# Percolation Theories for Quantum Networks

Xiangyi Meng  Network Science Institute and Department of Physics,
Northeastern University, Boston, Massachusetts 02115, USA  Department of
Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
Xinqi Hu  School of Mathematical Sciences, Jiangsu University, Zhenjiang,
212013, China  Yu Tian  Nordita, Stockholm University and KTH Royal Institute
of Technology, Stockholm, SE-106 91, Sweden  Gaogao Dong  School of
Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, China  Renaud
Lambiotte  Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
Turing Institute, NW1 2DB, London, UK  Jianxi Gao  Department of Computer
Science, Rensselaer Polytechnic Institute, Troy, NY 12180, USA  Network
Science and Technology Center, Rensselaer Polytechnic Institute, Troy, NY
12180, USA  Shlomo Havlin<EMAIL_ADDRESS>Department of Physics, Bar-Ilan
University, Ramat Gan, 52900, Israel

###### Abstract

Quantum networks have experienced rapid advancements in both theoretical and
experimental domains over the last decade, making it increasingly important to
understand their large-scale features from the viewpoint of statistical
physics. This paper discusses a fundamental question: how can entanglement be
effectively and indirectly (e.g., through intermediate nodes) distributed
between distant nodes in an imperfect quantum network, where the connections
are only partially entangled and subject to quantum noise? We survey recent
studies addressing this issue by drawing exact or approximate mappings to
percolation theory, a branch of statistical physics centered on network
connectivity. Notably, we show that the classical percolation frameworks do
not uniquely define the network’s indirect connectivity. This realization
leads to the emergence of an alternative theory called “concurrence
percolation,” which uncovers a previously unrecognized quantum advantage that
emerges at large scales, suggesting that quantum networks are more resilient
than initially assumed within classical percolation contexts, offering
refreshing insights into future quantum network design.

Keywords: percolation, quantum network, entanglement distribution, critical
phenomena, networks of networks, hypergraph

##  I Introduction

Quantum information [1] is a fast-developing field that has transcended its
roots originally in quantum mechanics and information theory to other areas
like condensed matter physics [2], statistical physics [3, 4, 5], and network
science [6, 7]. At the core of quantum information lies the quantum bit, or
qubit, the basic quantum information carrier. Two qubits can be designed into
a relationship, called entanglement, which is an essential quantum resource
[8] for quantum computing. Yet, entanglement is notoriously fragile,
especially when qubits are spatially distant. Fortunately, by path routing and
adding in-between sites for replaying, entanglement between remote qubits may
eventually be established in an indirect way. Such an action, called
_entanglement distribution_ [9], is a fundamental benefit of quantum networks
(QN) [10, 11].

In general, a QN is a network representation of different parties (nodes) that
share entanglement (links) as connections. A significant part of our interest
lies in distributing entanglement between _two_ arbitrary nodes in the
network, a process we refer to as “entanglement transmission.” Entanglement
across different parties is essentially transmitted through quantum
communication protocols. Successful demonstrations of quantum communication
protocols have already been made on small-scale QN using diamond nitrogen-
vacancy centers [12, 13, 14] and ion traps [15, 16]. However, the big question
that looms is how to scale this to much larger networks. A large-scale,
practical QN would offer significant advantages for many industrial and
scientific applications. For example, financial institutions and governments
would benefit from quantum cybersecurity providing an unprecedented level of
secure communication. Researchers could also use networked quantum computers
to dramatically increase the simulation speed of the physical and chemical
processes of many interacting particles. Yet, if the individual channels
(links) along the routed path are too noisy, the entanglement transmission may
fail. Study of such dependence of “indirect” transmission ability on the noise
level of individual links requires tools from statistical physics and complex
network theories.

One of such theory that has proven to be useful is percolation theory [17, 18,
19, 20]. Percolation theory offers a mathematical framework for understanding
how networks behave when subjected to random processes (can be treated as a
form of noise), such as how water percolates through soil or how diseases
spread through populations. In the context of QN, percolation could provide
valuable insights into the robustness and efficiency of entanglement
distribution. By applying percolation theory, we can model and analyze the
network structure directly and identify the most effective ways to maintain
and distribute quantum entanglement across it. This lays the groundwork for
examining QNs through the lens of _statistical physics_ and opens up new
avenues for understanding the upper limits of entanglement distribution in
these networks.

In this work, we will explore and summarize the developments of the QN
framework and how a mapping to percolation offers unique tools for dissecting
the problem of entanglement transmission. Specifically, we will show that the
mapping to percolation theories—and a definition of how a combination of
pairwise edges combines into indirect connectivity—are, indeed, _not_ unique.
A new, alternative percolation-like theory termed _concurrence percolation_
[21] emerges, and it underlies an unexpected “quantum advantage,” revealing
that QNs are more robust than we initially thought within the classical
percolation framework. Moreover, the finding is scalable with network size and
adaptable to different network topologies, suggesting a macroscopic
improvement over classical considerations from a statistical physics
perspective.

This paper focuses on the comparison between classical percolation and
concurrence percolation when mapped based on QN. It is structured as follows:
In Section II, we give a definition of the QN theoretical framework as well as
its possible generalizations to other QN-based structures (e.g., hypergraphs).
In Section III, we briefly review the concept and definition of percolation
theory and, in particular, how it relates to network connectivity at large
scales. In Sections IV and V, we will focus on the discoveries that the new
concurrence percolation theory surpasses the traditional percolation theory
(which we refer to as “classical percolation” for comparison). In Section VI,
we will delve into the algorithms developed for calculating concurrence
percolation. Finally, in Section VII, we will discuss the open questions and
practical implications of the findings, both theoretically and for real-world
communications.

Figure 1: A pure-state quantum network (QN) consists of nodes (purple) and
links (gray line). Each node comprises a collection of qubits (gray dots) that
are entangled with qubits belonging to other nodes, and each link represents a
bipartite entangled pure state $\left|\phi\right\rangle$ connecting the two
qubits at its endpoints. This QN model can be extended to (a) $d$-dimensional
qudits, which allow higher bandwidth for transmitting information, or to (b)
tensor networks by employing linear transformations $\mathcal{T}_{i}$ at each
node $i$. (c) Moreover, the QN can be adapted to higher-order graphs, where
each link manifests as a hyperedge, denoting a multipartite entangled pure
state.

##  II Quantum networks (QN)

As in traditional network theories, a QN resembles a topological graph or a
graph-like structure, comprising nodes and links. This paper primarily focuses
on a pure-state version of QN (Fig. 1) [10]. The QN is defined based on the
following three principles:

  1. 1.

Each _node_ (purple) comprises a collection of qubits (gray dots) that are
entangled with qubits belonging to other nodes.

  2. 2.

Each _link_ (gray line) represents a bipartite entangled pure state
$\left|\phi\right\rangle$ connecting the two qubits at its endpoints.

  3. 3.

A _weight_ $\theta$ is assigned to each link to characterize the degree of the
link’s entanglement.

Using the Dirac notation, a link, which corresponds to a bipartite entangled
pure state connecting two nodes (e.g., Alice and Bob), can be written as
$\left|\phi\right\rangle=\cos\theta\left|00\right\rangle+\sin\theta\left|11\right\rangle$.
Here, w.l.o.g., the weight parameter $\theta$ is constrained within the range
$0\leq\theta\leq\pi/4$, ensuring that $\cos\theta\geq\sin\theta$ is satisfied.
In this notation, the first “$0$” in $\left|00\right\rangle$ and the first
“$1$” in $\left|11\right\rangle$ represent the two possible states of Alice’s
qubit, $\left|0\right\rangle_{\text{Alice}}$ and
$\left|1\right\rangle_{\text{Alice}}$, respectively. Similarly, the second
“$0$” and “$1$” represent the two possible states of Bob’s qubit,
$\left|0\right\rangle_{\text{Bob}}$ and $\left|1\right\rangle_{\text{Bob}}$.
The entanglement between Alice’s and Bob’s qubits is evident by the presence
of only two terms $\left|00\right\rangle$ and $\left|11\right\rangle$ in
$\left|\phi\right\rangle$, while $\left|01\right\rangle$ and
$\left|10\right\rangle$ are absent. This implies that upon measuring the state
$\left|{\phi}\right\rangle$ in the
${\left|0\right\rangle,\left|1\right\rangle}$ basis from either Alice’s or
Bob’s side, the state will randomly collapse to either $\left|00\right\rangle$
or $\left|11\right\rangle$. Consequently, if Alice’s (or Bob’s) measurement
yields “$0$,” it guarantees that the other party’s measurement result will
also be “$0$.” This highlights a correlation feature that can be harnessed for
communication in the quantum realm. Similar to how correlation in classical
communication is measured using mutual information, we can quantify this
quantum correlation using quantum mutual information [1], which is given by
$-2\cos^{2}\theta\ln\left(\cos^{2}\theta\right)-2\sin^{2}\theta\ln\left(\sin^{2}\theta\right)$.
The quantum mutual information reaches its maximum value when $\theta=\pi/4$,
which corresponds to a maximally entangled state,
$\left|\phi_{\perp}\right\rangle=\sqrt{1/2}\left|00\right\rangle+\sqrt{1/2}\left|11\right\rangle$,
commonly referred to as a Bell state or a singlet [10].

The entire QN, comprising many links, can be regarded as a huge pure state
$\left|\text{QN}\right\rangle$—the tensor product of all the individual
bipartite pure states associated with each link. Consequently, the QN solely
focuses on “quantum noise,” which comes from the fact that when
$\theta<\pi/4$, the link exhibits only partial entanglement. This partial
entanglement, when employed in quantum communication tasks such as quantum
teleportation, leads to errors in the teleported qubits, affecting the QN’s
overall communication capacity. Yet, as a pure state, the QN does not involve
any classical noise (i.e., mixed states). This makes
$\left|\text{QN}\right\rangle$ an excellent medium for examining quantum
phenomena without the interference of classical noise. Thus, this “minimalist”
construction of the QN can serve as an ideal framework for investigating
quantum theories and concepts on large scales.

At present, the choice to define nodes as collections of qubits rather than
individual qubits may appear arbitrary. What is the physical meaning of a node
as a collection of qubits? And what about the qubits that belong to the same
node—are they also entangled?

To answer these questions, it is crucial to comprehend the concept of
_locality_. While the theoretical framework of quantum mechanics is inherently
nonlocal, practical implementations of quantum information technologies often
necessitate considering a “distant lab” paradigm [22]. In such scenarios, when
a quantum system is distributed among multiple spatially distant parties or
laboratories, it becomes unrealistic to assume the feasibility of executing
global quantum operations. Instead, the parties are typically constrained to
apply quantum operations exclusively to their respective subsystems (in their
own “labs”), rather than collectively to the global system. This subset of
quantum operations is known as local operations (LO).

For example, given the entangled state
$\left|\phi\right\rangle=\cos\theta\left|00\right\rangle+\sin\theta\left|11\right\rangle$
between Alice and Bob, Alice may apply local unitary transformation on her
qubit (e.g., a rotation
$\\{\left|0\right\rangle,\left|1\right\rangle\\}\to\\{\frac{\left(\left|0\right\rangle+\left|1\right\rangle\right)}{\sqrt{2}},\frac{\left(\left|0\right\rangle-\left|1\right\rangle\right)}{\sqrt{2}}$),
and Bob may apply the same transformation as well. This yields a new state
$\left|\phi\right\rangle\to\left|\phi^{\prime}\right\rangle=\cos\theta\frac{\left(\left|0\right\rangle+\left|1\right\rangle\right)\left(\left|0\right\rangle+\left|1\right\rangle\right)}{2}+\sin\theta\frac{\left(\left|0\right\rangle-\left|1\right\rangle\right)\left(\left|0\right\rangle-\left|1\right\rangle\right)}{2}$.
Furthermore, LO also allows Alice or Bob to locally measure their qubits as
well, resulting in the random collapse of $\left|\phi\right\rangle$ to one of
its eigenstates. However, Alice and Bob cannot transform their state globally
and obtain a singlet,
$\left|\phi\right\rangle\to\left|\phi_{\perp}\right\rangle=\sqrt{1/2}\left|00\right\rangle+\sqrt{1/2}\left|11\right\rangle$.
This is not counted as LO.

On top of LO, Alice and Bob are also free to communicate classical information
(CC), sharing their results of quantum measurements. Together, this set of
operations is called the local operations and classical communication (LOCC).
The LOCC defines a set of strategies to share and manipulate quantum
information under the locality constraint. One of the most powerful theorems
in quantum information states that the average entanglement between two
parties can never be increased if only LOCC is allowed for the two parties.
This establishes the role of entanglement as a quantum resource, given that
LOCC is the “free operations” of the system [8].

Therefore, qubits belonging to the same node are not constrained by LOCC. They
can be freely entangled or disentangled as needed, but the entanglement is not
viewed as a resource for communication. Only the entanglement of qubits
belonging to different nodes matters. In other words, quantum networks are an
effective representation of the _fundamental constraints of locality_ ,
manifested by assigning qubits to different local compartments and
entanglement to inter-compartment connections.

