aainstitutetext: State Key Laboratory of Surface Physics, Department of
Physics, Center for Field Theory and Particle Physics, and Institute for
Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai
200433, China

# Characterizing the ambiguity in topological entanglement entropy

Yingcheng Li<EMAIL_ADDRESS>
###### Abstract

Topological entanglement entropy (TEE), the sub-leading term in the
entanglement entropy of topological order, is the direct evidence of the long-
range entanglement. While effective in characterizing topological orders on
closed manifolds, TEE is model-dependent when entanglement cuts intersect with
physical gapped boundaries. In this paper, we study the origin of this model-
dependence by introducing a model-independent picture of partitioning the
topological orders with gapped boundaries. In our picture, the entanglement
boundaries (EBs), i.e. the virtual boundaries of each subsystem induced by the
entanglement cuts, are assumed to be gapped boundaries with boundary defects.
At this model-independent stage, there are two choices one has to make
manually in defining the bi-partition: the boundary condition on the EBs, and
the coherence between certain boundary states. We show that TEE appears
because of a constraint on the defect configurations on the EBs, which is
choice-dependent in the cases where the EBs touch gapped boundaries. This
choice-dependence is known as the ambiguity in entanglement entropy. Different
models intrinsically employ different choices, rendering TEE model-dependent.
For $D(\mathbb{Z}_{2})$ topological order, the ambiguity can be fully
characterized by two parameters that respectively quantifies the EB condition
and the coherence. In particular, calculations compatible with the folding
trick naturally choose EB conditions that respect electric-magnetic duality
and set specific parameter values.

##  1 Introduction

Topologically ordered phases of matter, or topological orders for short, are
exotic gapped phases in two spatial dimension that exhibit long-range
entanglementChen  _et al._ (2010). This property of entanglement can be
captured by the entanglement entropy of the system. For gapped matter phases
in two dimension, their ground states usually have entanglement entropy that
yields the area-law, i.e the entanglement entropy is proportional to the
length of the entanglement cut; but for the ground state of a topological
order on a sphere, there is an extra sub-leading term that survive at
arbitrary long range. This sub-leading term is called the Topological
Entanglement EntropyKitaev and Preskill (2006) (TEE).

Topological entanglement entropy can be systematically and analytically
studied using the effective theories of topological orders, which are
generally classified into two different kinds: the Chern-Simons (CS) theory
Witten (1989), including the $K$-matrix theoryWen (2010); Hung and Wan (2013),
and the lattice models such as the Levin-Wen (LW) modelLevin and Wen (2005);
Hung and Wan (2012); Hu  _et al._ (2017, 2018a); Wang  _et al._ (2022) and the
quantum double (QD) modelKitaev (2003); Cong  _et al._ (2017a); Hu  _et al._
(2013); Bullivant  _et al._ (2017); Hu  _et al._ (2018b). In the lattice
models, the overall Hilbert space is inherently structured as a tensor product
of the Hilbert spaces of local degrees of freedom (d.o.f.), and hence it is
straightforward to bi-partition the lattice models and calculate entanglement
entropyLevin and Wen (2006); Hung and Wan (2015); Luo  _et al._ (2016); Chen
_et al._ (2018); Hu and Wan (2019). In the CS theory, however, one requires
certain methods such as the edge theory approachWen  _et al._ (2016); Lou  _et
al._ (2019); Shen  _et al._ (2019) to derive entanglement entropy. For
topological orders on closed manifolds, calculating TEE using different models
gives same results. Nevertheless, for topological orders on open manifolds
with gapped boundaries, different theories may produce different TEE when the
entanglement cuts touch gapped boundaries. For example, consider the
$\mathbb{Z}_{2}$ toric code on a disk with a gapped boundary and bi-partition
the disk in the way shown in Fig. 1. The calculation in the QD model shows
that TEE varies with the boundary condition of the physical boundaryChen  _et
al._ (2018), while the calculation in the $K$-matrix theory shows that TEE is
independent of the boundary conditionShen  _et al._ (2019). It requires
clarification what information is encoded in TEE and what leads to the model-
dependence in these open-cut cases.

Figure 1: Topological order on a disk with a gapped boundary. (a) The disk is
divided into two subsystems, and the entanglement cut intersects with the
physical boundary. One calculate the entanglement by partially tracing out
either of the two subsystems and calculate the von Neumann entropy of the
reduced density matrix. (b) Each subsystem is a disk with two boundary
segments: a segment of physical boundary and a segment of the virtual
entanglement boundary (thick blue line). In our picture, the entanglement
boundaries are gapped boundaries with boundary defects (dark blue dots).

In this paper, we focus on the topological orders that allow gapped boundaries
and study the origin of this model-dependence in Abelian topological orders
via a model-independent picture of bi-partitioning the topological order.
Suppose we have a topological order on a two-dimensional manifold, and divide
the manifold into two subsystems P and Q. For each subsystem, the entanglement
cuts serve as virtual entanglement boundaries (EBs), and the Hilbert space of
each subsystem is spanned by the boundary states describing EBs. In our
picture, the EBs are assumed to be gapped boundaries, and the EB states
describe different configurations of boundary defects on the EBs. These
boundary defects are anyonic objects that can be model-independently
characterized by the bulk topological order and the gapped boundary condition
(GBC)Kong (2014); Cong  _et al._ (2017b); Shen and Hung (2019), rendering the
EB states and the corresponding subsystem Hilbert spaces model-independent.
The TEE, as we will see in our picture, results from a constraint on each EB
that only captures the universal properties of the boundary states.

Given the subsystem Hilbert spaces, the global state can be decomposed into a
summation of tensor product states of the two subsystem Hilbert spaces. At the
model-independent level, there are two choices one has to make manually in
defining the decomposition: the boundary conditions on the EBs, and the
coherence between the boundary states. The former specifies the explicit form
of the EB states, while the latter determines the eigenstates and the
eigenvalues of the reduced density matrix. This dependence on the manual
choices is the ambiguity in entanglement entropyCasini  _et al._ (2014). The
question is, whether different choices change the universal properties of the
boundary states? We answer this question in Abelian topological orders by
introducing an operator version of the EB constraint and using this operator
constraint to pick out the universal properties. It turns out that when the
EBs touch gapped boundaries, the TEE will explicitly depend on the EB
conditions and the coherence between the two junctions connecting the EB and
physical boundaries. Different effective theories have different intrinsic
choices of the EB condition and the coherence, rendering TEE model-dependent.
Particularly, the lattice models choose the EB conditions that condense
specific anyons, while the $K$-matrix theory, which employs the folding trick,
naturally choose EB conditions that respect electric-magnetic (EM) dualityWang
_et al._ (2020) and set a specific coherence pattern between certain boundary
states. For the $\mathbb{Z}_{2}$ toric code, we introduce a general definition
of the decomposition with two parameters that respectively characterize the EB
condition and the coherence. These two parameters fully quantifies the
ambiguity of TEE in the $\mathbb{Z}_{2}$ toric code.

Once the ambiguity is characterized, the universal properties encoded in TEE
is then transparent. For each EB, there are two individual facts that
contribute to TEE: the total topological charge that goes through the EB and
the defect species that are allowed on the EB. While the latter exhibits
choice-dependnce, the former is choice-independent and contributes a ln$d_{x}$
term to TEE, where $d_{x}$ is the quantum dimension of a bulk anyon or a
boundary defect in different cases. For a generic state, there will also be a
classical term encoding the probability of finding certain basis states, as
already recognized in ref Wen  _et al._ (2016).

This paper is organized as follows. In section 2, we review some useful facts
about gapped boundaries and boundary defects of topological orders. We will
see the constraint on the boundary defect configurations in a general picture.
In section 3, we focus on the closed EB cases and introduce the model-
independent picture. We will first demonstrate the procedure of calculating
TEE, and then discuss the operator version of the constraint. In section 4, we
use our picture to investigate the open EB cases with the simplest choice of
EB conditions. In section 5, we will show the general definition of the state
decomposition for $\mathbb{Z}_{2}$ toric code. We will reproduce the result
calculated in the CS theory by employing choices that respect EM duality. We
comment on some important points and conclude in section 6.

##  2 Review of gapped boundaries and boundary defects

In this section, we review some facts about the gapped boundaries of
topological orders and clarify the notations that we will use later. A review
about the basic properties of anyons can be found in the Appendix A. For a
topological order, we use Greek alphabets $\alpha,\beta,\gamma,\cdots$ to
label the species of bulk anyons. The defects on a specific gapped boundary,
also called boundary anyons, are labeled by lower Latin alphabets
$a,b,c,\cdots$. The trivial anyon is denoted as $1$. The set of bulk anyons is
denoted as $\mathcal{C}$, and the set of boundary anyons is denoted as
$\mathcal{D}$.

A gapped boundary is characterized by a Lagrangian subsets $\mathcal{L}$ of
$\mathcal{C}$, i.e. a maximal set of anyons that are self-bosons and have
mutually trivial braiding statistics. All the anyons in $\mathcal{L}$ can
freely annihilate or create at the boundary. This mechanism is called the
anyon condensationKong (2014). Other anyons in $\mathcal{C}$ are confined at
the boundary, and will decompose into boundary anyons if moved onto the
boundary. The relation between the bulk anyons $\alpha\in\mathcal{C}$ and the
boundary anyons $a\in\mathcal{D}$ can be expressed by a matrix $W$,

$\alpha=\bigoplus_{a\in\mathcal{D}}W_{\alpha,a}a,$ (1)

where the matrix elements $W_{\alpha,a}$ are non-negative integers. If
$\alpha\in\mathcal{L}$, then $W_{\alpha,1}\neq 0$. This decomposition process
commute with fusion, i.e.,

$\sum_{\gamma}N_{\alpha\beta}^{\gamma}=\sum_{a,b}N_{ab}^{c}W_{\alpha,a}W_{\beta,b}.$
(2)

For a certain bulk anyon $\beta$, there can be more than one $b$ satisfying
$W_{\beta,b}\neq 0$. We say that the anyon $\beta$ splits at the boundary. We
also have the following two relations for quantum dimensions of the bulk
anyons and the boundary anyons,

$\displaystyle D=\sqrt{\sum_{\alpha}d_{\alpha}^{2}}=\sum_{a}d_{a}^{2}$ (3)
$\displaystyle d_{\alpha}=\sum_{a}W_{\alpha,a}d_{a}.$ (4)

There is a constraint on the configurations of boundary defects on a boundary.
To see it, we consider a simple case of a disk with no anyons in the bulk but
some anyons on the boundary. We can measure the total topological charge of
the boundary by creating a pair of ‘test anyons’, let one of them travel
through a trajectory that is very close to the boundary and then annihilate
with its partner, as shown in the Fig. 2a. This closed anyon loop, as one find
immediately, is contractible and detects nothing. Hence, the gapped boundary
should have total charge $1$, and consequently all the boundary anyons should
fuse to $1$. A general case is depicted in Fig. 2b. Suppose the boundary
anyons fuse to a non-trivial anyon $a$, then the test anyon loop should
recognize the boundary as a bulk anyon $\alpha$ that satisfies
$W_{\alpha,a}\neq 0$. In this sense, the gapped boundaries are treated on the
same footing as bulk anyonsLi  _et al._ (2019). Conversely, if the bulk anyons
on a disk fuse to $\alpha$, the boundary total charge should be $a$ that
satisfies $W_{\alpha,a}\neq 0$. If $\alpha$ splits on the boundary, the
boundary anyons can fuse to any of the $a$’s decomposed from $\alpha$.

