranteed that the minimal entanglement
wedge will find the true geometric wedge. As a result the entropy will be an
approximation of the true RT entropy. This is visible by the mutual
information no longer remaining zero.

Figure 22: Approximations of the RT entropy of an increasing boundary region
$A$. In addition, the mutual information between central qubit and $A$ and the
moment of inclusion in geometric wedge of the central bulk qubit. On the left
a one layer holographic pentagon code with a random boundary and on the right
a two layer pentagon code with connected boundary.

#### Error Corrected Entropy

Similar to the second part of Section 2.3, it can also be investigated how the
entropy of the holographic state changes with boundary size. Since every bulk
qubit will now be maximally entangled, the algorithm of the minimum wedge
will, for boundary A, add one to the size of entanglement wedge
$|\gamma_{A}^{*}|$ whenever a bulk qubit is encapsulated. Furthermore, for
$A^{c}$ all bulk qubits encapsulated between $A$ and $\gamma_{A^{c}}^{*}$,
after finishing the greedy algorithm on the compliment of the boundary, will
add one to the size of $|\gamma_{A^{c}}^{*}|$. Consequently, the entropy of
the boundary that may be calculated with this the wedge size will also change
form its RT Formula (19) equivalent to the Von Neumann equivalent of an error
correcting network (21) for which the $S_{bulk}(\rho_{\epsilon_{A}})$ term is
just the amount of bulk qubits encapsulated by the wedge and boundary.
Additionally, the $S_{bulk}$ term of the central qubit mutual information is
no longer zero but one.

In Figure 23 this entropy as a function of an increasing connected boundary
region is visible for the one layer pentagon- and the one layer single qubit
code. Additionally, the mutual information between the central qubit and the
boundary region in function of increasing boundary size is plotted as well.
Lastly, the moment of inclusion in the geometric wedge, determined by the
minimum entanglement wedge algorithm, is also marked. Here it is visible that
for both the pentagon code and the single qubit code, the entropy graphs are
no longer symmetrical. Furthermore, the entropy of the whole boundary region
will no longer be zero but the total amount of bulk qubits in the tensor
network. This is expected as each bulk qubit is maximally entangled with the
entire boundary through a series of five-qubit tensors. Most crucially, the
mutual information between the central bulk qubit and the boundary no longer
remains zero for this corrected entropy. For both the pentagon- and the single
qubit code it is visible that mutual information remains at zero until over
half of the boundary qubits are included in the region, which is when the
boundary is larger than 10 and 12.5 for pentagon- and single qubit code
respectively. At this moment, the following happens: firstly, the central bulk
qubit is included in the minimum entanglement wedge of $A$. Consequently, the
geodesic as a result of the minimum entanglement algorithm on $A$, which is
the geometric geodesic $\gamma_{A}$, and the geometric geodesic as a result of
$A$ and the central bulk, $\gamma_{A,bulk}$, will be the same from this point
and forward. As the cut of the central bulk does not need to be added in
$|\gamma_{A,bulk}|$, it will be smaller than $|\gamma_{A}|$ by one. This
results in the observed mutual information of two.

Figure 23: The corrected entropy of an increasing connected boundary region
$A$ in function of the size of that boundary. In addition, the mutual
information between central qubit and $A$ and the moment of inclusion in
geometric wedge of the central bulk qubit. On the left the one layer
holographic pentagon code and on the right the one layer single qubit code.

When this happens, for both the pentagon- as the single qubit code, the true
geometric wedge, equal to the minimum entanglement wedge, will include the
central qubit.

Figure 24: Approximations of the corrected entropy of an increasing boundary
region $A$. In addition, the mutual information between central qubit and $A$
and the moment of inclusion in geometric wedge of the central bulk qubit. On
the left a one layer holographic pentagon code with a random boundary and on
the right a two layer pentagon code with connected boundary.

In Figure 24, similar to the previous section, the same graphs are visible for
both random boundary addition and a two layer pentagon code. However, since a
random boundary was used, the wedge size only provides an upper bound for the
entropy according to Formula (17). Moreover, it is not guaranteed that the
minimal entanglement wedge finds the true geometric wedge for the two layer
pentagon code. Therefore the right most graph in Figure 24 is also an
approximation.

Nonetheless, nearly the same behaviour as Figure 23 is visible in Figure 24.
The graphs will fluctuate and end on the amount of bulk qubits in the system.
The randomness on the boundary is visible in the left graph of Figure 24, as
the entropy now moves in a straight line. This graphs also shows a different
behaviour in mutual information with respect to geometric inclusion. While the
mutual information will still rise to be two, it will no longer do it at half
the boundary qubits. Moreover the inclusion in the geometric wedge will be
even later. However, this mutual information is the result of an approximation
of the RT formula, which does not guarantee accurate mutual information. In
the next section, the entropy and mutual information for this particular
tensor network with the same random boundary will be precisely determined with
the explicit Von Neumann entropy using the holographic state.
The two layer pentagon code for a connected region does show the same
behaviour. The mutual information will rise to two at the same time as over
half the boundary qubits are included and the geometric wedge encapsulates the
the central bulk qubit at the same time.