###  II.1 Qudit-based quantum networks

A natural extension of QN is to use more general $d$-dimensional “qudits”
(qutrits, ququarts, etc.) instead of qubits (Fig. 1). Each link, as a
bipartite pure state of qudits, can be written as

$\left|\boldsymbol{\lambda}\right\rangle=\sum_{j=1}^{d}\sqrt{\lambda_{j}}\left|jj\right\rangle.$
(1)

Here, $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{d}\geq 0$ and
$\sum_{j=1}^{d}\lambda_{j}=1$. In this generalization, the weight of each link
is no longer a single number but a list of non-negative numbers known as
Schmidt numbers, denoted as
$\boldsymbol{\lambda}=\left(\lambda_{1},\lambda_{2},\cdots,\lambda_{d}\right)$.
When $d=2$, the bipartite pure state reduces to qubits, where
$\lambda_{1}=\cos^{2}\theta$ and $\lambda_{2}=\sin^{2}\theta$.

The consideration of qudit-based QN offers both theoretical and practical
advantages. Theoretically, a $d$-dimensional qudit inherently carries
$\log_{2}d$ times more information than a qubit. Therefore, as the value of
$d$ increases, a single carrier can transmit more information, increasing the
bandwidth. This enhanced capability is also evident in the robustness of
entanglement for entangled states of qudits. Indeed, even when some
coefficients are erased ($\lambda_{j}\to 0$) from $\boldsymbol{\lambda}$ in
the presence of noise, the pure state
$\left|\boldsymbol{\lambda}\right\rangle$ can still remain entangled, as long
as the two largest Schmidt numbers $\lambda_{1}$ and $\lambda_{2}$ remain
positive.

In experiments, qubit systems are commonly realized using two-level atoms or
superconducting states. However, isolating these two levels from other nearby
levels can be challenging. By including nearby levels and increasing the
potential dimension $d$, the experimental design may become more feasible. In
fact, several experiments have employed qudits to achieve better performance,
including applications in quantum scrambling [23] and superdense coding [24].

###  II.2 Quantum networks are the basis of tensor networks

There is also an interesting and deep connection between QN and tensor
networks (Fig. 1), the latter being a familiar and powerful tool in condensed
matter physics, mostly used for the purpose of facilitating computations and
simulations in quantum physics and materials science. To be specific, tensor
networks are designed to _efficiently_ represent many-body quantum states
[25]. These quantum states, which are essentially large, high-dimensional
tensors in mathematical terms, can be factorized into smaller tensors using
tensor networks. In particular, tensor networks are useful for representing
the ground state of quantum systems, which typically exhibit strong ordering
compared to excited states. This strong ordering often means that entanglement
does not grow very fast with the length scale, which, in turn, allows for
easier and more efficient factorization of the corresponding ground state.

To delve deeper into the concept, note that a general $N$-body quantum state
reads

$\left|\Psi\right\rangle=\sum^{D}_{x_{1},x_{2},\cdots,x_{N}=1}T^{x_{1}x_{2}\cdots
x_{N}}\left|x_{1}\right\rangle\left|x_{1}\right\rangle\cdots\left|x_{N}\right\rangle,$
(2)

which lives in a $D^{N}$-dimensional Hilbert space that is the tensor product
of $N$ “single-body” Hilbert spaces ($i=1,2,\cdots,N$). Each space is spanned
by a basis of $D$ vectors, $\left|x_{i}\right\rangle$, where
$x_{i}=1,2,\cdots,D$. The complex tensor $T$ that stores the coefficients is
exponentially large ($\sim D^{N}$), effectively preventing direct computations
of the quantum state $\left|\Psi\right\rangle$’s characteristics. However,
there may be a significant level of redundancy in the coefficients present
stored in the tensor. Consider an example where every entry in $T$ can be
fully factorized, such that $T=a\otimes b\otimes c\otimes\cdots$ where
$a,b,c,\cdots$ are $D$-dimensional vectors. In this case, it becomes
unnecessary to store the entire tensor or perform calculations on it. Rather,
it suffices to simply store the vectors $a,b,c,\cdots$ of which the total size
is $DN$.

This, indeed, is how a tensor network works—by leveraging different ways of
factorization that can be depicted through different graphical network
structures [25, 26]. Among various tensor networks, the matrix product state
(MPS) is among the most researched [27]. In computer science, it is often
called the tensor-train network [28]. The MPS is commonly utilized to
represent one-dimensional many-body quantum states. When extended to higher
dimensions, this becomes what is known as the projected entangled pair state
(PEPS) [29]. More involved tensor network structures, such as the multiscale
entanglement renormalization ansatz (MERA) [30], are also routinely used to
study critical quantum systems.

For example, a MPS representation of Eq. (2) can be written as

$\left|\Psi\right\rangle=\sum^{D}_{x_{1},x_{2},\cdots,x_{N}}\text{tr}\left\\{A_{1}^{x_{1}}A_{2}^{x_{2}}\cdots
A_{N}^{x_{N}}\right\\}\left|x_{1}\right\rangle\left|x_{1}\right\rangle\cdots\left|x_{N}\right\rangle,$
(3)

where for each single-body $i$, a set of $D$ different matrices,
$A_{i}^{1},A_{i}^{2},\cdots,A_{i}^{D}$, are introduced. Each matrix is of size
$d\times d$, where $d$ is called the bond dimension. Thus, the total number of
parameters is $NDd^{2}$, which is linear in $N$. For a sufficiently large $d$,
the MPS has enough degrees of freedom to exactly represent any tensor $T$.
However, it is frequently observed that a small $d$ can approximately, if not
perfectly, reproduce $T$. This occurs when the information stored in $T$
scales linearly with $N$, a condition often found in the ground states of one-
dimensional noncritical quantum systems.

Intriguingly, the MPS offers a new physical perspective—the valence-bond
picture [29]. To be specific, we map each single-body Hilbert space (spanned
by $\left|x_{i}\right\rangle$) to a physical site and assume that there are
two $d$-dimensional qudits located at each site. For every two neighboring
sites ($1\leftrightarrow 2,2\leftrightarrow 3,\cdots,N\leftrightarrow 1$), two
qudits from each site are fully entangled, forming a “valence bond” that can
be written as an unnormalized maximally entangled state,

$\left|\psi\right\rangle=\sum^{d}_{j=1}\left|j\right\rangle\left|j\right\rangle\rightarrowtail\psi=\begin{pmatrix}1&0&\cdots\\\
0&1&\cdots\\\ \vdots&\vdots&\ddots\end{pmatrix}.$ (4)

Here, the state is also represented (matricized) into the matrix form $\psi$.
Combining this with Eq. (3), we obtain

$\displaystyle\left|\Psi\right\rangle$ $\displaystyle=$
$\displaystyle\sum^{D}_{x_{1},x_{2},\cdots,x_{N}}\text{tr}\left\\{A_{1}^{x_{1}}\psi
A_{2}^{x_{2}}\psi\cdots
A_{N}^{x_{N}}\psi\right\\}\left|x_{1}\right\rangle\left|x_{1}\right\rangle\cdots\left|x_{N}\right\rangle$
(5) $\displaystyle=$
$\displaystyle\left(\mathcal{T}_{1}\otimes\mathcal{T}_{2}\otimes\cdots\otimes\mathcal{T}_{N}\right)\left|\psi\right\rangle^{\otimes
N},$

where
$\mathcal{T}_{i}=\sum_{x_{i}=1}^{D}\sum_{j,j^{\prime}=1}^{d}\left(A_{i}^{x_{i}}\right)_{jj^{\prime}}\left|x_{i}\right\rangle\left\langle
j,j^{\prime}\right|$ represents a linear transformation acting on the two
qudits (labeled by $j$ and $j^{\prime}$) on-site $i$. The valence-bond picture
is now evident: In this picture, the many-body state is not the primary
entity. Instead, it is built upon something more fundamental—a network of
qudits and “valence bonds.” The linear transformations $\mathcal{T}_{i}$ are
then employed on top of it to form the tensor network.

Note that this fundamental network $\left|\psi\right\rangle^{\otimes N}$ is a
one-dimensional (periodic) quantum network consisting of maximally entangled
states, making it remarkably suitable to be _generalized_ to partially
entangled states [Eq. (1)]. This can be achieved by replacing $\psi$ in Eq.
(4) by

$\psi=\begin{pmatrix}\lambda_{1}&0&\cdots\\\ 0&\lambda_{2}&\cdots\\\
\vdots&\vdots&\ddots\end{pmatrix}.$ (6)

The physical meaning of inserting such a partially entangled state is that
since LO cannot increase the entanglement, the entanglement between
neighboring sites will be upper bounded by the amount of entanglement in
$\psi$.

The valence-bond picture is not limited to MPS but can be generalized to
arbitrary tensor networks. Indeed, suppose $A_{i}$ at site $i$ does not denote
a set of matrices but a set of tensors,
$A_{i}^{1},A_{i}^{2},\cdots,A_{i}^{D}$, each having entries
$\left(A^{x_{i}}_{i}\right)_{jj^{\prime}j^{\prime\prime}\cdots j^{(k)}}$
labeled by $k$ subscripts $j,j^{\prime},j^{\prime\prime},\cdots,j^{(k)}$. Each
subscript denotes a qudit on-site $i$. The site has $k$ qudits in total,
indicating that the corresponding node $i$ has degree $k$ (i.e., $k$ incident
links) in the QN. The linear transformation then becomes

$\mathcal{T}_{i}=\sum_{x_{i}=1}^{D}\sum_{j,j^{\prime},\cdots,j^{(k)}=1}^{d}\left(A^{x_{i}}_{i}\right)_{jj^{\prime}\cdots
j^{(k)}}\left|x_{i}\right\rangle\left\langle
j,j^{\prime},\cdots,j^{(k)}\right|,$ (7)

and the many-body state is expressed as

$\displaystyle\left|\Psi\right\rangle=\left(\mathcal{T}_{1}\otimes\mathcal{T}_{2}\otimes\cdots\otimes\mathcal{T}_{N}\right)\left|\text{QN}\right\rangle,$
(8)

where $\left|\text{QN}\right\rangle$ represents the entire QN considered as a
huge pure state (Fig. 1).

###  II.3 Multipartite quantum networks

Our attention has been narrowed to bipartite entanglement. However, a complete
QN framework should take _multipartite_ entangled states into account. This is
since multipartite entangled states have a specialized and unique role in
certain quantum communication applications, such as secret sharing [31].
Although multipartite entanglement has been widely explored, we still lack a
unified, clear method to precisely detect, measure, and define it. For
example, even with an entangled state of just three qubits, there exist two
non-equivalent forms of genuine tripartite entanglement. The first is known as
the GHZ class, characterized by five real parameters, $\alpha$, $\beta$,
$\gamma$, $\delta$, and $\theta$, and can be expressed as [32]

$\displaystyle\left|\phi_{\text{GHZ}}\right\rangle\propto\cos\delta\left|000\right\rangle+\sin\delta
e^{i\theta}\left(\cos\alpha\left|0\right\rangle+\sin\alpha\left|1\right\rangle\right)\left(\cos\beta\left|0\right\rangle+\sin\beta\left|1\right\rangle\right)\left(\cos\gamma\left|0\right\rangle+\sin\gamma\left|1\right\rangle\right).$

The second form, called a W class, has a general representation as [32]

$\displaystyle\left|\phi_{\text{W}}\right\rangle=\sqrt{a}\left|001\right\rangle+\sqrt{b}\left|010\right\rangle+\sqrt{c}\left|100\right\rangle+\sqrt{d}\left|000\right\rangle,$
(10)

with the real parameters $a,b,c>0$ and $d=1-a-b-c\geq 0$. Both the GHZ and W
classes represent a level of correlation that goes beyond just pairwise
interactions, meaning that a measurement on any single qubit among the three
will instantaneously affect the outcomes of the other two. Despite this,
states within one class cannot be converted to those in the other class using
LOCC. As a result, we cannot directly compare the degree of entanglement of
states belonging to different classes. This represents a fundamentally
challenging quantum “three-body problem” that complicates the practical
applications of multipartite entanglement. For example, a W state may
outperform in certain applications, while in others, a GHZ state may be more
effective. Note that states belonging to the W class are characterized by only
three real d.o.f., whereas the GHZ class requires five. Hence, a generic
tripartite state typically belongs to the GHZ class.