Figure 2: A topological order with a gapped boundary. The blue dots on the
boundary represent for boundary anyons. (a) A test anyon loop near the
boundary. The anyon loop will find that the boundary has total topological
charge $1$. (b) The test anyon loop detect a non-trivial bulk anyon $\alpha$.
One can shrink the gapped boundary into a point, during which the boundary
anyons are fused to a single anyon $a$. This single anyon $a$ reside at the
point should satisfy $W_{\alpha,a}\neq 0$.

A topological order can have multiple Lagrangian subsets. Each Lagrangian
subset corresponds to a certain gapped boundary condition (GBCs). We use upper
Latin alphabets $A,B,C,\cdots$ to denote the GBCs of a topological order. The
junctions between the boundaries with different GBCs also host anyonic
defects. We will call them junction anyons hereinafter. We denote the set of
allowed junction anyons between two boundaries with GBC $A$ and $B$ as
$\mathcal{D}^{A|B}=\\{a^{A|B},b^{A|B},...\\}$. The number of elements in
$\mathcal{D}^{A|B}$ equals
$\sum_{\alpha}W^{A}_{\alpha,1^{A}}W^{B}_{\alpha,1^{B}}$, where $W^{A}$
($W^{B}$) is the $W$ matrix on the boundary $A$ ($B$) and $1^{A}$ ($1^{B}$) is
the trivial anyon on the boundary $A$ ($B$). The junction anyons also subject
to fusion rules,

$a^{A|B}\times b^{B|C}=\bigoplus_{c^{A|C}}N_{a^{A|B}b^{B|C}}^{c^{A|C}}\hskip
5.0ptc^{A|C}$ (5)

where $N_{a^{A|B}b^{B|C}}^{c^{A|C}}$’s are non-negative integers. If $A=B$,
i.e. the junction connects two boundaries with same GBC, the set of allowed
junction anyons then equals the set of boundary anyons
$\mathcal{D}^{A|A}=\mathcal{D}^{A}$. We use the term ‘boundary defects’ to
refer to both boundary anyons and junction anyons.

Like a bulk anyon can be detected by a bulk anyon loop, a boundary defect can
be detected by a half loop. Suppose an anyon $\gamma$ condenses at both $A$
and $B$, and a generic defect $c^{A|B}$ reside on the junction. One can create
an $\gamma$ at boundary $A$, let it go around the junction through the bulk,
and then annihilate at boundary $B$, as shown in Fig. 3a. The trajectory of
$\gamma$ forms a half loop, which generates a unitary transformation
$U_{\gamma,c^{A|B}}$ on the defect states. This unitary transformation
$U_{\gamma,c^{A|B}}$ is like the ‘braiding’ of a condensed anyon and a
boundary defect, and is analogous to the braiding of bulk anyons. If we
further annihilate the defect with its conjugate counterpart, we will result
in the half-linking introduced in ref Shen and Hung (2019) and Kapustin and
Saulina (2011).

There is also a constraint on the configurations of boundary defects on an
open boundary segment. Suppose we have a disk with two boundary segments with
GBCs $A$ and $B$, and there is no anyon in the bulk or on the boundary $B$.
Only boundary $A$ and the two junctions bear anyons. We can then detect the
total topological charge of the boundary segment $A$ via the half-loop
detection: we create an anyon $\gamma\in\mathcal{L}^{B}$ on the boundary $B$
near one of the junctions, let $\gamma$ travel very close to boundary $A$
though the bulk and annihilate near the other junction on boundary $B$, as
shown in Fig. 3b. Again, one find immediately that this half loop is
contractible and detects nothing. Therefore, all the defects, including the
anyons on boundary $A$ and the two junction anyons, should fuse to

$j_{0}^{B|A}\times x_{1}^{A}\times x_{2}^{A}\cdots\times j_{1}^{A|B}=1^{B},$
(6)

where $j_{i}^{B|A}$ denotes the anyons reside on the two junctions and
$x_{i}^{A}$ denotes the anyons on boundary $A$.

Figure 3: A topological order with multiple boundary segments. The orange dots
represent for junctions between boundaries. (a) The half loop that detects the
boundary defect. The orange dot represents for a generic boundary defect. (b)
A test half loop near the boundary. The half loop will find that the boundary
has total topological charge $1$. (c) The half loop may detect a non-trivial
boundary defect $a^{A|B}$. If we shrink the blue boundary segment into a
point, through which the boundary defects fuse to a single defect, we get the
same picture as (a).

More general cases involve multiple boundary segments, where the boundary
anyons and the junction anyons may fuse to a generic boundary defect, as
illustrated in Fig. 3c. We will see that the constraint on the configurations
of boundary defects plays a crucial role in deriving the entanglement entropy
and eventually leads to TEE.

Before we proceed, we would like to collect some useful facts about the
$\mathbb{Z}_{2}$ toric code, which we would heavily use in the rest of this
paper. There are $4$ kinds of anyons in the $\mathbb{Z}_{2}$ toric code, which
are labelled as $\\{1,e,m,\epsilon\\}$. The anyon $\epsilon$ can be viewed as
a composition of anyon $e$ and $m$. All the four anyons are self-conjugate and
have quantum dimension $1$. The remaining non-trivial fusion rules are given
by

$e\times m=\epsilon;\hskip 15.0ptm\times\epsilon=e;\hskip
15.0pte\times\epsilon=m.$ (7)

The toric code has two Lagrangian subsets $A=\\{1,m\\}$ and $B=\\{1,e\\}$. The
bulk anyons $\\{e,\epsilon\\}$ ($\\{m,\epsilon\\}$) are identified on the
boundary with GBC $A$ ($B$) and become the same boundary anyon $e^{A}$
($m^{B}$). The junction between two boundaries with different GBC can only
bear one kind of anyon $\sigma^{A|B}$. The fusion rules yields

$\displaystyle\sigma^{A|B}\times m^{B}=\sigma^{A|B};\hskip
10.0pt\sigma^{B|A}\times e^{A}=\sigma^{B|A};$
$\displaystyle\sigma^{A|B}\times\sigma^{B|A}=1^{A}+e^{A};\hskip
10.0pt\sigma^{B|A}\times\sigma^{A|B}=1^{B}+m^{B}.$

We have $d_{e^{A}}=d_{m^{B}}=1$ and $d_{\sigma^{A|B}}=\sqrt{2}$.

##  3 The boundary defect picture: closed EBs

In this section, we introduce the boundary defect picture in the cases where
the EBs are closed loops. In our picture, the EBs are assumed to be gapped
boundaries, and the Hilbert space of each subsystem is spanned by the states
of different configurations of boundary anyons. We will focus on two cases:
the unique ground state of a topological order on a sphere, and the ground
states on a torus. The result we will derive in this section is consistent
with that in ref Wen  _et al._ (2016), where the authors used the CS theory to
calculate the TEE and discussed general cases. Then, we will discuss the
operators of the subsystem for Abelian topological orders. Some contractible
anyon loops in the perspective of a global observer are cut by the EBs and
become half loops in each subsystem, as shown in Fig. 4. These half loops will
create or annihilate boundary defects on the EBs. There is a constraint on
these bi-partitioned operators, which will help us to distinguish the TEE from
the local contributions of sub-leading terms.

###  3.1 Case: ground state on a sphere

We start from the simplest case: the ground state of a topological order on a
sphere, as shown in Fig. 4. We assume that the EB have a fixed GBC and omit
the superscript that labels the GBC for the moment. Suppose the length of the
EB is $L$, and for simplicity we suppose the points that can host boundary
anyons on the EB are discrete and homogeneously distributed on the EB. The
spacing between two adjacent such points is fixed to be $\delta$, and hence
there are totally $nL/\delta$ such points on the EB. 111For lattice models,
the spacing $\delta$ is naturally fixed, and therefore $n$ is finite; for
continuous theories, $\delta$ is in principle zero, and $n$ goes to infinity.
Technically, when calculating entanglement entropy in continuous theories, one
may choose a cutoff and also fix $\delta$ to be finite. We assume $\delta$ is
fixed and finite hereinafter. For subsystem P, the EB anyon states are written
as

$|{x_{1},x_{2},\cdots,x_{n}}\rangle_{\textrm{P}},$ (8)

where $x_{i}\in\mathcal{D}$ denotes the boundary anyon on the $i$-th point.

Figure 4: The topological order on a sphere is divided into two subsystems P
and Q. We suppose there are $n=L/\delta$ discrete points homogeneously
distributed on the EB that can host boundary anyons. A contractible anyon loop
(dark blue line) is also divided into two half loops; in the perspective of a
subsystem observer, such half loops will create or annihilate boundary anyons
on the EB.

These states are not norm-$1$; instead, we have

$\langle
x_{1},x_{2},\cdots,x_{n}|x_{1},x_{2},\cdots,x_{n}\rangle=\sqrt{d_{x_{1}}d_{x_{2}}\cdots
d_{x_{n}}}.$ (9)

See Appendix B for details. The local Hilbert space of the subsystem P is

$\mathcal{H}_{\textrm{P}}=\textrm{span}\\{|{x_{1},x_{2},\cdots,x_{n}}\rangle_{\textrm{P}}\\},$
(10)

and also the local Hilbert space for subsystem Q is

$\mathcal{H}_{\textrm{Q}}=\textrm{span}\\{|{\bar{x}_{1},\bar{x}_{2},\cdots,\bar{x}_{n}}\rangle_{\textrm{Q}}\\},$
(11)

where $\bar{x}_{i}$ is the conjugate anyon of $x_{i}$. Given the local Hilbert
spaces, the ground state of the sphere can then be expressed in terms of the
tensor product of the states in $\mathcal{H}_{\textrm{P}}$ and
$\mathcal{H}_{\textrm{Q}}$,

$|{\psi}\rangle=\sum_{\\{x_{i}\\}}\frac{1}{\sqrt{\Psi}}|{x_{1},x_{2},\cdots,x_{n}}\rangle_{\textrm{P}}|{\bar{x}_{1},\bar{x}_{2},\cdots,\bar{x}_{n}}\rangle_{\textrm{Q}},$
(12)

where the summation runs over all allowed configurations of the boundary
anyons $\\{x_{i}\\}$ and $\Psi$ is a normalization factor to be determined.
This definition of tensor product decomposition is artificial and not unique
at the model-independent level. In section 3.4, we will see that the TEE in
this case does not depend on the choice of this definition. As discussed in
section 2, all the boundary anyons should fuse to $1$ due to the constraint on
the configurations of the boundary anyons. Therefore, the summation runs over
all configurations where the boundary anyons fuse to $1$222For boundary anyons
whose quantum dimension is greater than $1$, there might be multiple fusion
channels for these anyons to fuse to $1$. The states of these different fusion
channels are orthogonal to each other and hence should be summed
individually.. As such, we can fix the normalization factor to be

$\Psi=\sum_{\prod x_{i}=1}|\langle
x_{1},x_{2},\cdots,x_{n}|x_{1},x_{2},\cdots,x_{n}\rangle|^{2}=\sum_{\prod
x_{i}=1}d_{x_{1}}d_{x_{2}}\cdots d_{x_{n}}=D^{n-1},$ (13)

where $D=\sum_{a\in\mathcal{D}}d_{a}$. See Appendix C for a detailed
verification of the last equality. We can now partial-trace the subsystem Q
and acquire the reduced density matrix of the subsystem P,