#### Mutual Information wedge

In this last section the viability of the mutual information wedge is
discussed. The mutual information wedge that is proposed can be defined as the
wedge that includes all bulk qubits where the mutual information between the
boundary and the bulk qubit is two. This wedge has the benefit that it is
dependent on the state in the bulk. This is visible in Figure 21, where the
mutual information for the central bulk qubit is zero and therefore never
included. From the examples of Figure 23 and Figure 24, it does seem that
there is some correlation between inclusion in the geometric wedge and mutual
information. For this corrected entropy, the inclusion of greedy entanglement
wedge is associated to rise in mutual information. However, this might be an
artifact of the connected boundary subregions.

The validity of this wedge will be investigated using the random boundary
corresponding to the graph of Figure 24 (left). Here, the mutual information
will not be calculated using the wedge size but rather the Von Neumann entropy
of the holographic state. This allows to determine the entropy precisely for
both the random disconnected region $A$ and the union of $A$ and the central
bulk qubit. Consequently, the mutual information will also be correct.
In Figure 25, this graph of the Von Neumann entropy and mutual information is
visible. Because of the difficult nature of these computations, the entropy
has only been determined up to a boundary size of twelve. Nonetheless, it is
already visible that mutual information rises sooner than the inclusion in the
geometric wedge. The rise in mutual information occurs at a boundary size of
eight and corresponds for this example to the first boundary where three legs
of one of the outer five-qubit tensors is included. However, an investigation
of different examples proves that this correspondence does not occur in all
examples. Additionally, while the RT upper bound on the entropy of $A$,
visible in Figure 24, is reached, the rise in mutual information is sooner
than the approximate graph. Therefore, it can be stated that entropy of the
union of $A$ and the central bulk qubit does not follow its wedge upper bound.

Figure 25: The Von Neumann entropy of an increasing random boundary region $A$
of a one layer pentagon code. In addition, the mutual information between
central qubit and $A$ and the moment of inclusion in geometric wedge of the
central bulk qubit.

Figure 25 indicates that a rise in mutual information is not a valid
indication for the inclusion in the geometric wedge. Moreover, the rise in
mutual information is not even associated to the inclusion in the entanglement
wedge. While this has been confirmed by explicitly working out the example, it
is also visible in the graph directly since the entanglement wedge for a
subregion of only eight qubits will never include the bulk qubit. As a result,
a mutual information wedge which includes all bulk qubits with mutual
information of two is not a valid candidate wedge to assure operator
reconstructability. The associated rise in mutual information has only been
found in an increasing connected boundary region where the boundary starts on
the outer leg of a tensor and moves to first include that tensor. Lastly, no
central qubit operator was reconstructable on that random boundary subregion
until it was included in the geometric wedge.

##  4 Conclusion

The tensor network framework has different wedges for boundary subregions that
enclose the bulk qubits for which logical operators are reconstructable on
these subregions. These were investigated by performing Monte-Carlo
simulations of the boundary erasure and central qubit inclusion for the HaPPY
five-qubit toy tensor networks [1]. The simulations indicated that, while the
causal wedge will not have any detectable threshold probability, the greedy
entanglement wedge will have detectable threshold probabilities. The existence
of these detectable threshold probabilities were strongly influenced by the
structure of the tensor network. This is highlighted by the fact that a
threshold probability was found for the single qubit code, but none was
detected for the holographic pentagon code. Simulations of tensor networks
with a non-uniform boundary showed its limited influence on a threshold
probability. As shown by the stacking of tensor networks, the existence of
such threshold probability seems to be an inherent property of the structure
of the tensor network rather than the fraction
$\frac{N_{bulk}}{N_{boundary}}$. Furthermore, the minimum entanglement wedge
was introduced as the minimum of the greedy algorithms on the boundary and on
the compliment. This showed good results in approximating the true geometric
wedge, but lost its guaranteed operator reconstructability. An explicit
example of operator reconstruction beyond the greedy entanglement wedge was
found using the stabilizer elements of the five-qubit code. Lastly, the
possibility of a wedge based on mutual information was rejected. Only for
particular connected boundary subregions was the inclusion of the central
tensor by the geometric wedge associated to a rise in mutual information.

## Acknowledgment

I would like to thank my family and friends for their continued love and
support. Particularly my mother, father and brother for keeping my spirits
high and being always there for me. Additionally, I thank my brother-in-arms,
Maxime, for the many insightful conversations and discussions. I would also
like to thank my supervisors, Juan and Philip, for without their guidance and
wisdom this thesis would not have been possible. I thank my promotor Prof. Dr.
Craps and the VUB for allowing me to do this research. I thank the authors of
the HaPPY paper [1], for their clear and understandable language enabling me
to recreate and expand on their research. Finally, I also thank the open
source Python libraries Qiskit and TensorNetworks [26, 27]. Much of the code
used can be found on https://github.com/VicVanderLinden/AdS-CFT_HaPPY.