Traditionally, each link in a network is also “bipartite,” connecting exactly
two nodes. As a result, to study a QN consisting of multipartite entangled
states, it is essential to go beyond “bipartite” network theory and consider
multipartite entangled states as higher-order interactions [33, 34]. These can
be mathematically represented as “hyperedges” of _hypergraphs_ [35]. Here,
Fig. 1 shows an example of a hypergraph-based QN consisting of hyperedges in
the form of

$\displaystyle\left|\phi_{\text{GHZ}}\right\rangle=\cos\delta\left|000\cdots\right\rangle+\sin\delta\left|111\cdots\right\rangle,$

which represents a special case of the GHZ class [Eq. (II.3)]. Of course, this
is only _one_ specific form of multipartite entangled states, characterized by
the sole parameter $\delta$. Yet it illustrates the necessity of representing
these as hypergraphs and studying them through higher-order network theories,
such as higher-order percolation theories (see Section III.3 for a brief
review).

##  III Percolation of complex network

Percolation theory, serving as a foundational model for investigating
disordered systems [17, 36, 18, 37], is mainly concerned with understanding
the geometric connectivity of random media. Constructing a percolation model
is straightforward: Take, for example, a square lattice (or a lattice of any
shape) in which each link is randomly either present with a probability $p$ or
absent with a probability $1-p$. In a real-world application, one could
consider the present links as electrical conductors and the absent ones as
insulators [20]. The electrical current would then flow solely through the
conductor links. When $p$ is small, almost no paths exist that connect the
lattice’s two distant boundaries (e.g., the left and right boundaries in the
square lattice). However, as $p$ grows, various conduction pathways begin to
emerge. A phase transition [38] is eventually triggered when $p$ crosses a
critical threshold, labeled as $p_{\text{th}}$, effectively changing the
composite material from an insulating to a conducting state. At this point,
the probability of a path connecting the two distant boundaries becomes
greater than zero. (This specific probability of connecting distant boundaries
is termed the “sponge-crossing probability,” about which we will delve into
more details in Section V.3).

This phenomenon of a phase transition between two phases of different
_connectivity_ is prevalent in real-life scenarios. An illustrative example
from biology is the spread of epidemics [39, 40, 41, 42, 43, 44, 45, 46, 47].
In its most basic manifestation, an epidemic commences with an infected
individual who, with a probability denoted as $p$, can transmit the infection
to its nearest neighbors over time until it propagates extensively. A
comparable methodology can be applied to model forest fires, where the
probability of a burning tree igniting its nearest neighbor tree in the
subsequent time step replaces the infection probability [48, 49]. Another
notable application of this concept can be found in polymerization processes
within chemistry, where the activation of bonds between small branched
molecules leads to the formation of larger molecules [50, 51]. This
transformation is known as a gelation transition. An illustrative example of
this gelation process can be observed when boiling eggs. Percolation theory
has a wide range of other applications as well, spanning fields such as
disordered quantum systems [52, 53, 54, 55, 56, 57], material science [58, 59,
60], geophysics [61, 62, 63, 64, 65], social dynamics [66, 67, 68, 69], and
infrastructures [70, 71, 72, 73, 74, 75, 76].

###  III.1 Percolation of single-layer networks

Percolation theory is closely associated with a wide range of concepts of
critical phenomena, including scaling laws, fractals, self-organization
criticality, and renormalization, holding significance across diverse
statistical physics disciplines [18]. The traditional characterization of
phase transition in percolation hinges on the statistical properties of
_clusters_ near $p_{\text{th}}$. For $p<p_{\text{th}}$, only finite clusters
exist. As $p>p_{\text{th}}$, a unique, infinite cluster emerges. A crucial
parameter is $P_{\infty}$, signifying the relative size of the infinite
cluster, which exhibits a power-law near $p_{\text{th}}$ [20]:

$P_{\infty}\sim\left|p-p_{\text{th}}\right|^{\beta}.$ (11)

The parameter $P_{\infty}$ serves as a measure of order within the percolation
system and can be identified as the order parameter.

If we exclude the infinite cluster (if it exists), then the rest of the finite
clusters follow a distribution:

$n_{s}\sim s^{-\tau}e^{-s/s^{*}}.$ (12)

Here, $s$ is the cluster size, and $n_{s}$ is the number of clusters of size
$s$. At criticality, the characteristic size $s^{*}$ diverges:

$s^{*}\sim\left|p-p_{\text{th}}\right|^{-1/\sigma}.$ (13)

Consequently, the tail of the distribution $n_{s}$ becomes a power law,
$n_{s}\sim s^{-\tau}$.

The mean cluster size, i.e., how large a finite cluster is on average, also
diverges:

$\langle
s\rangle\sim\sum_{s}s^{2}n_{s}\sim\left|p-p_{\text{th}}\right|^{-\gamma},$
(14)

with the same exponent $\gamma$ above and below $p_{\text{th}}$.

Finally, the correlation length $\xi$, defined as the average distance between
two sites on the same finite cluster, also diverges:

$\xi\sim\left|p-p_{\text{th}}\right|^{-\nu},$ (15)

again, with the same exponent $\nu$ above and below $p_{\text{th}}$.

These exponents, namely $\beta$, $\tau$, $\sigma$, $\gamma$, and $\nu$,
encapsulate the critical behavior of key quantities associated with the
percolation transition and are collectively referred to as the critical
exponents. Notably, they satisfy the _scaling relations_ :

$\beta=\frac{\tau-2}{\sigma}\text{ and }\gamma=\frac{3-\tau}{\sigma}.$ (16)

It is worth emphasizing that these exponents exhibit universality, meaning
they remain invariant irrespective of the specific structural attributes of
the lattice (e.g., square or triangular) or the type of percolation (site or
bond). Instead, their values are solely determined by the dimensionality of
the lattice. Besides, at the critical point, $\xi$ and $s^{*}$ also follow a
relation,

$s^{*}\sim\xi^{d_{f}}.$ (17)

The exponent $d_{f}$ is often called the fractal dimension [20],
characterizing the structure of the infinite cluster at the critical point.
Assuming the dimension of the system is $d$, there is another relation between
critical exponents, called the _hyperscaling relation_ ,

$d_{f}=d-\frac{\beta}{\nu}.$ (18)

Thus, the fractal dimension of the infinite cluster at $p_{\text{th}}$ is not
a new independent exponent but depends on $\beta$, $\nu$ and $d$.

Finally, for complex network structures, similar critical exponents following
Eqs. (11)-(15) can also be identified. For example, in scale-free networks
[77, 6, 78, 79, 80], which are characterized by a power-law distribution
$P(k)\sim k^{-\lambda}$ of its degree $k$, the values of critical exponents
depend on the power-law exponent $\lambda$, as outlined in Table 1. As an
essential process inherently associated with the notion of connectivity in
networked systems, percolation has been generalized to models that go beyond
undirected networks, with studies dedicated to directed networks [81],
temporal networks [82], and, as we discuss in more detail in the next
sections, networks of networks and hypergraphs.

$\lambda$ | $\beta$ | $\gamma$ | $\nu$ | $\sigma$ | $\tau$ | $d_{f}$
---|---|---|---|---|---|---
(2, 3) | 1/(3-$\lambda$) | -1 | ($\lambda$-1)/(3-$\lambda$) | (3-$\lambda$)/($\lambda$-2) | (2$\lambda$-3)/($\lambda$-2) |
(3, 4) | 1/($\lambda$-3) | 1 | ($\lambda$-1)/($\lambda$-3) | ($\lambda$-3)/($\lambda$-2) | (2$\lambda$-3)/($\lambda$-2) | 2($\lambda$-2)/($\lambda$-3)
(4, $\infty$) | 1 | 1 | 3 | 1/2 | 5/2 | 4
Table 1: The critical exponents for the classical percolation transition in
scale-free networks [83, 84].

###  III.2 Percolation of networks of networks

In many real-world systems, an individual network is one component within a
much larger complex network of interdependent networks [85, 86, 87, 88, 89,
90]. In interdependent networks, the failure of nodes in one network leads to
the failure of dependent nodes in other networks, which may cause further
damage to the first network, leading to cascading failures and possibly
catastrophic consequences. In 2010, Buldyrev et al. studied the percolation of
two fully interdependent networks subject to cascading failures based on a
generating function formalism. They found a surprising first-order
discontinuous phase transition, dramatically different from the second-order
continuous phase transition in single-layer networks [91], as shown in Fig. 2.
Later, Parshani et al. studied two partially interdependent networks and found
that the percolation transition changes from first to second order as the
coupling strength decreases [92]. Considering a malicious attack, Huang et al.
developed a mathematical framework for understanding the percolation of two
interdependent networks under targeted attack, later extended to targeted
attack on partially interdependent networks [93]. Each node in one network may
depend on multiple nodes in another network. Therefore, Shao et al. proposed a
theoretical framework for understanding the percolation of interdependent
networks with various support and dependence relationships [94]. The study of
interdependence between networks also led researchers to realize that other
types of interactions are important. One example closely related to
interdependence is antagonistic interactions [95]. Here, for a node to be
active, the antagonistic node in another network has to be active, as can
happen if each pair of nodes competes for some limited resource. This concept
of antagonistic networks was later extended to model competition in dynamic
networks, where highly complex phenomena were observed [96]. Considering that
more than two networks may depend on one another, Gao et al. developed the
analytical framework to study the percolation of a network formed by $n$
interdependent networks [97, 98], which was later extended to the study of
targeted attacks on high-degree nodes [99, 100]. Baxter et al. studied the
percolation of multiplex networks, which can be considered as the percolation
of tree-like network of networks in Ref. [101]. Liu et al. developed a
theoretical framework based on generating functions and percolation theory to
understand the percolation of interdependent directed networks [102, 103]. In
the past decade, we have witnessed fruitful results and discoveries related to
the percolation of networks of networks [104, 105, 106, 107, 108, 109, 110,
111, 112, 113, 114, 115, 116, 117], as well as multilayer networks and
interconnected networks.

Figure 2: Schematic demonstration of first and second order percolation
transitions. In the second order case, the giant component is continuously
approaching zero at the percolation threshold $p=p_{\text{th}}$. In the first
order case the giant component approaches zero discontinuously. The figure is
from [85].

In general, the percolation of networks of networks extends that of single-
layer networks. For example, when the $n$ interdependent Erdős–Rényi (ER)
networks form a tree-like topology and have the same average degree $\bar{k}$,
$\bar{k}_{i}=\bar{k}$ ($i=1,2,...,n$), the giant connected component in each
layer, $P_{\infty}$, as a function of $\bar{k},p,$ and $n$ follows [86] :

$P_{\infty}=p[1-\exp(-\bar{k}P_{\infty})]^{n}.$ (19)

Note that for Eq. (19), the particular case $n=1$ is the known ER second order
percolation law for a single-layer network [118, 119, 120]. When $n\geq 2$,
the system shows a first-order phase transition. Using the generating
function, we obtain that $p_{\text{th}}$ and $P_{\infty}|_{p\to
p_{\text{th}}^{+}}$ satisfy

$p_{\text{th}}=-\frac{w}{\bar{k}[1+1/(nw)]^{n-1}},$ (20)

and

$P_{\infty}|_{p\to p_{\text{th}}^{+}}=-(w+1/n)/\bar{k}.$ (21)

where $w$ is given by $w=W_{-}(-1/n\exp(-1/n))$, and $W_{-}(x)$ is the
smallest of the two real roots of the Lambert equation $\exp(W_{-})W_{-}=x$.
For $n=1$ we obtain the known ER results $p_{\text{th}}=1/\bar{k}$, and
$P_{\infty}|_{p\to p_{\text{th}}^{+}}=0$ at $p=p_{\text{th}}$. Substituting
$n=2$ in Eqs. (20) and (21), we obtain the exact results derived by Buldyrev
et al. [91].

###  III.3 Percolation of hypergraphs

Hypergraphs generalize graphs by allowing that interactions, the hyperedges,
connect an arbitrary number of vertices [121]. Hypergraphs, and, up to a
certain extent, simplicial complexes, offer more flexibility to model
interacting systems, and they have become popular models of many real-world
networks over recent years [122, 123]. For example, more than two molecules
can participate in some reactions [124, 125], and group interactions also
frequently occur for collaborations of scientific papers [126, 127]. It has
been shown that higher-order interaction may significantly change the physical
properties of dynamical processes from those on ordinary networks with only
pairwise connections [128, 122, 33, 129]. However, there are only a few works
exploring the robustness or the percolation of hypergraphs [130, 131, 132,
133, 134, 135]. Specifically, Coutinho et al. introduced two generalizations
of core percolation to hypergraphs, and offered analytical solutions to
certain types of random hypergraphs accordingly [130]. Sun and Bianconi later
proposed a general framework for accessing hypergraph robustness, and further
characterized the critical properties of simple and higher-order percolation
processes [131]. Sun et al. also considered a paradigmatic type of higher-
order interactions, triadic interactions, where a node regulates the
interaction between two other nodes, and provided a general theory, accurately
predicting the full phase diagram on random graphs [132]. More recently,
Bianconi and Dorogovtsev have further developed a theory for hyperedge and
node percolation on hypergraphs, and showed that, in contrast to ordinary
networks, the node and hyperedge percolation problems for hypergraphs strongly
differ from each other [136].