$\rho_{\textrm{P}}=\sum_{\prod x_{i}=1}\frac{1}{D^{n-1}}\langle
x_{1},x_{2},\cdots,x_{n}|x_{1},x_{2},\cdots,x_{n}\rangle|{x_{1},x_{2},\cdots,x_{n}}\rangle\langle{x_{1},x_{2},\cdots,x_{n}}|.$
(14)

For an observer in the subsystem P, the classical probability of finding a
state $|{x_{1},x_{2},\cdots,x_{n}}\rangle_{\textrm{P}}$ is
$\frac{1}{D^{n-1}}d_{x_{1}}d_{x_{2}}\cdots d_{x_{n}}$. The two Hilbert spaces
are isomorphic to each other, and verifying the entanglement entropy using
either of the reduced density matrix gives equal results. We will omit the
subscript P and simply call the Hilbert spaces subsystem Hilbert spaces
hereinafter. The entanglement entropy is verified to be

$S_{\textrm{EE}}=-\sum_{\prod
x_{i}=1}\frac{d_{x_{1}}d_{x_{2}}...d_{x_{n}}}{D_{n-1}}\textrm{ln}\frac{d_{x_{1}}d_{x_{2}}...d_{x_{n}}}{D_{n-1}}=-n\sum_{a\in\mathcal{D}}\frac{d_{a}^{2}}{D}\textrm{ln}\frac{d_{a}}{D}-\textrm{ln}D.$
(15)

See Appendix C for a detailed verification of the last equality. The term
proportional to $n=L/\delta$ is the area-law contribution, and the
$-\textrm{ln}D$ is the TEE, which is a direct consequence of the constraint on
the boundary anyons. To see this, one can assume that the constraint is absent
and repeat the calculation above; the result of entanglement entropy would be
the area-law term alone. The constraint rules out some boundary anyon
configurations and consequently makes all the points on the EB correlated with
each other no matter how far away they are from each other. This correlation
is a straightforward demonstration of the long-range entanglement.

One more comment here is that we did not specify any GBC during the analysis.
In this case, different GBCs give same TEE, because Eq. (3) holds for every
GBC. We will see that in the cases where the entanglement cuts are open
segments, the choice of GBC matters.

###  3.2 Case: ground states on a torus

We now consider a specific ground state of a topological order on a torus
where an anyon $\alpha$ threads through the torus, as shown in Fig. 5.

Figure 5: (a) Bi-partitioning a torus. The torus is in the grouns state where
a bulk anyon $\alpha$ threads through the torus. (b) The subsystem is still a
disk. (c) The subsystem is a cylinder. A bulk anyon $\alpha$ thread through
the cylinder.

If subsystem P is still a disk (as shown in Fig. 5b), the entanglement entropy
is the same as that in the case of a sphere. If the subsystem is a cylinder
with two EBs (as shown in Fig. 5c), both EBs will have non-trivial total
topological charge. According to the discussion in section 2, one of the EBs
has total charge $\alpha$ and the other has $\bar{\alpha}$. There is no
further correlation between the two EBs. Suppose the two EBs have length
$L_{1}=n_{1}\delta$ and $L_{2}=n_{2}\delta$, and for simplicity, we assume the
two EBs have the same GBC. The subsystem Hilbert space is spanned by all the
states of boundary anyon configurations that survives the constraint,

$\mathcal{H}_{\alpha,\bar{\alpha}}=\bigoplus_{a,b}\mathcal{H}_{a,b},$ (16)

where $\mathcal{H}_{a,b}$ is spanned by the states where the anyons on the two
EBs respectively fuse to $a$ and $b$, and $a$ ($b$) runs over all allowed
boundary anyons satisfying $W_{\alpha,a}\neq 0$ ($W_{\bar{\alpha},b}\neq
0$)333If $W_{\alpha a}>1$, we should also count the multiplicity of $a$.. The
reduced density matrix is then

$\rho_{\alpha}=\frac{1}{\Psi_{\alpha}}\sum_{a,b}\sum_{\prod x_{i}=a,\prod
y_{i}=b}|{x_{1},x_{2},\cdots,x_{n_{1}};y_{1},y_{2},\cdots,y_{n_{2}}}\rangle\langle{x_{1},x_{2},\cdots,x_{n_{1}};y_{1},y_{2},\cdots,y_{n_{2}}}|,$
(17)

where $x_{i}$ and $y_{i}$ denotes the anyons respectively on the two EBs, and
the first summation runs over all $a,b$ satisfying $W_{\alpha a}\neq 0$ and
$W_{\bar{\alpha}b}\neq 0$. We can then fix the normalization factor,

$\displaystyle{\Psi_{\alpha}}=$
$\displaystyle\sum_{a}W_{\alpha,a}\sum_{b}W_{\bar{\alpha},b}\sum_{\prod
x_{i}=a}\sum_{\prod y_{i}=b}$ $\displaystyle|\langle
x_{1},x_{2},\cdots,x_{n_{1}}|x_{1},x_{2},\cdots,x_{n_{1}}\rangle|^{2}|\langle
y_{1},y_{2},\cdots,y_{n_{2}}|y_{1},y_{2},\cdots,y_{n_{2}}\rangle|^{2}$ (18)
$\displaystyle=$
$\displaystyle\sum_{a}W_{\alpha,a}\sum_{b}W_{\bar{\alpha},b}\sum_{\prod
x_{i}=a}\sum_{\prod y_{i}=b}d_{x_{1}}d_{x_{2}}\cdots
d_{x_{n_{1}}}d_{y_{1}}d_{y_{2}}\cdots d_{y_{n_{1}}}$ (19) $\displaystyle=$
$\displaystyle\sum_{a}W_{\alpha,a}\sum_{b}W_{\bar{\alpha},b}d_{a}d_{b}D^{n_{1}+n_{2}-2}=d_{\alpha}d_{\bar{\alpha}}D^{n_{1}+n_{2}-2}=d_{\alpha}^{2}D^{n_{1}+n_{2}-2}.$
(20)

where the third equality uses the formula introduced in Appendix C and the
forth equality uses Eq. (4). The entanglement entropy follows to be,

$\displaystyle S_{\textrm{EE}}=$
$\displaystyle-\sum_{a}W_{\alpha,a}\sum_{b}W_{\bar{\alpha},b}\sum_{\prod
x_{i}=a}\sum_{\prod y_{i}=b}$
$\displaystyle\frac{d_{x_{1}}d_{x_{2}}...d_{x_{n_{1}}}d_{y_{1}}d_{y_{2}}...d_{y_{n_{1}}}}{d_{\alpha}^{2}D^{n_{1}+n_{2}-2}}\textrm{ln}\frac{d_{x_{1}}d_{x_{2}}...d_{x_{n_{1}}}d_{y_{1}}d_{y_{2}}...d_{y_{n_{1}}}}{d_{\alpha}^{2}D^{n_{1}+n_{2}-2}}$
(21) $\displaystyle=$
$\displaystyle-\sum_{a}W_{\alpha,a}\sum_{b}W_{\bar{\alpha},b}\frac{d_{a}d_{b}}{d_{\alpha}^{2}}(n_{1}\sum_{x\in\mathcal{D}}\frac{d_{x}^{2}}{D}\textrm{ln}\frac{d_{x}}{D}+n_{2}\sum_{a\in\mathcal{D}}\frac{d_{x}^{2}}{D}\textrm{ln}\frac{d_{i}}{D}-2\textrm{ln}\frac{D}{d_{\alpha}})$
(22) $\displaystyle=$
$\displaystyle-(n_{1}+n_{2})\sum_{x\in\mathcal{D}}\frac{d_{x}^{2}}{D}\textrm{ln}\frac{d_{x}}{D}-2\textrm{ln}\frac{D}{d_{\alpha}}.$
(23)

Again, the term proportional to $n_{1}+n_{2}$ is the area-law contribution,
and the $-2\textrm{ln}\frac{D}{d_{\alpha}}$ is TEE. The factor $2$ is due to
the two EBs. Each EB contribute to $-\textrm{ln}\frac{D}{d_{\alpha}}$ due to
the constraint on this EB, and the ln$d_{\alpha}$ term is due to the anyon
$\alpha$ that threads through the torus.

For a generic ground state
$|{\psi}\rangle=\sum_{\alpha}\psi_{\alpha}|{\psi_{\alpha}}\rangle$ (we abuse
the notation of $\psi_{k}$ to be both the label of the ground state
$|{\psi_{k}}\rangle$ with an anyon $k$ threads through the torus and its
probability amplitude in the generic ground state $|{\psi}\rangle$.), the
entanglement entropy will have an extra term due to the superposition of
$|{\psi}\rangle$. Now, since it is possible to find the total charge of the
EBs to be any of the anyons with $\psi_{\alpha}\neq 0$, the subsystem Hilbert
space is then

$\mathcal{H}=\bigoplus_{\alpha}\mathcal{H}_{\alpha,\bar{\alpha}},$ (24)

where $\mathcal{H}_{\alpha,\bar{\alpha}}$ is the space defined in Eq. (16) and
$\alpha\in\mathcal{C}$ runs over all bulk anyon species with
$\psi_{\alpha}\neq 0$. The two states that respectively belong to
$\mathcal{H}_{\alpha}$ and $\mathcal{H}_{\beta}$ are orthogonal to each other
because the EB total charge is different. The reduced density matrix is then

$\rho=\bigoplus_{\alpha}|\psi_{\alpha}|^{2}\rho_{\alpha},$ (25)

where $\rho_{\alpha}$ is the density matrix derived in Eq. (17). The TEE
follows to be

$S_{\textrm{TEE}}=-2\sum_{\alpha}|\psi_{\alpha}|^{2}\textrm{ln}\frac{D}{d_{\alpha}}+\sum_{\alpha}|\psi_{\alpha}|^{2}\textrm{ln}|\psi_{\alpha}|^{2}.$
(26)

The $\sum_{\alpha}|\psi_{\alpha}|^{2}\textrm{ln}|\psi_{\alpha}|^{2}$ is the
classical von Neumann entropy due to the superposition of the ground basis
$|{\psi_{\alpha}}\rangle$ and enhances the mixture of the mixed state as
expected. For the TEE of a generic state on an arbitrary closed manifold, one
can expand the state in terms of the states where each EB have a single total
charge. Then, it is straightforward to write down the TEE that composes of
three parts: the $-\textrm{ln}D$ term multiplies the number of disconnected
EBs, the ln$d_{\alpha}$ terms of each EB, and the classical term due to the
superposition.