## References

  * [1] Pastawski F., Yoshida B., Harlow D. and Preskill J. Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence [Online]. 2015 [Available on 19 February 2023.] https://arxiv.org/abs/1503.06237
  * [2] Feynman, R. Simulating Physics with Computers International Journal of Theoretical Physics, Volume 21, Nos. 6/7, 7 May 1982
  * [3] Schumacher, B. Quantum coding Physical Review A (Atomic, Molecular, and Optical Physics), Volume 51, Issue 4, April 1995, pp.2738-2747
  * [4] Arute, F., Arya, K., Babbush, R. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019). https://doi.org/10.1038/s41586-019-1666-5
  * [5] IBM Quantum Quantum Composer User Guide [Online]. 2022 [Available on 19 February 2023.] https://quantum-computing.ibm.com/composer/docs
  * [6] Einstein, A, B Podolsky, N Rosen. Can Quantum-Mechanical Description of Physical Reality be Considered Complete? Physical Review, Volume 47, Issue 10, May 1935, pp. 777-780
  * [7] Griffiths, D., & Schroeter, D. (2018). Introduction to Quantum Mechanics (3rd ed.). Cambridge: Cambridge University Press. doi:10.1017/9781316995433
  * [8] R. Larose Quick Seminar VI: Quantum States and Partial Trace [Online]. 2021 [Available on 2 May 2023.] https://www.ryanlarose.com/uploads/1/1/5/8/115879647/quic06-states-trace.pdf
  * [9] Shor, P. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer [Online]. 2022 [Available on 20 February 2023.] arXiv:quant-ph/9508027v2.
  * [10] Gottesman, D. Stabilizer Codes and Quantum Error Correction [Online]. 2022 [Available on 19 February 2023.] arXiv:quant-ph/9705052v1.
  * [11] Nielsen, M. A., Chuang, I. L. Quantum Computation and Quantum Information: 10th Anniversary Edition. [Online]. 2022 [Available on 19 February 2023.] http://mmrc.amss.cas.cn/tlb/201702/W020170224608149940643.pdf.
  * [12] Jahn, A. and Eisert, J. Holographic tensor network models and quantum error correction: A topical review [Online]. 2022 [Available on 30 March 2023.] https://arxiv.org/pdf/2102.02619.pdf.
  * [13] Hawking, S. W  Particle creation by black holes. Commun.Math. Phys. 43, 199–220 (1975). https://doi.org/10.1007/BF02345020
  * [14] L. Susskind, The world as a hologram  J. Math. Phys., vol. 36, pp. 6377–6396, 1995, arXiv:hep-th/9409089. doi:10.1063/1.531249.
  * [15] G. ’t Hooft, “Dimensional reduction in quantum gravity,” Conf. Proc., vol. C930308, pp. 284–296, 1993, arXiv:gr-qc/9310026.
  * [16] J. M. Maldacena, The large N limit of superconformal field theories and supergravity Int. J. Theor. Phys., vol. 38, pp. 1113–1133, 1999, arXiv:hep-th/9711200. doi:10.1023/A:1026654312961.
  * [17] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from the anti-de Sitter space/conformal field theory correspondence Phys. Rev. Lett., vol. 96, p. 181602, 2006. doi:10.1103/PhysRevLett.96.181602.
  * [18] A. Deger, J. Pachos AdS/CFT Correspondence with a 3D Black Hole Simulator [Online]. 2022 [Available on 4 April 2023.] https://arxiv.org/abs/2211.15305.
  * [19] Jafferis, D., Zlokapa, A., Lykken, J.D. et al. Traversable wormhole dynamics on a quantum processor. Nature 612, 51–55 (2022). https://doi.org/10.1038/s41586-022-05424-3
  * [20] S. Antonini, G. Bentsen, C. Cao, J. Harper, S. Jian and B. Swingle Holographic measurement and bulk teleportation [Online]. 2022 [Available on 14 April 2023.] https://arxiv.org/abs/2209.12903.
  * [21] Yoshida, B. Quantum gravity from quantum error-correcting codes? [Online]. 2015 [Available on 4 April 2023.] https://quantumfrontiers.com/author/beniyoshida/
  * [22] V. Hubenya, M. Rangamania, E. Tonnib Global properties of causal wedges in asymptotically AdS spacetimes [Online]. 2013 [Available on 10 April 2023.] https://arxiv.org/abs/1306.4324
  * [23] D. Harlow The Ryu-Takayanagi Formula from Quantum Error Correction [Online]. 2016 [Available on 4 May 2023.] https://arxiv.org/abs/1607.03901.
  * [24] T. Faulkner, A. Lewkowycz, J. Maldacena Quantum corrections to holographic entanglement entropy [Online]. 2013 [Available on 4 May 2023.] https://arxiv.org/abs/1307.2892.
  * [25] M. Grandjean AdS/CFT Toy Models. Operator Reconstruction, Holgraphic Entropy and Nearest Neighbor Interactions. [Online]. 2023 [Not yet publicly available.]
  * [26] Qiskit contributors Qiskit: An Open-source Framework for Quantum Computing [Online]. 2023 [Available on 4 May 2023.] 10.5281/zenodo.2573505.
  * [27] The TensorNetwork Authors TensorNetwork [Online]. 2023 [Available on 4 May 2023.] https://tensornetwork.readthedocs.io/en/latest.