##  IV Classical percolation in quantum networks

Why is percolation theory useful in the study of QN? The roots of this
interest can be traced back to a 2007 paper [10]. In the seminal work, the
authors first proposed a mapping between percolation theory and a particular
entanglement transmission scheme, which they discovered and accordingly termed
as the classical entanglement percolation (CEP) scheme. Within this context,
an entanglement transmission _scheme_ refers to a (possibly infinite) series
of quantum communication protocols that may be applied collectively to a QN
for distributing entanglement between two nodes. This pioneering discovery has
opened up a new approach to studying QN from a _statistical physics_
perspective, with a focus on understanding the large-scale, collective
characteristics of the entanglement transmission task and how they are
influenced by the topology of the QN.

###  IV.1 Classical entanglement percolation (CEP)

As previously noted, the LOCC cannot increase the average entanglement.
However, it does not mean that one cannot use LOCC as a form of “gambling”—to
enhance the entanglement with a certain probability $p$, even though it might
reduce the entanglement with probability $1-p$. This principle forms the
foundation of the CEP scheme. To be specific, the CEP scheme involves two
steps [10]. First, we “gamble” to enhance the entanglement of each link,
aiming to obtain a singlet (maximally entangled state) with a probability of
$p$. The optimal probability for this is referred to as the singlet conversion
probability, given by $p=2\sin^{2}\theta$. Second, if a path of links
connecting the source ($s$) and target ($t$) has all been converted to
singlets, then a specific protocol known as entanglement swapping can be
applied. This protocol converts every two singlet links sharing a common node
(Relay, R) into a single singlet linking the two end nodes. For example, if
there is a singlet between Alice and the Relay, and another between the Relay
and Bob, the entanglement swapping protocol can merge the two into one singlet
between Alice and Bob. By applying this protocol recursively along the singlet
path connecting $s$ and $t$, we arrive at a final singlet between $s$ and $t$,
fulfilling the transmission task.

Equipped with these concepts, the mapping between CEP and (classical)
percolation theory is straightforward. The probability $p=2\sin^{2}\theta$
represents the probability for links to be present or absent. The CEP scheme
succeeds if $s$ and $t$ are connected after the random percolation process is
applied. Furthermore, this connection implies a nontrivial critical threshold
for the CEP scheme on infinitely large QN. Specifically, when
$2\sin^{2}\theta$ falls below the percolation threshold $p_{\text{th}}$, $s$
and $t$ are almost certainly disconnected if they are infinitely apart. Hence,
$p_{\text{th}}$, which solely depends on the network topology, serves as a
metric of the overall capacity of the QN in the context of CEP.

The CEP scheme represents a great simplification of the QN entanglement
transmission task to a pure percolation problem. Nevertheless, the CEP is not
necessarily the optimal scheme. Indeed, even when $2\sin^{2}\theta\leq
p_{\text{th}}$, there might still be other schemes that can fulfill the
transmission task, as we will explore in the following sections.

###  IV.2 Quantum entanglement percolation (QEP)

It is expensive to obtain a singlet from a partially entangled state given its
“gambling” nature. Even worse, the swapping protocol spends all the singlets
along a path and converts them into just one singlet. This process leads to a
waste of singlets and causes the inefficiency of the CEP scheme. Naturally,
this leads to the question: Is it necessary to convert _every_ link into a
singlet? As we will see, the answer is negative, paving the way for the QEP
scheme [10].

The QEP scheme is based on the discovery that given two partially entangled
states between three parties, Alice–Relay–Bob, there exists a LOCC protocol
that can yield a _higher_ probability of obtaining a singlet between Alice and
Bob. This probability is higher than obtaining two singlets (Alice–Relay,
Relay–Bob) individually and followed by a swapping protocol. Indeed, the
optimal probability is found to be
$\min\\{2\sin^{2}\theta_{\text{AR}},2\sin^{2}\theta_{\text{RB}}\\}$ [137],
outperforming the probability
$\left(2\sin^{2}\theta_{\text{AR}}\right)\left(2\sin^{2}\theta_{\text{RB}}\right)$
achieved by CEP.

What about three partially entangled states between four parties
(Alice–Relay1–Relay2–Bob)? Unfortunately, the optimal conversion probability
does not intuitively simplify to
$\min\\{2\sin^{2}\theta_{\text{A}\text{R}_{1}},2\sin^{2}\theta_{\text{R}_{1}\text{R}_{2}},2\sin^{2}\theta_{\text{R}_{2}\text{B}}\\}$,
but takes on a much more complicated form. This forbids us to generalize the
optimal result to larger scale. For readers who wish to delve deeper, further
details can be found in Ref. [138].

Even though we cannot generalize the optimal result, we can still extend the
_improvement_ , by bypassing one relay every other step. This gives rise to
the QEP scheme, which avoids the need to create singlets for every link,
bypassing (half of) the Relays. But this approach is not without its trade-
offs, especially on a large scale. Since the Relays are bypassed, the QN
misses out on the potential connectivity to other paths through the Relays.
Thus, it still remains a question whether QEP can achieve a lower critical
threshold than CEP, fulfilling the entanglement transmission task at infinite
scales. The authors of the 2007 paper [10] demonstrated that this is indeed
achievable for specific topologies, such as a “double”-honeycomb topology,
where there are two links between every two adjacent nodes on a hexagonal
network. The QEP scheme is equivalent to adding a preprocessing step,
modifying the network into a triangular structure and thereby reducing the
percolation threshold.

Note that despite being referred to as QEP, the mapping of the scheme is still
aligned with classical percolation theory from a statistical physics point of
view. The quantum aspect of this process is confined primarily to the
preprocessing step, which is executed only at the local scale. Additionally,
the QEP does not yield the optimal result either [138], which leaves open the
question of whether a more effective entanglement transmission scheme might
exist. It would be intriguing if this new scheme were guided by a
fundamentally different statistical physical theory distinct from classical
percolation. We will show that such a theory does exist.

Figure 3: Entanglement transmission on quantum networks can be understood from
two different mappings. (a) Classical percolation, where each link is present
or absent with probability $p$ or $1-p$, respectively. Only the paths in which
all links are fully present contribute to the connectivity. This forms the
basis of the classical/quantum entanglement percolation (CEP/QEP) schemes
[10], with the goal of securing a singlet between source $s$ and target $t$
through a “gambling” approach. (b) Concurrence percolation, where every path
contributes to the connectivity. This mapping forms the basis of the
deterministic entanglement transmission (DET) scheme [139], where the aim is
not to obtain a singlet probabilistically but to establish a partially
entangled state deterministically.

##  V Concurrence percolation in quantum networks

###  V.1 No need to establish singlets

The mapping of CEP/QEP to classical percolation is essentially established on
the necessity that _two nodes must be connected by at least one path of
singlets_. However, we have seen that in the QEP scheme, it is not mandatory
for all links along the path to be converted to singlets. By bypassing some of
the links, a more efficient scheme might be realized.

This observation leads us to a natural question: _why not stop establishing
singlets at all?_ In other words, we would bypass not just some, but all
links, resulting in a final state between the source $s$ and target $t$ that
remains only partially, rather than maximally, entangled. This scenario is, of
course, approachable, considering that one can always “downgrade” a singlet to
a partially entangled state with no cost [140]. What we truly seek, however,
is a trade-off where we can achieve a much higher probability of obtaining
such a partially entangled state instead of a singlet. By carefully weighing
the compromise (having only partial entanglement) against the benefit (a
significantly higher conversion probability), we might discover a more
advantageous scheme for entanglement transmission overall. This revised
approach challenges the conventional thinking in terms of classical
percolation and could lead to new opportunities in developing new schemes on
QN.

###  V.2 Deterministic entanglement transmission (DET)

Based on the above ideas, a new scheme named the deterministic entanglement
transmission (DET) scheme is introduced [139]. The DET approaches the
entanglement transmission task from a completely distinct perspective: the
scheme demands that the conversion probability throughout the process always
equals _one_. In other words, rather than “gambling” to increase the links’
entanglement, we operate directly on partially entangled states in a
deterministic fashion. The aim of DET is to maximize the final (partial)
entanglement under the constraint of determinacy, contrasting with CEP/QEP’s
objective of always acquiring a singlet with high (but not unit) probability.

The DET involves two quantum communication protocols: The first is a
continuation of the swapping protocol [141, 137]. However, here the swapping
protocol operates directly on partially entangled states. It can be shown that
given entanglements $\theta_{\text{AR}}$ and $\theta_{\text{RB}}$ in the A–R–B
configuration, one can tune the swapping protocol such that it
deterministically yields a final state between A and B, having a new
entanglement $\theta_{\text{AB}}$ that satisfies $\sin
2\theta_{\text{AB}}=\sin 2\theta_{\text{AR}}\sin 2\theta_{\text{RB}}$ [139].
The second protocol is the entanglement concentration protocol [142, 143].
This protocol takes two links between A and B (with entanglement $\theta_{1}$,
$\theta_{2}$, respectively) as input. At the expense of the two links, a new
link that has a higher entanglement $\theta$ is produced between A and B,
where $\cos\theta=\cos\theta_{1}\cos\theta_{2}$ or $\sqrt{1/2}$, whichever is
the largest.

The DET scheme is founded on the generalization of these two protocols to
global scales. This is possible since the swapping and concentration protocols
are fully equivalent to the series/parallel rules, respectively, as often
employed in circuit network analysis [139]. Therefore, the DET scheme becomes
applicable if the network topology between the source $s$ and target $t$ is
series-parallel [144]. In other words, the network can be fully reduced to a
link between $s$ and $t$ using only series and parallel reductions. Note that
there are already fast network algorithms designed specifically for detecting
series-parallel networks and determining their feasible reductions, which will
be explored further in the next section.

One of the most intriguing characteristics of the DET scheme is that, when
applied to infinitely large series-parallel networks, a threshold similar to
the CEP threshold is observed [21], below which the DET can never produce
nonzero entanglement between $s$ and $t$. This threshold, however, is _lower_
than the CEP threshold, demonstrating a “quantum advantage” over the classical
scheme. The existence of such a threshold suggests that the DET may be
globally governed by a statistical physics theory. In subsequent sections, we
will explore this statistical theory in details.

###  V.3 Concurrence percolation theory

To establish the statistical theory, recall that in classical percolation
theory, given a regular lattice, one can define a “sponge-crossing” quantity,
$P_{\text{SC}}$, as the probability that there is an open path connecting two
far-apart boundaries [145]. When the lattice becomes infinitely large (the
number of nodes $N\to\infty$), it is known that a minimum value of $p$ exists,
below which $P_{\text{SC}}$ becomes zero in the thermodynamic limit:

$\displaystyle
p_{\text{th}}=\text{inf}\left\\{p\subset[0,1]|\text{lim}_{N\rightarrow\infty}P_{\text{SC}}>0\right\\}.$
(22)

This minimum value coincides with the traditionally defined percolation
threshold $p_{\text{th}}$ in Section III, which is based on the size of the
infinite cluster. As a result, Eq. (22) offers an alternative definition of
$p_{\text{th}}$. In the special case of two-dimensional square lattices,
Kesten proved that the “sponge-crossing” probability $P_{\text{SC}}$ of
connecting the left and right boundaries is strictly zero until $p>1/2$. Thus,
$p_{\text{th}}=1/2$ [145].

Moreover, the existence of such a critical threshold is not limited to regular
lattices but can also be observed for complex network topologies. All we need
to do is to generalize $P_{\text{SC}}$ from being defined between two apparent
boundaries to two arbitrary sets of nodes, denoted $S$ and $T$. It is
reasonable to believe that there still exists a nontrivial ConPT threshold
$p_{\text{th}}$ for this generalized $P_{\text{SC}}$, as long as the minimum
length of all paths connecting $S$ and $T$ increases with the network size
$N$. We contract the two sets $S$ and $T$ into two “mega” nodes, which amounts
to erasing the internal network topologies of $S$ and $T$, and then calculate
the “sponge-crossing” probability $P_{\text{SC}}$ between them. This provides
us a definitive way of calculating $P_{\text{SC}}$ for arbitrary network
topology and inferring $p_{\text{th}}$ from Eq. (22).

How to derive $p_{\text{th}}$? First, we need to know how $P_{\text{SC}}$
manifests as a function of $p$. The exact expression of $P_{\text{SC}}$ can be
calculated by basic addition and multiplication rules of probability measures
[18]. In general, $P_{\text{SC}}$ bears the form as a ratio of two large
polynomials of $p$ (i.e., meromorphic in $p$), which quickly becomes complex
when the number of links becomes large. Nevertheless, when the network
topology between $S$ and $T$ is series-parallel, then $P_{\text{SC}}$ can be
decomposed into the iteration of two connectivity rules, namely, the series
and parallel rules (Table 2). these rules establish the probability of _at
least_ one path connecting the two ends.