###  3.3 The operators of the subsystem

As mentioned before, some anyon loops seen by a global observer are cut by the
EBs, becoming operators that transform the states in the subsystem Hilbert
space. These operators can be classified into two types: the ones with both
end points on the same EB, and the ones whose end points are on different EBs.
For the second type, the collective behavior of these operators are already
captured by the test anyon loops and contribute to the total topological
charge of the boundary; for the first type, they create or annihilate boundary
defects on the EBs and hence transforms a boundary anyon configuration into
another. In fact, for Abelian topological orders, the first type of operators
yield to a constraint444The non-Abelian cases involve non-invertible operators
and hence are completely different. We will report the story elsewhere.. Let
us focus on the $D(\mathbb{Z}_{2})$ topological order on a sphere, where we
only have the first type of operators. There are four kinds of anyon loops
respectively correspond to the four kinds of anyons $\\{1,e,m,\epsilon\\}$.
Suppose the EB condenses $e$. Any two points $x_{1}$ and $x_{2}$ on the EB are
connected by a half loop; each half loop induces two operators: the operator
$O_{0}^{x_{1},x_{2}}$ induced by the half loop of anyons $\\{1,e\\}$, and a
nontrivial operator $O_{1}^{x_{1},x_{2}}$ induced by the half loop of anyons
$\\{m,\epsilon\\}$. Since the anyons $\\{1,e\\}$ condense on the EB, the
operators $O_{0}$ do not change the boundary defect configuration and hence
are identities in the subsystem Hilbert space. The anyons $\\{m,\epsilon\\}$
are confined at the EB, and hence the operators $O_{1}$ creates or annihilates
defects on the EB,

$O_{1}^{x_{1},x_{2}}|{x_{1},\cdots,x_{2}}\rangle=|{\tilde{x}_{1},\cdots,\tilde{x}_{2}}\rangle,$
(27)

where $\tilde{x}_{i}=m^{\\{1,e\\}}\times x_{i}$, namely the anyon type on the
point $x_{i}$ is flipped. Since, according to the fusion rule,

$O_{1}^{x_{1},x_{2}}O_{1}^{x_{2},x_{3}}=O_{1}^{x_{1},x_{3}},$ (28)

the $O_{1}$ operators are not independent from each other. We can pick up a
generating set by employing the following choice. We arrange the EB into a
circle and use an angle $\theta$ to label each point on the cut, as
illustrated in Fig. 6c. A generating set $\mathcal{K}$ consists of the $O_{1}$
operators that connects two points with fixed angle difference $2\pi\theta$,

$\mathcal{K}_{\theta}=\\{O_{1}^{x_{1},x_{2}}|x_{1}-x_{2}=2\pi\theta\\}.$ (29)

where $\theta\in(0,1)$ is a fixed irrational number. For each boundary point,
there are exactly two operators in the set $\mathcal{K}$ that can create or
annihilate an anyon on the point. We fix the $\theta$ to be irrational because
otherwise the set cannot generate all the operators.555In discrete lattice
models, it is more convenient to define the generating set as the operators
connecting the two adjacent boundary points. In the continuum limit, however,
the ‘adjacent’ point of a certain point is not well defined. This is the
reason why we need the irrational number $\theta$ here, which makes our choice
of the generating set also works in the continuum limit..

Figure 6: (a) The first type of operators, where each operator has both its
endpoints on the same EB. (b) The second type operators, where each operator
has only one endpoint on an EB. (c) Choice of the generating set. We choose a
reference point on the EB to have $x_{0}=0$. Other points are labeled by the
central angles with respect to the reference point.

All the operators in $\mathcal{K}$ are mutually commute, and hence a state of
certain boundary defect configuration can be expressed as

$\prod_{x_{i}}^{\textrm{all points on
EB}}(O^{x_{i},x_{i}+\theta})^{k_{x_{i}}}|{0}\rangle,$ (30)

where $k_{x_{i}}=0$ or $1$ for all $x_{i}$ and $|{0}\rangle$ is the boundary
state with no defect. If, for applicability, we stick to the assumption that
there are $n$ points on the EB hosting boundary anyons, then there are $n$
operators in the set $\mathcal{K}$, and the expression (30) has $2^{n}$
different combinations of operators. But this is not the end, the operators in
$\mathcal{K}$ yield an extra constraint,

$\prod_{x_{i}}^{\textrm{all points on EB}}O^{x_{i},x_{i}+\theta}=1.$ (31)

This constraint is the operator version of the constraint on the EB. The
number of independent expressions generated by $\mathcal{K}$ is halved,
leaving $2^{n-1}$ independent configurations. The $2^{-1}$ factor finally
becomes $S_{\textrm{TEE}}=-\textrm{ln}2$. The construction above is easily
generalized to all Abelian topological orders, where we have $k-1$ constraints
for a topological order with $k$ species of boundary anyons.

The operator constraint implies that the boundary anyon fluctuation due to the
first type anyon loops should all fuse to $1$, leaving the total charge of the
boundary fully determined by the second type anyon loops. In this sense, we
can separately take the $-$ln$D$ term and the ln$d_{\alpha}$ term as the
contribution of the first type anyon loops and the second type anyon loops
respectively.

###  3.4 Local vs. Topological

The operator constraint serves as a good criterion to distinguish TEE from
locally originated contributions. In particular, the tensor product
decomposition in Eq. (12) is not unique at the model-independent level. One
can use other definition of the decomposition, which will lead to off-diagonal
terms in the reduced density matrix Eq. (14). Such off-diagonal terms make
some points on the EB coherent with each other and will contribute to the
entanglement entropy. If the coherence violates the homogeneity of the EB, the
contribution will be a sub-leading term. For example, for the ground state of
a $D(\mathbb{Z}_{2})$ on a sphere, we can pick up two points $x_{p}$ and
$x_{q}$ on the EB and make them coherent manually. The reduced density matrix
will have extra off-diagonal terms between the states
$|{x_{1},\cdots,x_{p},\cdots,x_{q},\cdots,x_{n}}\rangle$ and
$|{x_{1},\cdots,\tilde{x}_{p},\cdots,\tilde{x}_{q},\cdots,x_{n}}\rangle$.
Then, there will be a
$\frac{1+\lambda}{2}\textrm{ln}\frac{1+\lambda}{2}+\frac{1-\lambda}{2}\textrm{ln}\frac{1-\lambda}{2}$
term in the entanglement entropy, with $\lambda$ quantifying the strength of
the coherence (See Appendix D for a detailed derivation of this term.). This
kind of terms due to inhomogeneity of the EB is neither related to the EB
constraint nor topologically invariant. In the effective theories, such terms
are naturally circumvented because the EBs are chosen to be homogeneous. At
model-independent level, there is a more fundamental way to distinguish these
terms from TEE: by checking the operator constraint. We add or remove
coherence in the density matrix while keeping the operator constraint
unchanged, until we reach a homogeneous EB. Then, the sub-leading term derived
from the homogeneous EB is TEE. For the construction we introduced in this
section, namely the EBs are closed loops and have fixed GBCs, the operator
constraint is always independent of the coherence.

We require the EBs to be homogeneous in the the rest of this paper, such that
the locally originated sub-leading terms are avoided.

##  4 The boundary defect picture: open EBs

In this section, we investigate the cases where the EBs touches gapped
boundaries. In this section, we adopt the simplest EB condition where each EB
have a fixed GBC. Unlike the cases of closed entanglement cuts, different GBCs
on the EB will give different TEE. This ambiguity happens because the anyon
species that can be hosted at the junction varies when changing the GBC.

###  4.1 Case: ground state on a disk

We first consider the simplest case: the ground state of a topological order
on a disk. Each subsystem is a disk with two boundary segments, as shown in
Fig. 7. We suppose that the GBCs of the physical boundary and the EB are $A$
and $B$. While there is no anyon on the physical boundary segment, there can
be anyons on the EB and the two junctions. The subsystem Hilbert space
$\mathcal{H}^{A|B}$ is then spanned by

$|{x_{1},x_{2},\cdots,x_{n};z_{1},z_{2}}\rangle,$ (32)

where $x_{i}\in\mathcal{D}^{B}$ represents for the anyon on the $i$-th point
on the EB, and $z_{i}\in\mathcal{D}^{A|B}$ labels the anyon on the $i$-th
junction. According to the discussions in section 2, one can measure the total
charge of the EB by a half loop near the EB, and the result in this case would
be a trivial total charge $1^{A}$. Therefore, the constraint on the boundary
defects is

$z_{1}\times x_{1}\times\cdots\times x_{n}\times z_{2}=1^{A}.$ (33)

Figure 7: Bi-partitioning a disk. We suppose there are $n$ discrete points on
the EB segment (excluding the two junctions) that can host boundary anyons.

The reduced density matrix follows to be

$\displaystyle\rho^{B=\\{1,m\\}}=\frac{1}{\Psi}\sum_{z_{1}(\prod
x_{i})z_{2}=1^{A}}d_{z_{1}}d_{x_{1}}\cdots
d_{x_{n}}d_{z_{2}}|{x_{1},x_{2},\cdots,x_{n};z_{1},z_{2}}\rangle\langle{x_{1},x_{2},\cdots,x_{n};z_{1},z_{2}}|.$
(34)

The TEE depend on the choice of the GBC of the EB. To be concrete, we focus on
the $D(\mathbb{Z}_{2})$ topological order. Suppose both the physical boundary
and the EB have GBC $A=B=\\{1,m\\}$. Then, $\mathcal{H}^{A|B}$ is $2^{n+1}$
dimensional, and it is straightforward to find that $\Psi=2^{n+1}$. The
entanglement entropy follows to be

$S_{\textrm{EE}}^{B=\\{1,m\\}}=-\sum_{z_{1}(\prod
x_{i})z_{2}=1^{A}}\frac{d_{z_{1}}d_{x_{1}}...d_{x_{n}}d_{z_{2}}}{2^{n+1}}\textrm{ln}\frac{d_{z_{1}}d_{x_{1}}...d_{x_{n}}d_{z_{2}}}{2^{n+1}}=-(n+1)\textrm{ln}\frac{1}{2}.$
(35)

On the other hand, if the EB have GBC $B=\\{1,e\\}$, then
$\mathcal{H}^{A|B}=$span$\\{|{x_{1},\cdots,x_{n};\sigma,\sigma}\rangle\\}$ is
$2^{n}$ dimensional. The reduced density matrix is then

$\rho^{B=\\{1,e\\}}=\frac{1}{\Psi}\sum_{\\{x_{i}\\}}d_{\sigma}d_{x_{1}}\cdots
d_{x_{n}}d_{\sigma}|{x_{1},x_{2},\cdots,x_{n};\sigma,\sigma}\rangle\langle{x_{1},x_{2},\cdots,x_{n};\sigma,\sigma}|.$
(36)

The summation runs over all configurations of $\\{x_{i}\\}$ because
$\sigma\times(\prod x_{i})\times\sigma=1^{A}+e^{A}$ automatically satisfies
the constraint. Again, $\Psi=2^{n+1}$, and the entanglement entropy is

$S_{\textrm{EE}}^{B=\\{1,e\\}}=-\sum_{\\{x_{i}\\}}\frac{\sqrt{2}d_{x_{1}}...d_{x_{n}}\sqrt{2}}{2^{n+1}}\textrm{ln}\frac{\sqrt{2}d_{x_{1}}...d_{x_{n}}\sqrt{2}}{2^{n+1}}=-n\textrm{ln}\frac{1}{2}.$
(37)

Immediately, one wonders which term is the area-law contribution, and which
term is TEE? Note that we assumed there are $n$ points on the boundary that
can host boundary anyons, and the spacing between two adjacent points is
$\delta$. When the EB is open, we have $L=(n+1)\delta$, and hence the area-law
term should be proportional to $(n+1)$. We therefore recognize that
$S_{\textrm{TEE}}^{B=\\{1,m\\}}=0$ and
$S_{\textrm{TEE}}^{B=\\{1,e\\}}=-$ln$2$.

It is worth mentioning that in ref Chen  _et al._ (2018), the authors used the
QD model to calculate the TEE of this case, and the area-law term is taken to
be $(n+2)$ln$2$. In the QD model, the entanglement cut can be viewed as a
gapped boundary with ‘charge’ excitations (excitations reside on vertices) on
it, which is the $e$ anyon in $D(\mathbb{Z}_{2})$. Therefore, calculating the
entanglement entropy using the QD models is equivalent to employing the EB
condition $B=\\{1,m\\}$ in our picture. The $(n+2)$ factor takes the two
junction points on an equal footing with other points on the EB. Nevertheless,
the two junctions are naturally different from other points; choosing the
$(n+2)$ln$2$ term to be area-law is not proper. We will see in section 5 that
a homogeneous open EB should have area-law term proportional to $(n+1)$.