Table 2: Connectivity rules that define the classical/concurrence percolation
theories.

|  Classical |  Concurrence
---|---|---
Series rule |  $p=p_{1}p_{2}\cdots$ |  $c=c_{1}c_{2}\cdots$
Parallel rule |  $1-p=$
$\left(1-p_{1}\right)\left(1-p_{2}\right)\cdots$ |  $\frac{1+\sqrt{1-c^{2}}}{2}=\max\\{\frac{1}{2},$
$\frac{1+\sqrt{1-c_{1}^{2}}}{2}\frac{1+\sqrt{1-c_{2}^{2}}}{2}\cdots\\}$
Higher-order rules |  Can be approximated by the _star-mesh transform_ , by the following two-step argument:

1\. The star-mesh transform can reduce an $N$-graph to an $(N-1)$-graph (right
panel) and is solvable by applying the series and parallel rules recursively
through a group of $N(N-1)/2$ coupled equations (see the Algorithms section
for details).
2\. Applying the transform consecutively on a network can reduce nodes one by one—and thus reduce any topology to two nodes, yielding the final (approximate) connectivity between them (bottom panel, i. $\to$ viii.). |
---|---

When the network topology between $S$ and $T$ is not series-parallel (such as
a bridge circuit [144]), higher-order connectivity rules are needed. Unlike
the series/parallel rules, these higher-order rules do not follow a general
form. That being said, there is a way to approximate (see the Algorithms
section for a detailed discussion about the nature of this approximation)
these higher-order rules using _only_ the series/parallel rules. This
technique is know as the star-mesh transform, originating from the study of
circuit analysis. We will include more details on this technique in the
Algorithms section.

It becomes clear that the series/parallel rules play a very special rule in
defining percolation connectivity. Given that the DET scheme is also founded
on series/parallel rules, a natural and intriguing question arises: can we
define a new statistical theory _reversely_ , starting directly from the exact
forms of series/parallel rules? Such a definition may not be complete, since
we do _not_ know the exact forms for higher-order rules (which may not even
have a closed mathematical form). Yet, using the star-mesh transform
technique, it may be possible that we can approximate these higher-order rules
using the series/parallel rules that we have known.

This is the motivation of the concurrence percolation theory. Recall that the
DET series/parallel rules are given by $\sin 2\theta=\sin 2\theta_{1}\sin
2\theta_{2}$ and
$\cos\theta=\max\\{\sqrt{1/2},\cos\theta_{1}\cos\theta_{2}\\}$, respectively.
Under a change of variable $c\equiv\sin 2\theta$, the rules can be rewritten
in the form presented in Table 2. Note that the new series rule in terms of
$c$ bears the same nominal form as the classical series rule in terms of $p$.
This variable $c$, indeed, has a physical meaning, known as the _concurrence_
, a specific measure of bipartite entanglement [146]. This inspires the new
theory to be termed “concurrence percolation.” In comparison to classical
percolation where the variable of interest is the probability $p$, in
concurrence percolation, choosing $c$ as the variable of interest is
appropriate.

After fixing all connectivity rules (series + parallel + star-mesh), an
analogous quantity, $C_{\text{SC}}$, referred to as the sponge-crossing
concurrence can be defined as the weighted sum of all paths in terms of this
new weight $c$ [21]. It is believed that a nontrivial threshold on $c$ also
exists:

$\displaystyle
c_{\text{th}}=\text{inf}\left\\{c\subset[0,1]|\text{lim}_{N\rightarrow\infty}C_{\text{SC}}>0\right\\},$
(23)

such that $c_{\text{th}}$ is the minimum value of the concurrence $c$ per
link, below which $C_{\text{SC}}$ becomes zero when $S$ and $T$ become
infinitely distant.

###  V.4 Results

_DET outperforms CEP_.—Utilizing the framework of concurrence percolation, we
can derive an essential and powerful result: the DET scheme always
_outperforms_ the CEP scheme on any series-parallel QN. To rigorously
demonstrate this comparative superiority, we rewrite both the classical and
concurrence series/parallel rules in terms of the entanglement variable
$\theta$ (Table 3). These rules correspond to the entanglement transmission
rules for CEP and DET, respectively (as illustrated in Fig. 4).

Table 3: Entanglement transmission rules for the CEP and DET schemes, derived from the classical and concurrence percolation rules (Table 2), respectively.  | CEP | DET
---|---|---
Series | $2\sin^{2}\theta=\left(2\sin^{2}\theta_{1}\right)\left(2\sin^{2}\theta_{2}\right)\cdots$ | $\sin 2\theta=\left(\sin 2\theta_{1}\right)\left(\sin 2\theta_{2}\right)\cdots$
Parallel | $\cos 2\theta=\left(\cos 2\theta_{1}\right)\left(\cos 2\theta_{2}\right)\cdots$ | $\cos\theta=\max\\{\sqrt{1/2},\left(\cos\theta_{1}\right)\left(\cos\theta_{2}\right)\cdots\\}$
Figure 4: The DET series/parallel rules outperform the CEP series/parallel
rules.

Now, for the series rule, we have

$\displaystyle c^{2}$ $\displaystyle=$ $\displaystyle\prod_{i}\left(\sin
2\theta_{i}\right)^{2}$ (24) $\displaystyle=$
$\displaystyle\prod_{i}\left(2\sin^{2}\theta_{i}\right)\left(2-2\sin^{2}\theta_{i}\right)$
$\displaystyle\geq$
$\displaystyle\left[\prod_{i}\left(2\sin^{2}\theta_{i}\right)\right]\left[2-\prod_{i}\left(2\sin^{2}\theta_{i}\right)\right]$
$\displaystyle=$ $\displaystyle p\left(2-p\right),$

where the inequality is supported by the subadditivity of
$f(x)=\ln(2-e^{-x})$, namely,

$f(x_{1}+x_{2}+\cdots)\leq f(x_{1})+f(x_{2})+\cdots$ (25)

for $x=-\ln\left(2\sin^{2}\theta\right)\geq 0$. This leads to

$1-c^{2}\leq 1-p\left(2-p\right)=\left(1-p\right)^{2}.$ (26)

This final inequality underscores that the $\theta$ obtained from the CEP
series rule (under a change of variable $p=2\sin^{2}\theta$) is never greater
than the $\theta$ obtained from the DET series rule (under a change of
variable $c=\sin 2\theta$).

For the parallel rule, similarly we have

$\displaystyle\frac{1}{2}+\frac{1}{2}\sqrt{1-c^{2}}$ $\displaystyle=$
$\displaystyle\prod_{i}\cos^{2}\theta_{i}$ (27) $\displaystyle=$
$\displaystyle\prod_{i}\left(\frac{1}{2}+\frac{1-2\sin^{2}\theta_{i}}{2}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}+\frac{1}{2}\prod_{i}\left(1-2\sin^{2}\theta_{i}\right)$
$\displaystyle=$ $\displaystyle 1-\frac{p}{2},$

where the inequality is supported by the subadditivity of
$f(x)=-\ln(1/2+e^{-x}/2)$ for $x=-\ln\left(1-2\sin^{2}\theta\right)\geq 0$.
This further leads to

$\frac{\sqrt{1-c^{2}}}{2}\leq 1-\frac{p}{2}-\frac{1}{2}=\frac{1-p}{2},$ (28)

which, again, underscores that the $\theta$ obtained from the CEP parallel
rule is never greater than the $\theta$ obtained from the DET parallel rule.
Together, it can be established that the DET rules consistently yield superior
results to those of the CEP rules, both in series and parallel configurations.
This underlines the potential of DET as a valuable tool in the ongoing
development of large-scale QN and adds a new dimension to our understanding of
quantum connectivity.

(a)

(b)

Figure 5: Classical percolation and concurrence percolation on the Bethe
lattice. 5 The Bethe lattice ($k=3$). 5 The sponge-crossing probability
$P_{\text{SC}}$ (brown) between sets $S$ (the root) and $T$ (the collection of
all leaf nodes) as a function of $\theta$. Driven by classical percolation, a
transition threshold is found at $\theta=\pi/6$, or
$p_{\text{th}}=2\sin^{2}\theta=1/2$. As a comparison, the sponge-crossing
concurrence $C_{\text{SC}}$ (red), driven by concurrence percolation, shows a
similar but lower threshold at $\theta=\pi/8$, or $c_{\text{th}}=\sin
2\theta=1/\sqrt{2}$. Moreover, a saturation point at
$\theta=0.633\left(\pi/4\right)$, or $c_{\text{sat}}\approx 0.838$ also
exists, beyond which we already have $C_{\text{SC}}=1$. This saturating
feature has no counterpart in classical percolation.

_Concurrence percolation threshold_.—In infinite-size QNs, both a classical
percolation threshold $p_{\text{th}}$ and a concurrence percolation threshold
$c_{\text{th}}$ exist. This leads us to the second insightful finding within
the realm of concurrence percolation: the prediction of a lower threshold
compared to what was known from earlier classical-percolation-theory-based
schemes, including CEP and its variants (such as QEP). What makes this result
particularly interesting is that the improvement exists across various network
topologies. Table 4 shows these findings, detailing tests conducted on
different network topologies, including the Bethe lattice (Fig. 5) as well as
other regular lattices such as the square, honeycomb, and triangular lattices
(Fig. 6). This consistency across multiple configurations underscores the
robustness of the concurrence percolation method, demonstrating its potential
to redefine our understanding of entanglement transmission within large-scale
QNs.

On series-parallel QN, this predicted concurrence percolation threshold is
readily achievable using the DET scheme. On general network topologies,
however, it is unknown if the higher-order connectivity rules produced by the
star-mesh transform is realizable by LOCC. They are only approximations of the
true LOCC-allowing rules. The study of the higher-order rules of concurrence
percolation remains a difficult task that could be handled by multipartite
strategies [147], QN routing [148, 149], or QN coding [150].

Table 4:  The concurrence percolation threshold is the lowest threshold compared to earlier known classical-percolation-theory-based schemes, including CEP and its variants. Particularly, for the Bethe lattice, one has $P_{\text{swap}}(k)=2x-x^{2}$, where $x(k)$ is the solution of $2x+x^{k}(kx-x-k-1)-(1-x)/(k-1)=0$ by the $q$-swapping strategy [151], and $P_{\text{GHZ}}(k)=y$, where $y(k)$ is the solution of $1-(1-y)\sum_{i=0}^{\lfloor k/2-1\rfloor}{\binom{2i}{i}4^{-i}(2y-y^{2})^{i}}-1/(k-1)=0$ where $\lfloor\cdot\rfloor$ is the floor function [147]. For the square and triangular lattices, QEP yields the same thresholds as of CEP.  [unit: $\left(\pi/4\right)^{-1}\theta$] |  Bethe lattice (degree $k$) |  Square |  Honeycomb |  Triangular
---|---|---|---|---
CEP [10] |  $(4/\pi)\sin^{-1}[1/{\sqrt{2\left(k-1\right)}}]$ |  $0.670$ |  $0.777$ |  $0.545$
QEP [10, 151, 138] |  $(4/\pi)\sin^{-1}\sqrt{P_{\text{swap}}(k)/2}$11footnotemark: 1 |  $0.670$ |  $0.761$ |  $0.545$
QEP-GHZ [147] |  $(4/\pi)\sin^{-1}\sqrt{P_{\text{GHZ}}(k)/2}$22footnotemark: 2 |  $0.584$ |  $0.745$ |  $0.481$
Concurrence [21] |  $(2/\pi)\sin^{-1}(1/{\sqrt{k-1}})$ |  $0.42(8)$ |  $0.51(8)$ |  $0.32(8)$

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6: Classical percolation and concurrence percolation on 6-6 the square
lattice, 6-6 the honeycomb lattice, and 6-6 the triangular lattice, The
sponge-crossing probability $P_{\text{SC}}$ (brown) and sponge-crossing
concurrence $C_{\text{SC}}$ (red) are defined between sets $S$ (the collection
of nodes on the left boundary) and $T$ (the collection of nodes on the right
boundary), as a function of $\theta$.

_Saturation_.—Concurrence percolation also differs from classical percolation
with the existence of a saturation point $c_{\text{sat}}$. Whenever $c\geq
c_{\text{sat}}$, the sponge-crossing $C_{\text{SC}}$ consistently equals one
[Fig. 5]. For example, basic calculations show that the exact value of the
saturation point for a Bethe lattice of degree $k$ is given by [21]

$c_{\text{sat}}=\sqrt{(1/2)^{1/k}-(1/4)^{1/k}}/\sqrt{(1/2)^{(k-1)/k}-(1/4)^{(k-1)/k}}.$
(29)

In contrast, in classical percolation, $P_{\text{SC}}$ equals one if and only
if $p=1$. This phenomenon originates from the anomaly of the parallel rule
(Table 2) being not a smooth function. The presence of this saturation point
unveils a new “quantum advantage” originating only from concurrence
percolation: _one can deterministically establish a singlet as long as the
entanglement in each link surpasses the saturation point._ This advantage
stands in contrast to schemes based on classical percolation, where a singlet
can only be established with certainty if each link is perfectly entangled.

(a)

(b)

(f)

(g)

(c)

(d)

(h)

(i)

(e)

(j)

Figure 7:  Critical phenomena of 7–7 classical percolation and 7–7 concurrence
percolation theories in the Bethe lattice.

_Critical exponents_.—Lastly, similar to classical percolation, concurrence
percolation also shows critical phenomena, marked by a set of dependent or
independent critical exponents [3]. However, it is important to note that
concurrence percolation is defined based on _connectivity rules_ (Table 2),
not _clusters_. As a result, one cannot simply deduce a traditional cluster-
based order parameter from the variable $c$ used in these rules. In fact, an
effective cluster could be defined using $c$, $c^{2}$ or any other power of
$c$. Altering the definition in this way essentially results in a variable
change in the connectivity rules but does not change the underlying physics.

In the absence of a cluster definition, the sole critical exponent that can be
determined is the dynamic thermal exponent, $\nu_{\text{dyn}}=z\nu$. This
exponent is tied solely to length dimensions, reflecting how the system’s
correlation length diverges as $c$ approaches $c_{\text{th}}$. Note that the
length in the context refers to chemical length, not the conventional
Euclidean length. The two length scales are related by the dynamic critical
exponent $z$ [19]. This is why $z\nu$, rather than $\nu$, is used in this
case.

Importantly, the dynamic thermal exponent $z\nu$ can be retrieved from finite-
size analysis [3]. Here, the idea is that the correlation length can be
replaced by the system’s finite length scale $l$ when
$\left|c-c_{\text{th}}\right|\to 0$. Therefore, near the critical threshold,
all dependence on $\xi$ can be deduced using $l$. The finite-size analysis
results for both classical and concurrence percolation for the Bethe lattice
are shown in Fig. 7. For concurrence percolation, it is found that
$C_{\text{SC}}$ follows a power law with an exponential cutoff as a function
of the number of layers $l$, $C_{\text{SC}}\sim l^{-1/2}\exp(-l/l^{*})$. Here,
$l^{*}$ diverges as a power law as $c$ approaches $c_{\text{th}}$ [Fig. 7].
The numerical value $z\nu=1.082(95)$ is obtained by fitting near
$|c-c_{\text{th}}|\sim 10^{-5}$ [Fig. 7].

Alternatively, $z\nu$ can also be determined by looking at the _finite-size_
critical threshold $c_{\text{th}}(l)$, which is defined as the turning point
of $C_{\text{SC}}$,

$c_{\text{th}}(l)=c|_{\partial^{2}C_{\text{SC}}/\partial{c}^{2}=0},$ (30)

which deviates from $c_{\text{th}}$ as a power law with respect to $l$ [Fig.
7]. Again, the numerical value $1/(z\nu)=0.99(5)$ is obtained near $l\sim
10^{4}$ [Fig. 7].

For general $k$, different $c_{\text{th}}$ and $c_{\text{sat}}$ are also
presented [Fig. 7], revealing two universal (i.e., independent of $k$) power
laws of $C_{\text{SC}}$ near $c\to c_{\text{th}}$ and $c\to c_{\text{sat}}$,
respectively, supported by numerical results (dots) on a finite Bethe lattice
of $l=500$. In particular, the power-law relation
$C_{\text{SC}}\sim|c-c_{\text{th}}|^{1/2}$ is reminiscent of the critical
exponent $\beta$ in classical percolation, which follows
$P_{\inf}\sim|p-p_{\text{th}}|^{\beta}$ [Eq. (11)]. Yet, as previously
discussed, $C_{\text{SC}}$ cannot be uniquely equated to a “cluster-based”
order parameter. Thus, it would be premature to assert that $\beta=1/2$
without accounting for certain nuances.

Note that the above results also have their counterparts in classical
percolation [Figs. 7-7] except for the saturation point $c_{\text{sat}}$. It
is found that the critical exponent $z\nu$ is the same for both classical and
concurrence percolation theories on the Bethe lattice. It thus remains unknown
whether the classical and concurrence percolation theories belong to the same
universality class or not.

In conclusion, the concept of concurrence percolation establishes a new theory
that governs the behavior of entanglement transmission across large-scale QN.
This novel theory brings in several unique characteristics that distinguish it
from classical percolation, thereby providing a refreshing and rich
perspective on QN. We believe that the theoretical framework set by
concurrence percolation may also open doors to practical applications, such as
more efficient entanglement transmission schemes or novel protocols for
quantum communication and computation. In essence, concurrence percolation not
only enriches our comprehension of the inherent complexity of QN but also
signifies a leap towards a more refined and versatile understanding of the
statistical physics that governs QN.

##  VI Algorithms

This section is dedicated to a comprehensive exploration of the fundamental
algorithms that have played a pivotal role in our investigation of concurrence
percolation theory. Not only are these algorithms instrumental in the analysis
and understanding of QN, but they also serve as essential tools for modeling
and simulating the complex behaviors within the network.

(a) Simple series QN.

(b) Simple parallel QN.

(c) Parallel-then-series QN.

(d) Series-then-parallel QN.

(e) Series-parallel QN.

(f) General QN.

Figure 8: Different QN topologies between $S$ and $T$. 8 Series. 8 Parallel. 8
Parallel-then-series. 8 Series-then-parallel. 8 Series-parallel. 8 Non-series-
parallel.

###  VI.1 Identification of series-parallel networks

Series-parallel networks were introduced by Duffin [144] as a mathematical
model of electrical networks, and a general version was introduced later by
Lawler [152] and Monma and Sidney [153] as a model for scheduling problems.
The classification of a network as series-parallel depends on the choice of
two specific nodes of interest [144]. Given two source and target nodes $S$
and $T$, the network topology can be grouped into different categories (Fig.
8). All topologies between $S$ and $T$ given in Figs. 8–8 are considered
series-parallel, except Fig. 8, due to an existing “bridge” in addition to
Fig. 8. Importantly, it is worth highlighting that many realistic complex
networks can be approximated as series-parallel. This is because, in infinite-
dimensional systems, cycles can typically be ignored through the Bethe
approximation [154].

It is known that when a “decomposition tree” (Fig. 9) for a series-parallel
graph is given, many problems, including those that are NP-hard for arbitrary
graphs [155, 156, 157, 158], can be solved in linear time. While series-
parallel networks continue to play an important role in various applications,
they have been extensively studied in their own right as well as in relation
to other optimization problems (cf. [159, 160, 161, 162]). We also refer to
[163] for more results in series-parallel graphs.

Series-parallel networks enjoy nice algorithmic properties. There is a fast
algorithm that determines whether any given network is series-parallel, and if
it is, also returns the decomposition tree that is suitable to be used in the
following applications [164]. Following this work, researchers have further
developed parallelized algorithms to determine the important class of series-
parallel networks [165, 166, 167, 168].

###  VI.2 Star-mesh transform

The star-mesh transform (also known as the Kron reduction [169]) was
originally developed as a circuit analysis technique for calculating the
effective resistance in resistor networks. The star-mesh transform replaces a
local star network topology by a mesh topology (a complete graph).
Importantly, the equivalent resistance between each pair of nodes remains
consistent before and after the transformation. Here, we generalize this idea
to offer an approximation method for percolation on networks. This approach
bears similarity to the real-space renormalization group (RG) methods used in
percolation theory. However, the star-mesh transform is more versatile,
applicable to various types of networks beyond regular lattices.

Figure 9:  Decomposition of a series-parallel network to the final base graph
(from i to viii). At each step, the links that the series rule and the
parallel rule are applied to are highlighted in orange and cyan, respectively.

A star-mesh transform [170] can be built upon only series and parallel rules
but not higher-order rules to map an $N$-node star graph to an $(N-1)$-node
complete graph, establishing a local equivalence (in terms of connectivity)
between the two graphs. Mathematically, we denote $\mathcal{G}(N)$ to be a
star graph with one root vertex and $N$ leaf vertices, where the weights of
the $N$ edges are from $\theta_{1}$ to $\theta_{N}$. And the $N$-complete
graph transformed from $\mathcal{G}(N)$ is denoted as
$\mathcal{G}^{\prime}(N)$, the weights of its $N(N-1)/2$ edges are
$(\theta_{12},\theta_{13},\cdots,\theta_{1N},\cdots,\theta_{N-1,N})$. The
equivalence between $\mathcal{G}(N)$ and $\mathcal{G}^{\prime}(N)$ are
formatted $N(N-1)/2$ independent equations:

$\displaystyle\text{seri}\left(\theta_{1},\theta_{2}\right)$ $\displaystyle=$
$\displaystyle c\left(1,2;\mathcal{G}^{\prime}\left(N\right)\right),$
$\displaystyle\text{seri}\left(\theta_{1},\theta_{3}\right)$ $\displaystyle=$
$\displaystyle c\left(1,3;\mathcal{G}^{\prime}\left(N\right)\right),$
$\displaystyle\cdots,$
$\displaystyle\text{seri}\left(\theta_{1},\theta_{N}\right)$ $\displaystyle=$
$\displaystyle c\left(1,N;\mathcal{G}^{\prime}\left(N\right)\right),$
$\displaystyle\cdots,$
$\displaystyle\text{seri}\left(\theta_{N-1},\theta_{N}\right)$
$\displaystyle=$ $\displaystyle
c\left(N-1,N;\mathcal{G}^{\prime}\left(N\right)\right),$ (31)

in which, $\text{seri}(\theta_{i},\theta_{j})$ is the series-sum of
$\theta_{i}$ and $\theta_{j}$ based on the series rule, and
$c(i,j;\mathcal{G}^{\prime}(N))$ is the net weight between vertices $i$ and
$j$ of the complete graph $\mathcal{G}^{\prime}(N)$.

Figure 10:  Demonstration of calculating the classical percolation between
nodes $1$ and $6$, in the following steps: (a) Original lattice. (b) and (c)
Series rules. (d) Star-mesh transform on the star graph (with edges
$4\leftrightarrow 1$, $4\leftrightarrow 3$, $4\leftrightarrow 6$), converting
it to a complete graph (with edges $1\leftrightarrow 3$, $3\leftrightarrow 6$,
$6\leftrightarrow 1$), then parallel rule for the double edges
$1\leftrightarrow 3$ and $3\leftrightarrow 6$. (e) Series rule for edges
$1\leftrightarrow 3$ and $3\leftrightarrow 6$, then parallel rule for edge
$1\leftrightarrow 6$.  Figure 11:  Demonstration of calculating the
concurrence percolation between nodes $1$ and $6$ (cf. Fig. 10).

To calculate the sponge-crossing percolation between the source $S$ and target
$T$ in a certain network, we approximately obtain the equivalent simplified
network by consecutively applying the star-mesh transform technique, where one
node is degraded for each application. Specifically, we arbitrarily choose a
vertex from $\mathcal{G}^{\prime}(N)$ (w.l.o.g., the last one, $N$) to be the
new root of a sub-star-graph of $\mathcal{G}^{\prime}(N)$ constructed from the
$N-1$ edges that connect the root to the other $N-1$ vertices. Next, we
transform this sub-star-graph $(\text{sub}\mathcal{G}^{\prime})(N-1)$ into a
$(N-1)$-complete graph, denoted by
$(\text{sub}\mathcal{G}^{\prime})^{\prime}(N-1)$, and combine it with what is
left untransformed,
$\mathcal{G}^{\prime}(N)\setminus\nobreak(\text{sub}\mathcal{G}^{\prime})(N-\nobreak
1)$. The new graph denoted as
$\text{Comb}\left(\mathcal{G}_{\alpha},\mathcal{G}_{\beta}\right)$ is derived
by setting each edge weight to be
$\theta_{ij}=\text{para}(\alpha_{ij},\beta_{ij})$, which is the parallel-sum
of $\alpha_{ij}\in\mathcal{G}_{\alpha}$ and $\beta_{ij}\in\mathcal{G}_{\beta}$
based on the parallel rule. Note that, because of the lack of closed-form
solution for concurrence percolation, we use Broyden’s root-finding algorithm
to numerically solve the $N(N-1)/2$ weights $\theta_{ij}$ that satisfy Eq.
(VI.2). In a word, we can calculate $c(i,j;\mathcal{G}^{\prime}(N))$ by first
solving a $(N-1)$-complete graph,

$c(i,j;\mathcal{G}^{\prime}(N))=c(i,j;\text{Comb}\left((\text{sub}\mathcal{G}^{\prime})^{\prime}(N-1),\mathcal{G}^{\prime}(N)\setminus(\text{sub}\mathcal{G}^{\prime})(N-1)\right)).$
(32)

By applying Eq. (32) one after the other on all but boundary nodes, the
network can be eventually reduced to two nodes and one link between them
(Table 2), the final weight $\theta$ of which shall be approximately
equivalent to the percolation of initial network. For demonstrative purposes,
here we show how the star-mesh transform works for both the classical
percolation (Fig. 10) and concurrence percolation (Fig. 11) on a small square
lattice.