Obviously, the TEE depends on the choice of EB condition because the junction
anyon species varies when employing different GBCs. This dependence is also
shown in the operator constraint. For the open cut cases, we choose the
generating set by employing the following convention. The EB is an open
segment with length $L$; a point on the EB can be represented by a number
$x_{i}\in[0,1]$, such that the distance between the point and $z_{1}$ is
$x_{i}L$, as shown in Fig. 8. We choose the generating set of the operators to
be all the operators that connects two points whose distance is $\theta$

$\displaystyle\\{O^{x_{i},x_{j}}|x_{j}=\left\\{\begin{array}[]{ll}(x_{i}+\theta)L&\textrm{if
$(x_{i}+\theta)L<L$ }\\\ (x_{i}+\theta-1)L&\textrm{if
$(x_{i}+\theta)L>L$}\end{array}\right.\\},$ (40)

where $\theta$ is again an irrational number. As such, the number of the
operators in the generating set equals the number of boundary points as
before. Multiplying these operators together results in, instead of the
identity in the cases of closed cuts, the operator that connects the two
junctions,

$\prod_{x_{i}}O^{x_{i},x_{i}+\theta}=O^{z_{1},z_{2}}.$ (41)

The dependence on the EB condition is encoded in $O^{z_{1},z_{2}}$, which
transforms a junction anyon configuration into another. If the subsystem
Hilbert space is $\mathcal{H}^{\\{1,e\\}|\\{1,m\\}}$, the operator
$O_{z_{1},z_{2}}$ is again an identity matrix, because there is only one type
of junction anyon allowed; for $\mathcal{H}^{\\{1,m\\}|\\{1,m\\}}$, however,
$O_{z_{1},z_{2}}$ is no longer an identity,

$O_{z_{1},z_{2}}:\hskip
5.0pt|{x_{1},\cdots,x_{n};z_{1},z_{2}}\rangle\leftrightarrow|{x_{1},\cdots,x_{n};\tilde{z}_{1},\tilde{z}_{2}}\rangle.$
(42)

The operator constraint also varies when changing the EB condition. A direct
observation is that adding coherence between the two junction points will
modify the explicit form of the operator $O_{z_{1},z_{2}}$ and accordingly the
value of TEE.

Figure 8: The convention of labelling the points on the EB. The generating set
consists of all operators with fixed $x_{j}-x_{i}$.

It is similar to derive the TEE for the case where $A=\\{1,e\\}$. We summarize
the results in the following table.

$S_{\textrm{TEE}}$ | A=$\\{1,e\\}$ | A=$\\{1,m\\}$
---|---|---
B=$\\{1,e\\}$ | 0 |  $-$ln$2$
B=$\\{1,m\\}$ |  $-$ln$2$ | 0

For a fixed EB condition, different GBCs on the physical boundary also gives
different TEE. In contrast, the TEE is $-\frac{1}{2}$ln$2$ regardless of the
physical GBC according to the calculation in the $K$-matrix theory. This
mismatch indicates that neither of the EB conditions we used in this section
is compatible with the EB condition in the $K$-matrix theory. We will treat
this mismatch in Section 5 by introducing a general construction of the
subsystem density matrix.

###  4.2 Case: ground states on a cylinder

Before introducing the general construction, we would like to first consider
the ground states on a cylinder. The cylinder cases are analogous to the torus
cases, in the sense that the total charge of an EB can be a non-trivial anyon.
Suppose the cylinder has GBCs $A$ and $A^{\prime}$. We focus on the specific
ground state that correspond to a confined anyon
$|{\psi_{a^{A|A^{\prime}}}}\rangle$, such that the total charge of both EBs is
$a^{A|A^{\prime}}\in\mathcal{D}^{A|A^{\prime}}$. We further suppose the EBs
has GBC $B$ and $B^{\prime}$, and there are $n$ ($m$) anyon points on the
boundary $B$ ($B^{\prime}$), as shown in Fig. (9).

Figure 9: (a) A cylinder is in the ground state labeled by a confined anyon
$a^{A|A^{\prime}}$. (b) Bi-partitioning a cylinder.

The subsystem Hilbert space is then

$\mathcal{H}^{A,A^{\prime}|B,B^{\prime}}=\textrm{span}\\{|{x_{1},\cdots,x_{n};z_{1},z_{2};y_{1},\cdots,y_{m};z_{3},z_{4}}\rangle\\},$
(43)

with the constraints

$\displaystyle z_{1}^{A|B}\times(\prod_{i}x_{i}^{B})\times
z_{2}^{B|A^{\prime}}=a^{A|A^{\prime}};\hskip
5.0ptz_{3}^{A|B^{\prime}}\times(\prod_{i}y_{i}^{B^{\prime}})\times
z_{4}^{B^{\prime}|A^{\prime}}=a^{A|A^{\prime}},$ (44)

where $x_{i}$’s denote the anyons on the EB $B$, $y_{i}$’s denote the anyons
on the EB $B^{\prime}$, and $z_{i}$’s respectively denote the anyons on the
four junction points. The reduced density matrix is then

$\rho_{a^{A|A^{\prime}}}=\sum_{\\{x_{i},y_{i},z_{i}\\}}\prod d_{z_{i}}\prod
d_{x_{i}}\prod
d_{y_{i}}|{x_{1},\cdots;z_{1},z_{2};y_{1},\cdots;z_{3},z_{4}}\rangle\langle{x_{1},\cdots;z_{1},z_{2};y_{1},\cdots;z_{3},z_{4}}|,$
(45)

where the summation runs over all boundary anyon configurations satisfying the
constraint Eq. (44). The resultant TEE is
$-\textrm{ln}D^{B}-\textrm{ln}D^{B^{\prime}}+2\textrm{ln}d_{a^{A|A^{\prime}}}$.
The ln$D^{B}$ and ln$D^{B^{\prime}}$ terms are analogous to the ln$D$ term in
the closed EB cases, but now they depend on the choice of EB conditions. The
$\textrm{ln}d_{a^{A|A^{\prime}}}$ term is analogous to the ln$d_{\alpha}$ term
in the closed EB cases because they both attribute to the total charge of the
EBs. A generic cylinder ground state can be expanded in terms of the basis
states $|{\psi_{a^{A|A^{\prime}}}}\rangle$, and the probability amplitude of
each basis state will again result in a classical part
$\sum_{a^{A|A^{\prime}}}|\psi_{a^{A|A^{\prime}}}|^{2}$ln$|\psi_{a^{A|A^{\prime}}}|^{2}$,
analogous to the classical part in Eq. (26).

##  5 A general state decomposition for $D(\mathbb{Z}_{2})$

As mentioned in before, TEE calculated in the $K$-matrix theory and the
lattice models does not match. In this section, we focus on the
$D(\mathbb{Z}_{2})$ topological order and introduce a general construction of
the state decomposition that addresses the ambiguity. As mentioned in Section
3.4, defining the decomposition amounts to constructing subsystem density
matrix. We construct a general form of the reduced density matrix that can
fully characterize the ambiguity.

###  5.1 The EM duality in the folding trick

The key idea of calculating TEE in the $K$-matrix theory is the folding trick.
In ref Shen  _et al._ (2019), the authors constructed the Ishibashi state that
describes the entanglement cut via unfolding the non-chiral topological order.
For example, for the $D(\mathbb{Z}_{2})$ on a disk, one can unfold the disk
into chiral $\mathbb{Z}_{2}$ topological order on a sphere, as shown in the
Fig. 10. The chiral $\mathbb{Z}_{2}$ anyon on the upper and the lower layer of
the squashed sphere can be respectively thought as the $e$ and $m$ anyons of
the $D(\mathbb{Z}_{2})$ in the bulk, and the $\epsilon$ anyon is the
composition of the $e$ and $m$.

Figure 10: (a) A chiral $\mathbb{Z}_{2}$ topological order on a sphere (left)
is squeezed into a $D(\mathbb{Z}_{2})$ topological order on a disk. The
equator on the sphere becomes the gapped boundary of the disk. The blue large
circle is the entanglement cut, which is also folded into an open line segment
on the disk. (b) The special points on the EB of the sphere are paired and
identified on the EB of the disk, except that the two orange points that
become the junctions on the disk.

Given the picture of the folding trick, we can understand why the area-law
term should be proportional to $n+1$. Suppose there are $n$ points that can
host boundary anyons on the EB of the disk, if we unfold the half disk into a
semi-sphere, the anyon points will split into $2n$ special points666The
unfolded EB is usually a gapless boundary described a CFT state, such that
there may not be boundary anyons on the unfolded EB. Therefore we simply call
these unfolded points ‘special points’.. Taking the two junctions into
consideration, there are in all $(2n+2)=\frac{2L}{\delta}$ special points on
the unfolded EB. The homogeneity of the EB then means that these $2n+2$
special points are on an equal footing, and hence the area-law term should be
proportional to $2n+2$.

If one regard the systems before and after the folding trick to be completely
equivalent, one can directly find the properties of the folded system from
that of the unfolded system. For example, since we already know that the TEE
of chiral $\mathbb{Z}_{2}$ topological order on a sphere is ln$\sqrt{2}$, one
may think the TEE of $D(\mathbb{Z}_{2})$ on a disk is also ln$\sqrt{2}$. This
folding process indicates that the EB condition in the $K$-matrix theory
respects EM duality: the EB treats equally the $\mathbb{Z}_{2}$ anyon on the
upper layer and the lower layer, which are respectively $e$ and $m$ in
$D(\mathbb{Z}_{2})$.