Since $c(i,j;\mathcal{G}^{\prime}(N))$ is calculable through recursions and
Eq. (32) involves a $(N-1)$-level star-mesh transform, the entire procedure is
a double recursion, the cost growing faster than exponential. But by
practically carrying out the recursive computation using symbolic expressions
in Mathematica and other numerical techniques, the solutions can be found
within a sufficiently small error range [21].

Note that the star-mesh transform functions as an approximation rather than an
exact representation of higher-order connectivity rules. To see this, consider
the example of classical percolation given in Fig. 10. Under the change of
variable $p\equiv 2\sin^{2}\theta$, the actual higher-order connectivity,
i.e., the probability of at least one path connecting nodes $1$ and $6$, can
be expressed as follows:

$\displaystyle
p_{34}\left[1-\left(1-p_{35}p_{56}\right)\left(1-p_{46}\right)\right]\left[1-\left(1-p_{12}p_{24}\right)\left(1-p_{13}\right)\right]$
(33) $\displaystyle+$
$\displaystyle\left(1-p_{34}\right)\left[1-\left(1-p_{13}p_{35}p_{56}\right)\left(1-p_{12}p_{24}p_{46}\right)\right]$
$\displaystyle\approx$ $\displaystyle 0.0799,$

where $p_{ij}\equiv 2\sin^{2}\theta_{ij}\approx 0.304$ represents the singlet
conversion probability for each link $i\leftrightarrow j$. The final
probability ($\approx 0.0799$) translates to a final entanglement
$\theta\approx 0.256\pi/4$, which is very close to the star-mesh approximation
result of $\theta\approx 0.25\pi/4$ (Fig. 10). The closeness of these values
supports our confidence that the star-mesh transform can offer a reasonably
accurate approximation. Also, note that the process of reducing a network
using star-mesh transforms is not unique in sequence. For example, in the
fourth step of Fig. 10, one can transform the star graph ($3\leftrightarrow
1$, $3\leftrightarrow 4$, $3\leftrightarrow 6$) instead. Different sequences
of reduction might lead to varying approximate results, but they tend to stay
close to each other and to the exact value [21].

###  VI.3 Fast numerical approximation for concurrence percolation

The heuristic approximation (star-mesh transform) used for higher-order
connectivity rules can be quite demanding in terms of computational resources.
To address this challenge, in this section, we discuss a more efficient
approach to calculate the sponge-crossing concurrence $C_{\text{SC}}$ [171].
This acceleration in computation is achieved through the incorporation of two
key simplifying approximations: the parallel approximation and the $S_{m}$
approximation (Fig. 12).

Figure 12:  Approximations for calculating the sponge-crossing concurrence
$C_{\text{SC}}$. (a) In the $S_{m}$ approximation, one defines $S_{m}$ as the
set which contains up to the $m$-th shortest paths (i.e., the shortest paths,
the $2$nd shortest paths, and so on up to the $m$-th shortest paths) between
$s$ and $t$ for all $s\in S$ and $t\in T$. (b) In the parallel approximation,
one treats all paths in $S_{m}$ as parallel and non-overlapping. Thus, the
network topology reduces to series-then-parallel, and the sponge-crossing
concurrence can be calculated using the series/parallel rules (Table 2).

####  VI.3.1 Parallel approximation

First, we introduce the parallel approximation: we treat all paths connecting
nodes of interest to be parallel, i.e., treating them as if they have no
overlap. For an arbitrary network with $N$ nodes and uniform concurrence $c$
per link, the parallel approximation $C_{\text{SC}}^{\prime}$ of the true
sponge-crossing concurrence $C_{\text{SC}}$ between two sets of nodes, $S$ and
$T$, is given by

$\displaystyle\frac{1+\sqrt{1-C_{\text{SC}}^{\prime 2}}}{2}$
$\displaystyle=\max\left\\{\prod_{l}\left(\frac{1+\sqrt{1-c^{2l}}}{2}\right)^{N_{l}},\frac{1}{2}\right\\},$
(34)

where $N_{l}$ is the total number of self-avoiding paths of length $l$ that
connect the source/target nodes $s$ and $t$ for all $s\in S$ and $t\in T$,
respectively. Equation (34) is the mathematical statement of the parallel
approximation, indicating that we are taking each of the $N_{l}$ paths to be
parallel and non-overlapping (Fig. 12).

Now, we will show that on series-parallel networks [144] the concurrence
calculated under the parallel approximation forms an _upper_ bound to the true
concurrence. First, we consider the case where our network is essentially
parallel, i.e., it can be expressed as the parallel combination of $k$
subnetworks, each with concurrence $c_{i}$. In this case, the parallel
approximation is exact:

$\displaystyle
C_{\text{SC}}^{\prime}=C_{\text{SC}}=\text{para}(c_{1},c_{2},\dots c_{k}).$

The more interesting case is that of an essentially series network, i.e., a
network that can be decomposed as a combination of subnetworks in series. We
consider an exemplary network that splits into $k$ branches, each with
concurrence $c_{p_{i}}$. The concurrence of the segment before branching is
$c_{s}$. Following the series and parallel rules (Table 2), the sponge-
crossing concurrence from the left of this network segment to the right is

$\displaystyle C_{\text{SC}}$
$\displaystyle=\begin{cases}c_{s}\left(2\sqrt{f(c_{p_{0}},\dots
c_{p_{k}})-f(c_{p_{0}},\dots c_{p_{k}})^{2}}\right)&f(c_{p_{0}},\dots
c_{p_{k}})>1/2,\\\ c_{s}&f(c_{p_{0}},\dots c_{p_{k}})\leq 1/2,\end{cases}$

where $f(c_{p_{0}},\dots
c_{p_{k}})=\prod_{i=1}^{k}g(c_{p_{i}})=\prod_{i=1}^{k}\left(\dfrac{1+\sqrt{1-c_{p_{i}}^{2}}}{2}\right)$.
Under the parallel approximation, the network is transformed so that the
concurrence of the segment is given by

$\displaystyle C^{\prime}_{\text{SC}}$
$\displaystyle=\begin{cases}2\sqrt{f(c_{s}c_{p_{0}},\dots
c_{s}c_{p_{k}})-f(c_{s}c_{p_{0}},\dots
c_{s}c_{p_{k}})^{2}}&f(c_{s}c_{p_{0}},\dots c_{s}c_{p_{k}})>1/2,\\\
1&f(c_{s}c_{p_{0}},\dots c_{s}c_{p_{k}})\leq 1/2.\end{cases}$

Since $c_{s}c_{p_{i}}\leq c_{p_{i}}$, it follows that $g(c_{s}c_{p_{i}})\geq
g(c_{p_{i}})$ and thus $f(c_{s}c_{p_{0}},\dots c_{s}c_{p_{k}})\geq
f(c_{p_{0}},\dots c_{p_{k}})$. After some calculations [171] one can show that
$C_{\text{SC}}^{\prime}\geq C_{\text{SC}}$.

Taken together, since every series-parallel network can be decomposed into
essentially series or parallel configurations, we have shown that
$C_{\text{SC}}^{\prime}$ is an upper bound for $C_{\text{SC}}$ on series-
parallel networks. Interestingly, as we will see, this upper bound seemingly
becomes tighter as the network becomes larger. We hence expect that a new
concurrence threshold on $C_{\text{SC}}^{\prime}$ can emerge, which should
numerically approach the true $c_{\text{th}}$ from below and match
$c_{\text{th}}$ in the thermodynamic limit $N\to\infty$.

####  VI.3.2 $S_{m}$ approximation

For most regular lattices and complex networks, however, it is a nontrivial
task to determine the number of paths $N_{l}$ of length $l$. When we look at
arbitrary networks, the calculation for the sponge-crossing concurrence is
essentially a path-counting problem which may require approximation as well.

Although the literature of path counting on graphs is rich and well studied,
we are not aware of any closed-form solutions for enumeration of self-avoiding
walks of arbitrary length for even the simplest network (like 2D lattices)
[172]. While approximate path enumerations exist for both 2D lattices [173]
and random networks [174], we find them impractical, since the concurrence
calculation is very sensitive to $N_{l}$ for small $l$. Based on this, if we
define $S_{m}$ as the set which contains up to the $m$-th shortest paths
(i.e., the shortest paths, the $2$nd shortest paths, and so on up to the
$m$-th shortest paths) between $s$ and $t$ for all $s\in S$ and $t\in T$, then
it is possible to approximate the sponge-crossing concurrence between $S$ and
$T$ using only these paths.

####  VI.3.3 Results

Figure 13: The sponge-crossing concurrence $C_{\text{SC}}$ for the Bethe
lattice under the parallel approximation. Results are shown for coordination
numbers (a) $k=3$ and (b) $k=4$. As the number of layers, $L$, in the network
become larger the numerical concurrence percolation threshold approaches the
analytical value, $\theta_{\text{th}}=(2/\pi)\sin^{-1}(1/{\sqrt{k-1}})$ . The
solid black lines represent the exact $C_{\text{SC}}$ for the Bethe lattice
(cf. Fig. 5).

In this section, using the $S_{m}$ and parallel approximations, we present
numerical results for calculating $C_{\text{SC}}$ in different networks of
large size $N$. From that, we can numerically estimate the finite-size
concurrence percolation threshold $c_{\text{th}}\equiv\sin
2\theta_{\text{th}}$, determining its position on the critical curve by
matching the corresponding sponge-crossing concurrence at the half point,
namely,

$c_{\text{th}}\equiv\sin 2\theta_{\text{th}}\approx c|_{C_{\text{SC}}=1/2}.$
(35)

Next, we show how to apply our approach to Bethe Lattice and 2D Square
Lattices.

_Bethe lattice_.—Given a finite Bethe lattice (i.e., a Cayley tree) [Fig. 5]
with $L$ layers and coordination number $k$ [19, 20], all paths from the root
node to any one of the boundary nodes have the same length, $L$. Since only
one path exists from the root node to any node on the boundary, the number of
paths of length $L$ is

$\displaystyle N_{L}=k(k-1)^{L-1}.$ (36)

There is no need to employ the $S_{m}$ approximation since all paths are
exactly known. Only the parallel approximation $C_{\text{SC}}^{\prime}$ of the
sponge-crossing concurrence $C_{\text{SC}}$ is to be calculated, which is
given by [following Eq. (34)]

$\displaystyle\frac{1+\sqrt{1-C_{\text{SC}}^{\prime 2}}}{2}$
$\displaystyle=\max\left\\{\left(\frac{1+\sqrt{1-c^{2L}}}{2}\right)^{N_{L}},\frac{1}{2}\right\\}.$
(37)

To solve for $c_{\text{th}}$, near $C^{\prime}_{\text{SC}}=0$ we let

$\displaystyle\left(\frac{1+\sqrt{1-c_{\text{th}}^{2L}}}{2}\right)^{N_{L}}$
$\displaystyle=1-\epsilon$ (38)

given an arbitrarily small positive $\epsilon$. This gives rise to

$\displaystyle
c_{\text{th}}^{2L}=1-\left[2\left(1-\epsilon\right)^{1/N_{L}}-1\right]^{2}\simeq-4N_{L}^{-1}\ln\left(1-\epsilon\right)+O(N_{L}^{-2}),$
(39)

and thus

$\displaystyle
c_{\text{th}}\simeq\left(\frac{4\epsilon}{k}\right)^{\frac{1}{2L}}\left(\frac{1}{k-1}\right)^{\frac{L-1}{2L}}\simeq\frac{1}{\sqrt{k-1}}$
(40)

in the limit of large $L$. The result is identical to the exact concurrence
percolation threshold for Bethe lattices (Table 4).

For validation purposes, numerical results of the parallel approximation
$C^{\prime}_{\text{SC}}$ versus the exact values $C_{\text{SC}}$ are shown in
Fig. 13. We see that as $L$ increases, the threshold $c_{\text{th}}$
approaches ${1}/{\sqrt{k-1}}$ from below, consistent with our theoretical
result. Hence, it is highly suggested that the parallel approximation can
correctly estimate the true concurrence percolation threshold $c_{\text{th}}$
in the limit $N\to\infty$.

It is known that a saturation point $c_{\text{sat}}<1$ also exists in the
Bethe lattice [21], namely, before $c$ reaches unity, $C_{\text{SC}}$ will
already reach unity at $c=c_{\text{sat}}$. It is obvious that
$c_{\text{sat}}\geq c_{\text{th}}$, given the monotonicity of the series and
parallel rules (Table 2). To see if we can recover $c_{\text{sat}}$ using the
parallel approximation too, let

$\displaystyle\left(\frac{1+\sqrt{1-c_{\text{sat}}^{2L}}}{2}\right)^{N_{L}}$
$\displaystyle=\frac{1}{2},$ (41)

set by $C^{\prime}_{\text{SC}}=1$. This yields

$\displaystyle
c_{\text{sat}}^{2L}=1-\left[2\left(1/2\right)^{1/N_{L}}-1\right]^{2}\simeq
4N_{L}^{-1}\ln 2+O(N_{L}^{-2}),$ (42)

and thus

$\displaystyle c_{\text{sat}}\simeq\left(\frac{4\ln
2}{k}\right)^{\frac{1}{2L}}\left(\frac{1}{k-1}\right)^{\frac{L-1}{2L}}\simeq\frac{1}{\sqrt{k-1}}.$
(43)

We see that the saturation point calculated using the parallel approximation
is equal to $c_{\text{th}}$ but different from the exact value [Fig. 5].