###  5.2 The general EB condition

The EM duality in the folding trick hints us to extend the EB condition to
include both GBCs of the $D(\mathbb{Z}_{2})$ topological order. If an EB have
a single GBC $\\{1,e\\}$ ($\\{1,m\\}$) as its EB condition, we use an extra
superscript e (m) to label the defect configuration states
$|{\\{x_{i}\\}}\rangle^{e}$ ($|{\\{x_{i}\\}}\rangle^{m}$) on this EB. The
Hilbert space spanned by $|{\\{x_{i}\\}}\rangle^{e}$
($|{\\{x_{i}\\}}\rangle^{m}$) is denoted as $\mathcal{H}^{e}$
($\mathcal{H}^{m}$). The general EB condition for a certain EB induces a
composite subsystem Hilbert space,

$\mathcal{H}^{\textrm{EM}}=\mathcal{H}^{e}\oplus\mathcal{H}^{m}.$ (46)

Namely, the general Hilbert space collects the states with different GBCs on
the EB. Let us focus on the $D(\mathbb{Z}_{2})$ on a disk with the physical
boundary having GBC $A=\\{1,e\\}$. We have
$\mathcal{H}^{e}=\mathcal{H}^{\\{1,e\\}|\\{1,e\\}}$ and
$\mathcal{H}^{m}=\mathcal{H}^{\\{1,e\\}|\\{1,m\\}}$, where
$\mathcal{H}^{\\{1,e\\}|\\{1,e\\}}$ is spanned by
$|{\\{x_{i}\\};z_{1},z_{2}}\rangle^{e}$, and
$\mathcal{H}^{\\{1,e\\}|\\{1,m\\}}$ is spanned by
$|{\\{x_{i}\\};\sigma,\sigma}\rangle^{m}$, with the superscript denoting which
Hilbert space the states belong to. A naive choice of the reduced density
matrix is a direct sum of the two reduced density matrix in Eq. (36) and Eq.
(36),

$\displaystyle\rho^{p}=$ $\displaystyle p\rho^{e}\otimes(1-p)\rho^{m}$ (47)
$\displaystyle=$ $\displaystyle p\sum_{z_{1}(\prod
x_{i})z_{2}=1}\frac{d_{z_{1}}d_{x_{1}}...d_{x_{n}}d_{z_{2}}}{2^{n+1}}|{x_{1},\cdots,x_{n};z_{1},z_{2}}\rangle\langle{x_{1},\cdots,x_{n};z_{1},z_{2}}|+$
(48)
$\displaystyle(1-p)\sum_{\\{x_{i}\\}}\frac{d_{\sigma}d_{x_{1}}...d_{x_{n}}d_{\sigma}}{2^{n+1}}|{x_{1},\cdots,x_{n};\sigma,\sigma}\rangle\langle{x_{1},\cdots,x_{n};\sigma,\sigma}|,$
(49)

where $p\in[0,1]$ is a parameter that cannot be model-independently
determined. The parameter $p$ is the classical probability of finding the EB
state in $\mathcal{H}^{e}$. For the lattice models, we have $p=0$, such that
the charges are confined and fluctuate on the EB. For the folding trick, we
should have $p=\frac{1}{2}$, such that the probability of finding the EB state
in $\mathcal{H}^{e}$ and $\mathcal{H}^{m}$ are equal. This equality is the
basic requirement of EM duality: the EB has $50\%$ chance condensing $e$ and
$50\%$ chance condensing $m$. This basic requirement is a necessary condition
for the $e$ anyons and the $m$ anyons being treated equally on the EB.

Nevertheless, this naive choice of the reduced density matrix $\rho^{p}$ can
not reproduce the result calculated in the $K$-matrix theory. The entanglement
entropy is verified to be

$\displaystyle S_{\textrm{EE}}=$ $\displaystyle\sum_{z_{1}(\prod
x_{i})z_{2}=1}\frac{pd_{z_{1}}d_{x_{1}}...d_{x_{n}}d_{z_{2}}}{2^{n+1}}\textrm{ln}\frac{pd_{z_{1}}d_{x_{1}}...d_{x_{n}}d_{z_{2}}}{2^{n+1}}$
(50)
$\displaystyle-\sum_{\\{x_{i}\\}}\frac{(1-p)d_{\sigma}d_{x_{1}}...d_{x_{n}}d_{\sigma}}{2^{n+1}}\textrm{ln}\frac{(1-p)d_{\sigma}d_{x_{1}}...d_{x_{n}}d_{\sigma}}{2^{n+1}}$
(51) $\displaystyle=$
$\displaystyle(n+1)\textrm{ln}2-p\textrm{ln}p-(1-p)\textrm{ln}(2-2p).$ (52)

For $p=\frac{1}{2}$, the sub-leading term is positive. If we look at the
operators of this subsystem, we will find two independent sets of operators
$O_{e}$ and $O_{m}$ that respectively transform the states within
$\mathcal{H}^{e}$ or $\mathcal{H}^{m}$; each set has its own operator
constraint, and no operator can transform states between $\mathcal{H}^{e}$ and
$\mathcal{H}^{m}$. This doubled operator constraints tell us that the whole
Hilbert space $\mathcal{H}^{\textrm{EM}}$ is too large. To reproduce the
results calculated in the $K$-matrix theory, the density matrix
$\rho^{p=\frac{1}{2}}$ should have have some zero eigenvalues, such that not
all the states in $\mathcal{H}^{\textrm{EM}}$ contribute to entanglement
entropy. We now construct such a mixed state.

###  5.3 The general density matrix

Technically, we can still work in the basis of
$|{x_{1},\cdots,x_{n};z_{1},z_{2}}\rangle$ and
$|{x_{1},\cdots,x_{n};\sigma,\sigma}\rangle$ and add coherence in the density
matrix $\rho^{p}$. According to the discussion in section 3.4, adding
coherence may result in local contributions to TEE. How to prevent from the
inhomogeneity while adding coherence?

The answer lies in the exact EM duality of the topological orders with gapped
boundaries Wang  _et al._ (2020). Mathematically, there is a map between the
allowed set of anyons on boundaries with different GBCs induced by the EM
duality. For the $D(\mathbb{Z}_{2})$ topological order, this map is an
isomorphism, namely $\mathcal{D}^{\\{1,e\\}}\cong\mathcal{D}^{\\{1,m\\}}$.
This isomorphism induces a natural map between the two Hilbert spaces
$\mathcal{H}^{e}$ and $\mathcal{H}^{m}$. Let us first see how this map work in
the case of closed EB.

####  5.3.1 Closed EB

Let us go back to the case of $D(\mathbb{Z}_{2})$ on a sphere, where
$\mathcal{H}^{e}$ and $\mathcal{H}^{m}$ are isomorphic to each other. It is
simple to establish the map

$|{x_{1},x_{2},\cdots,x_{n}}\rangle^{e}\leftrightarrow|{x_{1},x_{2},\cdots,x_{n}}\rangle^{m}.$
(53)

With this map, the basis states of the Hilbert space
$\mathcal{H}^{\textrm{EM}}$ can be decomposed into $2^{n-1}$ pairs, and the
reduced density matrix can be rearranged as

$\rho^{p}=\frac{1}{\Psi}\underbrace{\bordermatrix{&|{\\{x_{i}\\}}\rangle^{e}&|{\\{x_{i}\\}}\rangle^{m}\cr&p&0\cr&0&1-p}\oplus\cdots}_{2^{n-1}\textrm{
blocks}}$ (54)

A $|{\\{x_{i}\\}}\rangle$ stands for a certain defect configuration on the EB.
According to the discussions above, there are $2^{n-1}$ different
configurations; each of the $2^{n-1}$ blocks corresponds one configuration.
Adding coherence terms within each block in Eq. (54) but not between the
blocks prevents us from the local contributions. For the EB being homogeneous,
each block should receive the same coherence term. Therefore, the general form
of the reduced density matrix for $\mathbb{Z}_{2}$ topological order in closed
EB cases is

$\rho^{p}=\frac{1}{\Psi}\underbrace{\bordermatrix{&|{\\{x_{i}\\}}\rangle^{e}&|{\\{x_{i}\\}}\rangle^{m}\cr&p&c\cr&c^{*}&1-p}\oplus\cdots}_{2^{n-1}\textrm{
blocks}},$ (55)

where $c$ is the coherence term to be added. To reproduce the results
compatible with the folding trick, at least one of the eigenvalues of the
block should become zero, such that the dimension of the physical space is
reduced. As such, the only possible coherence term, in this simple case, is
then

$c=\sqrt{p(1-p)}e^{i\phi},$ (56)

where $\phi$ is a parameter that cannot be determined model-independently.
Diagonalizing this new density matrix gives

$\rho=\frac{1}{\Psi}\underbrace{\bordermatrix{&|{\phi^{+},\\{x_{i}\\}}\rangle&|{\phi^{-},\\{x_{i}\\}}\rangle\cr&1&0\cr&0&0}\oplus\cdots}_{2^{n-1}\textrm{
blocks}}$ (57)

and the $S_{\textrm{TEE}}=-$ln$2$ is independent of $p$. The eigenvectors of
this density matrix are
$|{\phi^{\pm},\\{x_{i}\\}}\rangle=\sqrt{(1-p)}|{\\{x_{i}\\}}\rangle^{e}\pm\sqrt{p}e^{i\phi/2}|{\\{x_{i}\\}}\rangle^{m}$,
where the vectors with plus (minus) sign has eigenvalue $1$ ($0$). In the
perspective of a local observer in the subsystem, what she sees is a classical
ensemble of these $|{\phi^{+},\\{x_{i}\\}}\rangle$ states. If we require the
EM duality, namely $p=\frac{1}{2}$, each $|{\phi^{+},\\{x_{i}\\}}\rangle$ is a
superposition of a $\mathcal{H}^{e}$ state and a $\mathcal{H}^{m}$ state with
equal probability. This equality is the advanced requirement of EM duality:
every states that a subsystem observer can see treat $e$ and $m$ equally. In
this simple case, this stronger requirement is recovered so long as the basic
requirement is fulfilled.

What happens to the operator constraint? In fact, the choice of the map is not
unique. Imposing any fixed combinations of $O_{e}$ ($O_{m}$) operators on the
left (right) hand side in expression (53) also gives valid maps. Using
different maps merely changes the explicit expression of
$|{\phi^{\pm}}\rangle$ but results in no physical consequence. Hence, the
operators $O_{e}$ and $O_{m}$ serve as ‘gauge’ transformations; only the
combinations of the two $O_{e}O_{m}$ transform a physical state, i.e. a
$|{\phi^{+},\\{x_{i}\\}}\rangle$ state seen by the subsystem observer, to
another. As such, the operator constraint of the physical states, now composed
of only $O_{e}O_{m}$ operators, is the same as that in Section 3.4. So does
TEE. The choice of the parameter $\phi$ here does not change the operator
constraint and lead to no physical consequence, either.

####  5.3.2 Open EB

Now we are ready to deal with the case of $D(\mathbb{Z}_{2})$ on a disk. Note
that $\mathcal{H}^{e}=\mathcal{H}^{\\{1,e\\}|\\{1,e\\}}$ has $2^{n+1}$
norm-$1$ vectors and $\mathcal{H}^{m}=\mathcal{H}^{\\{1,e\\}|\\{1,m\\}}$ has
$2^{n}$ norm-$\sqrt{2}$ vectors. The mapping between the two Hilbert spaces is
two-to-one,

$\displaystyle\\{|{\\{x_{i}\\};z_{1},z_{2}}\rangle^{e},|{\\{x_{i}\\};\tilde{z}_{1},\tilde{z}_{2}}\rangle^{e}\\}\leftrightarrow|{\\{x_{i}\\};\sigma,\sigma}\rangle^{m};$
(58)

With this map, the Hilbert space $\mathcal{H}^{e}\oplus\mathcal{H}^{m}$ can be
decomposed into $2^{N-1}$ triplets. The general form of the reduced density
matrix in this case is then

$\rho=\frac{1}{\Psi}\underbrace{\bordermatrix{&|{\\{x_{i}\\};z_{1},z_{2}}\rangle^{e}&|{\\{x_{i}\\};\tilde{z}_{1},\tilde{z}_{2}}\rangle^{e}&|{\\{x_{i}\\};\sigma,\sigma}\rangle^{m}\cr&p&c&d\cr&c^{*}&p&f\cr&d^{*}&f^{*}&2-2p}\oplus\cdots}_{2^{n}\textrm{
blocks}},$ (59)

where $c,d,e$ are coherence terms to be added. To fulfil the advanced
requirement of EM duality and vanish at least one eigenvalue for each block,
it turns out that

$p=\frac{1}{2};c=e^{i\phi};d=\frac{\lambda
e^{i\phi^{\prime}}}{\sqrt{2}};f=\frac{\lambda
e^{i\phi-i\phi^{\prime}}}{\sqrt{2}},$ (60)

where $\lambda$, $\phi$, and $\phi^{\prime}$ are some parameters that cannot
be determined model-independently. The eigenvalues of each block is
$\frac{1\pm\lambda}{2}$ and $0$. The eigenvectors for $\frac{1\pm\lambda}{2}$
are

$|{\phi^{\pm};\\{x_{i},z_{1},z_{2}\\}}\rangle\equiv\frac{1}{2}(|{\\{x_{i}\\};z_{1},z_{2}}\rangle^{e}+e^{i\phi}|{\\{x_{i}\\},\tilde{z}_{1},\tilde{z}_{2}}\rangle^{e}\pm
2^{\frac{1}{4}}e^{i\phi^{\prime}}|{\\{x_{i}\\},\sigma,\sigma}\rangle^{m}).$
(61)

Verifying TEE gives
$S_{\textrm{TEE}}=-\frac{1+\lambda}{2}\textrm{ln}(1+\lambda)-\frac{1-\lambda}{2}\textrm{ln}(1-\lambda)$.
For TEE to be $-\frac{1}{2}$ln$2$, the parameter $\lambda$ is $\pm 0.779944$.
The parameters $p$ and $\lambda$ completely characterize the ambiguity in the
open EB cases for $\mathbb{Z}_{2}$ topological order. These parameters suggest
that there are extra degrees of freedom in factorizing the extended
topological quantum field theory (TQFT) that underlies beneath the topological
order with gapped boundaries, and the folding trick naturally chooses a
specific values for $p$ and $\lambda$. While $p$ quantifies the probability of
finding the states in a certain GBC, the physical meaning of $\lambda$ awaits
to be clarified in future works.