Figure 14: The sponge-crossing concurrence $C_{\text{SC}}$ for 2D square
lattices under the $S_{m}$ and parallel approximations. (a) Sponge-crossing
concurrence $C_{\text{SC}}$ as a function of link’s entanglement $\theta$
under the $S_{3}$ approximation. The results of $S_{1}$ and $S_{2}$ are nearly
identical to $S_{3}$ and not plotted. (b) Numerical concurrence percolation
threshold $\theta_{\text{th}}$ under the $S_{m}$ approximation. As the
approximation order $m$ increases, $\theta_{\text{th}}$ approaches a constant
value. (c) $\theta_{\text{th}}$ for different size $N$. (d) Same as (c) but
for larger $N$. The results of $S_{3}$ are not shown because it becomes too
computationally intensive to calculate for $N>20^{2}$. As $N$ increases,
$\theta_{\text{th}}$ also approaches a constant value.  Figure 15: The speed-
up obtained by the approximations over the star-mesh transform. The figure
shows the computing time (in seconds) to calculate the sponge-crossing
concurrence between two nodes $s$ and $t$ on 2D square lattices with $N$
nodes, using the $S_{1}$ and $S_{2}$ approximation. In contrast to the star-
mesh transform, We can see that the approximations speed up the calculation
over the star-mesh transform approach by two orders of magnitude.

_Two-dimensional square lattice_.—In a 2D square lattice of $N$ nodes
($\sqrt{N}\in\mathbb{Z}$), the length of the $m$th shortest self-avoiding
path, between source and target nodes of coordinates $s=(x_{s},y_{s})$ and
$t=(x_{t},y_{t})$ ($1\leq x_{s},x_{t}\leq\sqrt{N}$ and $1\leq
y_{s},y_{t}\leq\sqrt{N}$), is simply

$l_{m}=|x_{s}-x_{t}|+|y_{s}-y_{t}|+2\left(m-1\right).$

Now, let $S$ and $T$ denote the left ($x_{s}=1$) and right ($x_{t}=\sqrt{N}$)
boundaries. Let $s=(1,y_{s})\in S$ and $t=(\sqrt{N},y_{t})\in T$. Under the
$S_{m}$ approximation, the total number of self-avoiding paths of length $l$
between $S$ and $T$ is given by

$\displaystyle
N_{l}\approx\sum_{y_{s}=1}^{\sqrt{N}}\sum_{y_{t}=1}^{\sqrt{N}}\delta_{l_{1}l}N_{l_{1}}(s\rightarrow
t)+\delta_{l_{2}l}N_{l_{2}}(s\rightarrow
t)+\cdots+\delta_{l_{m}l}N_{l_{m}}(s\rightarrow t),$ (44)

where $\delta_{ij}$ is the Kronecker delta. This approximation of $N_{l}$ is
then substituted into the parallel approximation [Eq. (34)] to calculate
$C_{\text{SC}}$ between $S$ and $T$.

For $m\leq 2$, it is possible to directly enumerate the $1$st and $2$nd
shortest self-avoiding paths between every pair of $s$ and $t$. For $m>2$,
however, it becomes difficult to write down a closed-form combinatorial
expression for $N_{l_{m}}(s\rightarrow t)$. A path enumeration algorithm is
thus needed. We treat paths of length $l_{m}$ with $m>2$ as deviations from
the $1$st and $2$nd shortest paths. For a given $m$, these deviations can only
take a finite number of shapes. Once we have identified these primitive
deviations, we must next identify positions in the lattice where these
deviations can be placed. Finally, we count the total number of paths by
counting the number of shortest paths between deviations [171].

For example, given every pair of source and target nodes $s$ and $t$, all
$3$rd-shortest paths ($m=3$) have either two single-step deviations or one
double-step deviation from the shortest path ($m=1$). For the case where we
have two single-step deviations, we first identify two sets of points, $D_{1}$
and $D_{2}$, where the first and second deviations can happen, respectively.
Then we calculate $N_{s,D_{1}}$ (the number of shortest paths from $s$ to
every point in $D_{1}$), $N_{D_{1},D_{2}}$ (the number of shortest paths from
every point in $D_{1}$ to every point in $D_{2}$), and $N_{D_{2},t}$ (the
number of shortest paths from every point in $D_{2}$ to $t$). The total number
of $3$rd-shortest paths is then given by $N_{l_{2}}(s\rightarrow
t)=N_{s,D_{1}}N_{D_{1},D_{2}}N_{D_{2},t}$.

The final numerical results of the sponge-crossing concurrence $C_{\text{SC}}$
are shown in Fig. 14. We see that the transition in the value of
$C_{\text{SC}}$ becomes sharper as the network size $N$ increases. Moreover,
for higher-order approximation $S_{m}$ and/or larger $N$, the numerical
threshold $\theta_{\text{th}}$ levels out at constant values that are very
close to those calculated using the star-mesh transform. For example, for
$N=8^{2}$, the numerical approximation yields $\theta_{\text{th}}\approx 0.4$,
closely mirroring the $\theta_{\text{th}}\approx 0.416$ result from the star-
mesh transform technique. This evidence underscores the viability of the new
approach for approximating the concurrence percolation threshold accurately.

One major benefit of this path-counting approach is its speed. As shown in
Fig. 15, the algorithm is over a hundred times faster than the heuristic star-
mesh transform. This substantial increase in computational speed facilitates
the extension of concurrence percolation threshold calculations to more
complex network topologies, such as random networks.

_Random networks_.—For random networks, the sponge-crossing concurrence is
simply defined as the concurrence between $S=\\{s\\}$ and $T=\\{t\\}$, each
set containing only one source (target) node $s$ ($t$), picked such that the
shortest path between $s$ and $t$ is equal to the diameter of the network. By
randomly generating and averaging $10^{2}$ network realizations of certain
sizes and degree distributions, the concurrence percolation threshold
$\theta_{\text{th}}$ is obtained.

The outcomes of this numerical approach, applied across different topologies
including the Erdős–Rényi (ER) [175] and the Barabási–Albert (BA) [77] random
networks at large scales ($N\sim 10^{4}$), are summarized in Table 5. These
findings shed light on the inherent capacities of large-scale complex QNs,
opening new avenues for exploration.

Network topology |  $(\pi/4)^{-1}\theta_{\text{th}}$ (fast approximation) |  $(\pi/4)^{-1}\theta_{\text{th}}$ [21]
---|---|---
Bethe Lattice ($L=100,k=3,S_{\infty}$) | 0.5 | 0.5
Bethe Lattice ($L=100,k=4,S_{\infty}$) | 0.39 | 0.3918
2D square ($N=8^{2},S_{9}$) | 0.40 | 0.416
2D square ($N=20^{2},S_{3}$) | 0.44 | n/a
2D square ($N=200^{2},S_{2}$) | 0.5 | n/a
ER ($N=10^{3},k=3,S_{5}$) | $0.6\pm 0.002$ | n/a
ER ($N=10^{3},k=4,S_{5}$) | $0.53\pm 0.0019$ | n/a
ER ($N=10^{4},k=2,S_{1}$) | $0.85\pm 0.0021$ | n/a
BA ($N=10^{3},z=5,S_{1}$) | $0.3\pm 0.0018$ | n/a
BA ($N=10^{4},z=1,S_{5}$) | $0.86\pm 0.0057$ | n/a
Table 5: Numerical concurrence percolation thresholds $\theta_{\text{th}}$
($\equiv 2^{-1}\arcsin{c_{\text{th}}}$) of different network topologies,
obtained by the $S_{m}$ and parallel approximations [171]. The results are
compared to those provided in Meng et al. [21] for the Bethe lattice and 2D
square lattice. Results on Erdős–Rényi (ER) and Barabási–Albert (BA) networks
are also reported.

##  VII Discussion and Conclusions

Distributing entanglement throughout a quantum network (QN) is a critical and
complex problem at the heart of quantum communications that has attracted
significant attention and studies. This field has been further enriched from a
statistical physics point of view, by the discovery of two percolation
theories (classical versus concurrence) that, at first glance, appear to be
remotely related but are, in fact, fundamentally distinct (Fig. 3). These
theories have not only deepened our understanding but also raised many new
questions for further exploration and potentially groundbreaking research. In
the following, we will outline and discuss some of the open questions that
have been brought to light by these developments:

  * •

_Optimality_. Does there exist an optimal scheme for entanglement
transmission? In the context of classical percolation, both classical
entanglement percolation (CEP) and quantum entanglement percolation (QEP) fall
short of yielding the optimal singlet conversion probability, especially as
network size scales up [10]. The deterministic entanglement transmission (DET)
[139], on the other hand, focuses on improving not the singlet conversion
probability but the entanglement that can be deterministically obtained. It
has been found that the DET optimizes the _average_ concurrence on either
series [Fig. 8] _or_ parallel topologies [Fig. 8], meaning that even when we
_relax the requirement of determinacy_ and consider the average entanglement
of general probabilistic outcomes, the DET results remain the optimal on
series or parallel topologies [139]. However, this result does not generalize
to general series-parallel topologies [Fig. 8], where DET may not always offer
the optimal average concurrence. This prompts us to ask how effective the DET
actually is across various QN topologies. Answering this question could
substantially deepen our comprehension of the maximum entanglement capacity of
QN.

  * •

_Universality._ As a statistical physics theory, the concurrence percolation
also exhibits a second-order phase transition near the threshold
$c_{\text{th}}$, similar to classical percolation near $p_{\text{th}}$. So,
what can be said about the universality of this phase transition? It has been
found that the thermal critical exponent $\nu$ remains the same on different
2D lattices (square versus honeycomb versus triangular), suggesting that
universality is likely at play near the critical threshold. Yet, the current
definition of percolation based on connectivity rules (Table 2) does not
clearly define an order parameter [21]. This lack of an order parameter hints
at a missing degree of freedom in the connectivity-based model. This omission
makes it challenging to determine other critical exponents besides $\nu$ (or
its dynamic counterpart, $z\nu$). Additionally, the existing data on $\nu$ (or
$z\nu$) do not allow us to distinguish between the universality classes of
concurrence percolation and classical percolation [21]. Thus, an open question
remains: are concurrence percolation and classical percolation simply two
facets of a single underlying statistical theory if we overlook short-range
details, or are they genuinely distinct theories at a macroscopic level?

  * •

_Experimental implementation._ One of the greatest challenges of quantum
information experiments is to achieve high-efficiency multi-body operations.
For instance, two-body quantum gates like CNOT are considerably more
challenging and less efficient to implement compared to single-body gates such
as the rotation gates RX, RY, and RZ. In fact, the number of two-body gates
often serves as a benchmark for gauging the computational complexity of
quantum algorithms. Interestingly, a practical QN offers an easier path to
scalability compared to universal quantum computers. This is since in a QN,
only local operations are allowed on qubits across different nodes. This
eliminates the need for complex gates like CNOT between qubits from different
nodes. This design constraint substantially simplifies QN implementation and
boosts its scalability. Recently, experimental feasibility of the
series/parallel rules of the DET scheme (Table 3) has also been demonstrated
on IBM’s quantum computation platform _Qiskit_ [176]. The series and parallel
rules typically perform with fidelity rates of $92.4\%$ and $78.2\%$,
respectively [139]. These rates are expected to improve, given ongoing
advancements in two-qubit gate fidelity [177, 178]. Compared to the CEP/QEP
schemes, the DET scheme has its advantages and drawbacks. On the upside, the
DET inputs/outputs are only partially entangled, which generally makes them
easier to produce and results in higher fidelity. On the flip side, circuit
parameters are input-dependent, requiring precise initial state estimations
through techniques like heralding [13] or tomography [179] before deployment.
This brings us to a crucial question: to what extent can we experimentally
scale the DET scheme for larger QNs? More importantly, given that the current
CEP/QEP/DET schemes focus solely on pure states, there is an urgent need to
extend these results to mixed states that are affected by noise—a vital step
for the practical implementation of QNs.

We have explored the far-reaching implications of entanglement transmission in
large-scale quantum networks, all through the lens of the percolation
frameworks. The presence of two distinct types of percolation—classical versus
concurrence—clearly suggests that the statistical landscape of quantum
networks is rather complex. As we look forward, we are enthusiastic that both
theoretical and experimental progress in the field of quantum networks will
enrich our understanding of this captivating area of study.

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