We end our analysis by commenting on the operator constraint of the physical
space. As in the case of closed EB, the choice of the two-to-one map is also
not unique, and therefore some of the operators are ‘gauged’, leaving one
constraint on physical operators. According to Eq. (41), the generating
operator multiplies to $O_{z_{1},z_{2}}$, whose explicit form will depend on
$\lambda$ if restricted to the transformation between the physical states.

##  6 Conclusion and outlook

In this paper, we introduced a model-independent picture to understand the
TEE. The TEE in general has three parts: a ln$d_{x}$ part due to the total
charges of the EBs, where $x$ can be a bulk anyon or a boundary defect; a
$-$ln$D$ part that characterized by the operator constraint; and a classical
part due to the superposition of basis states. It turns out that in closed EB
cases, the TEE is unambiguous and hence is good enough to characterize
topological orders. The TEE of open EB cases, particularly the $-$ln$D$ part,
depends on the EB condition and the choice of the mixed state. We introduced a
general constructions of the reduced denstiy matrix for the
$D(\mathbb{Z}_{2})$ topological order in both closed EB cases and open EB
cases. There are two key parameters in the construction: the probability $p$
of finding the EB in a certain GBC, and the coherence $\lambda$ between the
two junction points. To reproduce the compatibility with the folding trick, on
should adopt EM dual EB condition and set a certain value for $\lambda$.
Though we have focused on the $D(\mathbb{Z}_{2})$ topological order for
simplicity, our analysis generalize easily to all Abelian topological orders.

Figure 11: A composite system with topological orders (the blue region and the
orange region) on a torus separated by a gapped domain wall. If an
entanglement cut intersect with the domain wall, the intersection points will
also exhibit non-trivial behavior.

In this paper, we focus on the topological orders that are describable by both
lattice models and the CS theory. The primary reason for this restriction is
because the mismatch is observed only in these topological orders.
Nonetheless, our picture is model-independent, it is possible to generalize
our analysis to non-Abelian topological orders with gapped boundaries. For
non-Abelian topological orders, the operators are non-invertible and hence
requires different treatment. Studying non-Abelian cases might require the
notion of non-invertible symmetriesGaiotto  _et al._ (2015) and partial
electric-magnetic dualityHu and Wan (2020). Moreover, in the CS theory, the
EBs are treated as gapless boundaries described by Ishibashi states. The
discussion hints us that the gapped boundary and the gapless boundary of a
topological order may be related in some way; it is likely that they differ by
how the boundary points are coherent with each other. Moreover, the gapless
CFT boundaries are gappableLevin (2013), studying the compatibility of gapped
GBC and the folding trick will help us to further understand how CFT states
are folded into gapped boundary states. Finally, more general cases, such as
cases involve gapped domain wallsZhao  _et al._ (2023), are also worth
investigating. If the entanglement cut goes through the domain wall, as shown
in Fig. 11, the EB-domain wall junction will also have interesting properties.

###### Acknowledgements.

YL thanks Ling-yan Hung, Yidun Wan, Hongyu Wang, Yanyan Chen, and Yu Zhao for
helpful discussions. The author appreciates his supervisor Prof. Yidun Wan for
supporting him via the General Program of Science and Technology of Shanghai
No. 21ZR1406700, and Shanghai Municipal Science and Technology Major Project
(Grant No.2019SHZDZX01).

##  Appendix A Review of properties of anyons

We review the necessary fact about anyons that we will use in this paper.
Interested readers are invited to read the review by NayakNayak  _et al._
(2008) for a comprehensive introduction.

Anyons are fundamental excitations in topological orders. For a certain
topological order, we label the anyon species as
$\mathcal{C}=\\{\alpha,\beta,\gamma,\cdots\\}$. The anyons subject to fusion
rules,

$\alpha\times\beta=\sum_{\gamma}N^{\gamma}_{\alpha\beta}\gamma,$ (62)

where $N^{\gamma}_{\alpha\beta}$ are non-negative integers. Physically, fusing
two anyons means if the two anyons $\alpha$ and $\beta$ are close enough to
each other, they will behave like a superposition of $\gamma$ anyons. If we
are in a state where a set of anyons fuse to $\gamma$, then we say that the
total topological charge of this set of anyons is $\gamma$. The trivial anyon
$1$, also called the vacuum, is the identity of the fusion

$1\times\alpha=\alpha\times 1=\alpha.$ (63)

The conjugate anyon $\bar{\alpha}$ of an anyon $\alpha$ is defined as
$N^{1}_{\bar{\alpha}\alpha}\neq 0$. For an anyon $\alpha$, if for each
$\gamma$ there is only one $\beta$ that satisfies $N^{\gamma}_{\alpha\beta}=1$
and all other $N^{\gamma}_{\alpha\beta^{\prime}}=0$, the anyon $\alpha$ is
called Abelian. If all anyons in a topological order are Abelian, then the
topological order is called an Abelian topological order.

The Hilbert space of multiple anyons on a sphere is spanned by the possible
fusion channels of these anyons. We can denote the basis states of the
$n$-anyon Hilbert space using the following diagram,

$H_{n\textrm{ anyons}}=\textrm{span}\\{\Biggl{|}\begin{matrix}\leavevmode\hbox
to91.5pt{\vbox
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} {}{{}}{}
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{matrix}\Biggr{\rangle},\textrm{for
all allowed }\\{\beta_{i}\\}\\}$ (64)

where $\beta_{i}\in\mathcal{C}$ are internal degrees of freedom, and we impose
fusion rules on each vertex. One finds that the dimension of the Hilbert space
for Abelian anyons is always $1$.

The quantum dimension $d_{\alpha}$ of the anyon $\alpha$ can is the effective
dimension of the single anyon Hilbert space in the thermodynamic limit. The
quantum dimensions can be derived from the fusion rule,

$d_{\alpha}d_{\beta}=\sum_{\gamma}N^{\gamma}_{\alpha\beta}d_{\gamma}.$ (65)

One immediately finds that $d_{\bar{\alpha}}=d_{\alpha}$ and $d_{1}=1$, and
all Abelian anyons have quantum dimension $1$.

The most prominent property of the anyons is their non-trivial braiding
statistics. Exchanging the position of two anyons will give one a unitary
transformation on the internal Hilbert space of the anyons,

$\Biggl{|}\begin{matrix}\leavevmode\hbox to45.75pt{\vbox
to198.69pt{\pgfpicture\makeatletter\hbox{\hskip-217.125pt\lower-289.81943pt\hbox
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{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{232.4625pt}{-97.52249pt}\pgfsys@curveto{232.41pt}{-102.1575pt}{233.97pt}{-104.4975pt}{236.1075pt}{-106.08pt}\pgfsys@stroke\pgfsys@invoke{
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{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{242.31pt}{-109.47pt}\pgfsys@curveto{244.34999pt}{-110.7pt}{247.5525pt}{-113.7675pt}{247.5pt}{-120.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{{}}{}{{}}{}\pgfsys@moveto{239.25pt}{-121.5pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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162.0pt\cr\hbox{{$\displaystyle\beta_{i}$}}\cr}}}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{246.75pt}{-91.5pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{247.875pt}{-217.93442pt}\pgfsys@invoke{
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122.0pt\cr\hbox{{$\displaystyle\alpha_{i+1}$}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{225.75pt}{-91.5pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{matrix}\Biggr{\rangle}=\sum_{\beta^{\prime}_{i}}B^{\alpha_{i}\alpha_{i+1}}_{\beta_{i}\beta^{\prime}_{i}}\Biggl{|}\begin{matrix}\leavevmode\hbox
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{}{}{}\pgfsys@moveto{247.5pt}{-97.5pt}\pgfsys@lineto{247.5pt}{-112.5pt}\pgfsys@stroke\pgfsys@invoke{
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{}{{}}{}{{}}{}\pgfsys@moveto{254.25pt}{-114.3225pt}\pgfsys@stroke\pgfsys@invoke{
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152.43pt\cr\hbox{{$\displaystyle\beta^{\prime}_{i}$}}\cr}}}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{263.25pt}{-91.5pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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122.0pt\cr\hbox{{$\displaystyle\alpha_{i+1}$}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{240.75pt}{-91.5pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{matrix}\Biggr{\rangle},$
(66)

where $B$ stands for the transformation. For Abelian anyons, this unitary
transformation is merely a phase factor. A crucial point is that we can detect
anyons based on the non-trivial braiding statistics. Suppose we create two
pairs of anyons $\alpha$, $\bar{\alpha}$, $\beta$, and $\bar{\beta}$ on a
sphere. If we let the anyon $\beta$ wind around the anyon $\alpha$, and then
annihilate the two pairs $\alpha$ and $\bar{\alpha}$, $\beta$ and
$\bar{\beta}$, we will result into a matrix entry $S_{\alpha\beta}$.

$S_{\alpha\beta}=\Biggl{|}\begin{matrix}\leavevmode\hbox to54.63pt{\vbox
to108.69pt{\pgfpicture\makeatletter\hbox{\hskip-185.745pt\lower-184.81943pt\hbox
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{215.2125pt}{-86.12999pt}\pgfsys@curveto{217.8375pt}{-83.865pt}{221.25749pt}{-82.5pt}{225.0pt}{-82.5pt}\pgfsys@curveto{233.28749pt}{-82.5pt}{240.0pt}{-89.2125pt}{240.0pt}{-97.5pt}\pgfsys@curveto{240.0pt}{-105.78749pt}{233.28749pt}{-112.5pt}{225.0pt}{-112.5pt}\pgfsys@curveto{216.7125pt}{-112.5pt}{210.0pt}{-105.78749pt}{210.0pt}{-97.5pt}\pgfsys@curveto{210.0pt}{-94.50749pt}{210.87749pt}{-91.72499pt}{212.385pt}{-89.385pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{187.5pt}{-97.5pt}\pgfsys@lineto{188.87999pt}{-100.3725pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{187.5pt}{-97.5pt}\pgfsys@lineto{186.12pt}{-100.3725pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{210.0pt}{-97.5pt}\pgfsys@lineto{208.62pt}{-100.3725pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{210.0pt}{-97.5pt}\pgfsys@lineto{211.37999pt}{-100.3725pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}}{}\pgfsys@moveto{188.25pt}{-76.5pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{189.375pt}{-181.77777pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\tiny{\vbox{\halign{#\hfil\cr\cr\vskip
102.0pt\cr\hbox{{$\displaystyle\alpha$}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{231.75pt}{-76.5pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{232.875pt}{-184.06943pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\tiny{\vbox{\halign{#\hfil\cr\cr\vskip
102.0pt\cr\hbox{{$\displaystyle\beta$}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(67)

This matrix $S_{\alpha\beta}$ is an important observable that characterizes
the topological order and is called the topological S-matrix. Physically, one
can use this AB phase to distinguish anyons: one can create an anyon pair, let
them wind around the anyon to be recognized, and then annihilate the test
anyons to acquire the amplitude of this process. Repeat this process for all
$\alpha\in\mathcal{C}$ gives one a column in the S-matrix, which is enough to
recognize the anyon.

##  Appendix B Norm of the boundary anyon states

To understand the norm of the boundary anyon state, we consider the whole
space-time picture as shown in Fig. (12).

Figure 12: Caption

Suppose we have $n$ anyons on the boundary. They were created at the time
before the time slice we are focusing on. The worldline of these $n$ boundary
anyons can be arranged as shown in the Fig. (12a). The basis of the $n$-anyon
Hilbert space is also represented by these worldline diagrams, which is
exactly the basis vectors we introduced in Eq. ((64)). The Hermitian conjugate
of a $n$-anyon state is conversely represented by the worldline of the $n$
anyons that annihilate with each other in the future, as shown in Fig. (12b).
Taking the norm amounts to connecting the two diagrams and finding the
amplitude of the diagram

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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\tiny{\vbox{\halign{#\hfil\cr\cr\vskip 117.38pt\cr\hbox{{$\displaystyle
x_{n}$}}\cr}}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{221.205pt}{-77.1375pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{222.33pt}{-187.41249pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{\vbox{\halign{#\hfil\cr\cr\vskip
102.85pt\cr\hbox{{$\displaystyle\cdots$}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{matrix},$ (68)

which is $\sqrt{d_{x_{1}}d_{x_{2}}\cdots d_{x_{n}}}$ for all states in this
four-anyon Hilbert space.(See ref Nayak  _et al._ (2008) for details).

##  Appendix C Verifying the normalization factor and the entanglement entropy

We stick to the notation where $D$ is the total quantum dimension of the bulk
topological order, and the lower latin alphabets $a,b,x$ represent for
boundary anyons. We have $D=\sum_{a\in\mathcal{D}}d_{a}^{2}$, where
$\mathcal{D}$ is the set of boundary anyons.

  * •

$\sum_{\prod x_{i}=a}d_{x_{1}}d_{x_{2}}\cdots d_{x_{n}}=d_{a}D^{n-1}$ (69)

Proof. We prove this equation by induction. For $n=1$ this equation holds
trivially. Suppose the equation holds for a generic $n$; then, for the $n+1$
case,

$\displaystyle\sum_{\prod_{x_{i}}=a}d_{x_{1}}\cdots d_{x_{n}}d_{x_{n+1}}$
$\displaystyle=$
$\displaystyle\sum_{x_{n+1}}d_{x_{n+1}}\sum_{\prod_{x_{i}}^{i\leq
n}=\bar{x}_{n+1}a}d_{x_{1}}\cdots d_{x_{n}},$

where the second summation in the second line runs over all possible
combinations of $x_{1}\cdots x_{n}$ that fuses to any of the $c$’s in
$\bar{x}_{n+1}\times a=N^{c}_{\bar{x}_{n+1}a}c$. Then,

$\displaystyle\sum_{\prod_{x_{i}}=a}d_{x_{1}}\cdots d_{x_{n}}d_{x_{n+1}}$
$\displaystyle=$
$\displaystyle\sum_{x_{n+1}}d_{x_{n+1}}N^{c}_{\bar{x}_{n+1}a}\sum_{\prod_{x_{i}}^{i\leq
n}=c}d_{x_{1}}\cdots d_{x_{n}}$ $\displaystyle=$
$\displaystyle\sum_{x_{n+1}}d_{x_{n+1}}N^{c}_{\bar{x}_{n+1}a}d_{c}D^{n-1}$
$\displaystyle=$
$\displaystyle\sum_{x_{n+1}}d_{x_{n+1}}d_{x_{n+1}}d_{a}D^{n-1}$
$\displaystyle=$ $\displaystyle d_{a}D^{n},$

where in the third equality we used Eq. ((65)).

  * •

$\sum_{\prod x_{i}=a}\frac{d_{x_{1}}\cdots
d_{x_{n}}}{d_{a}D^{n-1}}\textrm{ln}\frac{d_{x_{1}}\cdots
d_{x_{n}}}{d_{a}D^{n-1}}=n\sum_{x\in\mathcal{D}}\frac{d_{x}^{2}}{D}\textrm{ln}\frac{d_{x}}{D}+\textrm{ln}\frac{D}{d_{a}}$
(70)

Proof.

$\displaystyle\sum_{\prod x_{i}=a}\frac{d_{x_{1}}\cdots
d_{x_{n}}}{d_{a}D^{n-1}}\textrm{ln}\frac{d_{x_{1}}\cdots
d_{x_{n}}}{d_{a}D^{n-1}}$ $\displaystyle=$
$\displaystyle\sum_{x_{n}}d_{x_{n}}^{2}\left[\sum_{\prod^{n-1}_{i=1}x_{i}=\bar{x}_{n}a}\frac{d_{x_{1}}\cdots
d_{x_{n-1}}}{d_{a}d_{x_{n}}D^{n-1}}\textrm{ln}d_{x_{1}}\cdots
d_{x_{n}}\right]+\textrm{ln}\frac{1}{d_{a}D^{n-1}}$ $\displaystyle=$
$\displaystyle\sum_{x_{n}}d_{x_{n}}^{2}\left[\sum_{\prod^{n-1}_{i=1}x_{i}=\bar{x}_{n}a}\frac{d_{x_{1}}\cdots
d_{x_{n-1}}}{d_{a}d_{x_{n}}D^{n-1}}\textrm{ln}d_{x_{1}}\cdots
d_{x_{n-1}}+\frac{1}{D}\textrm{ln}d_{x_{n}}\right]+\textrm{ln}\frac{1}{d_{a}D^{n-1}}$
$\displaystyle=$
$\displaystyle\sum_{x_{n}}d_{x_{n}}^{2}\left[\sum_{\prod^{n-1}_{i=1}x_{i}=\bar{x}_{n}a}\frac{d_{x_{1}}\cdots
d_{x_{n-1}}}{d_{a}d_{x_{n}}D^{n-1}}\textrm{ln}d_{x_{1}}\cdots
d_{x_{n-1}}\right]+\sum_{x}\frac{d_{x}^{2}}{D}\textrm{ln}d_{x}+\textrm{ln}\frac{1}{d_{a}D^{n-1}}$
$\displaystyle=$
$\displaystyle\sum_{x_{n}}d_{x_{n}}^{2}\left[\sum_{x_{n-1}}d_{x_{n-1}}^{2}\left(\sum_{\prod^{n-2}_{i=1}x_{i}=\bar{x}_{n}\bar{x}_{n-1}a}\frac{d_{x_{1}}\cdots
d_{x_{n-2}}}{d_{a}d_{x_{n}}d_{x_{n-2}}D^{n-1}}\textrm{ln}d_{x_{1}}\cdots
d_{x_{n-2}}\right)+\frac{1}{D^{2}}\textrm{ln}d_{x_{n-1}}\right]$
$\displaystyle\hskip
30.0pt+\sum_{x}\frac{d_{x}^{2}}{D}\textrm{ln}d_{x}+\textrm{ln}\frac{1}{d_{a}D^{n-1}}$
$\displaystyle=$
$\displaystyle\sum_{x_{n},x_{n-1}}d_{x_{n}}^{2}d_{x_{n-1}}^{2}\left[\sum_{\prod^{n-2}_{i=1}x_{i}=\bar{x}_{n}\bar{x}_{n-1}a}\frac{d_{x_{1}}\cdots
d_{x_{n-2}}}{d_{a}d_{x_{n}}d_{x_{n-1}}D^{n-1}}\textrm{ln}d_{x_{1}}\cdots
d_{x_{n-2}}\right]+2\sum_{x}\frac{d_{x}^{2}}{D}\textrm{ln}d_{x}+\textrm{ln}\frac{1}{d_{a}D^{n-1}}$
$\displaystyle\vdots$ $\displaystyle\hskip 5.0pt\textrm{repeat }n\textrm{
times}$ $\displaystyle\vdots$ $\displaystyle=$ $\displaystyle
n\sum_{x}\frac{d_{x}^{2}}{D}\textrm{ln}d_{x}+\textrm{ln}\frac{1}{d_{a}D^{n-1}}=n\sum_{x}\frac{d_{x}^{2}}{D}\textrm{ln}\frac{d_{x}}{D}+\textrm{ln}\frac{D}{d_{a}}.$

##  Appendix D The local term due to inhomogeneous EB

We focus on the $D(\mathbb{Z}_{2})$ toric code on a sphere. The Eq. ((12)) is
not the unique choice of how the global state is constructed by the states in
the subsystem Hilbert spaces. The ground state of the topological order on a
sphere can be instead decomposed as, for example,

$\displaystyle|{\psi}\rangle=\frac{1}{\sqrt{\Psi}}$ $\displaystyle\sum_{\prod
x_{i}=1}\sqrt{\lambda}(|{x_{pq}}\rangle+e^{i\phi/2}|{\tilde{x}_{pq}}\rangle)_{\textrm{A}}(|{\tilde{x}_{pq}}\rangle+e^{i\phi/2}|{x_{pq}}\rangle)_{\textrm{B}}$
$\displaystyle+\sqrt{(1-\lambda)}(|{x_{pq}}\rangle-e^{i\phi/2}|{\tilde{x}_{pq}}\rangle)_{\textrm{A}}(|{\tilde{x}_{pq}}\rangle-e^{i\phi/2}|{x_{pq}}\rangle)_{\textrm{B}},$

where
$|{x_{pq}}\rangle\equiv|{x_{1},\cdots,x_{p},\cdots,x_{q},\cdots,x_{n}}\rangle$
and
$|{\tilde{x}_{pq}}\rangle\equiv|{x_{1},\cdots,\tilde{x}_{p},\cdots,\tilde{x}_{q},\cdots,x_{n}}\rangle$
and $\lambda$ is a real number $\lambda\in[0,1]$. This decomposition assumes
extra coherence between the two points $x_{p}$ and $x_{q}$. The reduced
density matrix then becomes

$\rho=\frac{1}{\Psi}\underbrace{\bordermatrix{&|{\\{x_{pq}\\}}\rangle&|{\\{\tilde{x}_{pq}\\}}\rangle\cr&1&\lambda
e^{i\phi}\cr&\lambda e^{-i\phi}&1}\oplus\cdots}_{2^{n-1}\textrm{ blocks}}.$
(71)

Verifying the TEE of this reduced density matrix gives

$S=(n-1)\textrm{ln}2-\frac{1+\lambda}{2}\textrm{ln}\frac{1+\lambda}{2}-\frac{1-\lambda}{2}\textrm{ln}\frac{1-\lambda}{2}.$
(72)

There is an extra sub-leading term in the entanglement entropy, as we
mentioned in the main text.

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