# Calculating Concentratable Entanglement in Graph States

Alice R. Cullen  Pieter Kok  Department of Physics and Astronomy, The
University of Sheffield, Hounsfield Road, S3 7RH Sheffield, United Kingdom

###### Abstract

We propose a method to calculate the purity of reduced states of graph states
entirely within the stabilizer formalism, using only the stabilizer generators
for a given state. We apply this method to find the Concentratable
Entanglement of graph states, with examples including a single qubit within
any graph state, states proposed for use in quantum repeaters and states that
maximise and minimise the overall Concentratable Entanglement.

##  I Introduction

The quantum phenomenon of entanglement Einstein  _et al._ (1935); Schrödinger
(1935) that is present in quantum states associated with graphs Hein  _et al._
(2004) makes such states a useful resource in quantum information processing.
When used for Measurement Based Quantum Computation (MBQC) Briegel  _et al._
(2009) and the One-Way Quantum Computer Raussendorf and Briegel (2001);
Raussendorf  _et al._ (2003), the specific computation dictates the structure
of the graph state. Conversely, the entanglement structure of a graph state
determines its effectiveness in applications such as the quantum repeater
Briegel  _et al._ (1998); Zwerger  _et al._ (2012); Azuma  _et al._ (2015),
quantum error correction (QEC) codes Gottesman (1997); Schlingemann and Werner
(2001) and quantum secret sharing Markham and Sanders (2008). An entanglement
measure is required to investigate the structure of the multipartite
entanglement within a graph state.

Entanglement across the two subsystems within a bipartite quantum state is
well understood. An initial qualitative test of entanglement is the Peres-
Horodecki criterion Peres (1996); Horodecki  _et al._ (1996). A widely used
quantitative measure is entanglement of formation Bennett  _et al._ (1996),
which for pure states coincides with the von Neumann entropy of the reduced
state of either subsystem. The definition can be extended to mixed states by
considering ensembles of pure states. Entanglement of formation for pure
states can also be defined as a function of concurrence Hill and Wootters
(1997); a different entanglement measure that applies to pure states of two
qubits, which can be adapted to apply to bipartite systems Rungta  _et al._
(2001).

Entanglement in multipartite states is more difficult to characterise and
quantify, as the ways in which a state can be entangled increases with the
number of parties involved. A widely cited example of different kinds of
entanglement, first discussed by Dür et al. Dür  _et al._ (2000), is
demonstrated when considering three party entanglement, as the entanglement
present in a three qubit GHZ state differs from that in a three qubit W state.
It is not possible to transform either state to the other using Local
Operations and Classical Communication (LOCC). Several existing multipartite
measures are functions of bipartite entanglement measures Meyer and Wallach
(2002); Brennen (2003); Carvalho  _et al._ (2004), sometimes considered over
multiple bipartitions of the state; for example this approach is applied to
generalise concurrence to multipartite states Carvalho  _et al._ (2004).

A recently proposed multipartite entanglement measure is Concentratable
Entanglement Beckey  _et al._ (2021). When defined for different cuts of
qubits within a state, it relates to several existing measures Meyer and
Wallach (2002); Brennen (2003); Hill and Wootters (1997); Rungta  _et al._
(2001); Carvalho  _et al._ (2004); Foulds  _et al._ (2021); Wong and
Christensen (2001), making it a candidate for adoption as the universal
multipartite entanglement measure of pure states. Like other measures, one
approach to calculating Concentratable Entanglement is to consider
bipartitions of the multipartite quantum state. The purities of the
corresponding reduced density matrices are required to evaluate the
Concentratable Entanglement.

The multipartite entanglement of graph states has previously been investigated
in terms of Schmidt measure and Schmidt rank Hein  _et al._ (2004, 2006),
which can be challenging to compute. Concentratable Entanglement offers a new
measure that can be applied to graph states. Using the stabilizer formalism
Gottesman (1997), which gives a convenient and efficient mathematical
description of graph states, we have developed a method reliant only on the
stabilizer generators for determining reduced state purities and thus
Concentratable Entanglement for graph states. This allows for calculation of
the Concentratable Entanglement for examples such as single qubits in graph
states, for subsets of qubits within graph states proposed for use in quantum
repeaters and as a measure of the overall entanglement for any graph state.

##  II Background

We begin by outlining the mathematical background required to define
Concentratable Entanglement Beckey  _et al._ (2021), graph states within the
stabilizer formalism and how measurements of individual qubits affect a graph
state Hein  _et al._ (2004).

###  II.1 Concentratable Entanglement

Concentratable Entanglement Beckey  _et al._ (2021) can be used as a measure
of multipartite entanglement of a pure quantum state $\ket{\psi}$. For a state
of $n$ qubits with labels in $\mathcal{S}=\\{1,\ldots,n\\}$, the
Concentratable Entanglement can be calculated for the full set to determine
the overall entanglement, or it can be applied to any non-empty subset
$s\subseteq\mathcal{S}$ to investigate the structure of the entanglement
within the state. Operationally, the Concentratable Entanglement corresponds
to the probability of producing Bell pairs in a SWAP test Buhrman  _et al._
(2001). The Concentratable Entanglement of $\ket{\psi}$ can be defined in
terms of SWAP test outcomes, making it possible to efficiently calculate its
value using quantum computers and two copies of $\ket{\psi}$.

However, without the physical resource of a quantum computer, the definition
of the Concentratable Entanglement in terms of purities still offers a method
to determine the entanglement of a (subset of a) state. For a non-empty set of
qubit labels $s\subseteq\mathcal{S}$, the Concentratable Entanglement is
Beckey  _et al._ (2021)

$C_{\ket{\psi}}(s)=1-\frac{1}{2^{|s|}}\sum_{\alpha\in\mathcal{P}(s)}\text{Tr}\rho_{\alpha}^{2},$
(1)

where $\rho_{\alpha}$ is the density matrix of the reduced state with qubits
labelled by the set $\alpha$, which is a subset of the power set
$\mathcal{P}(s)$. This requires the calculation of up to $2^{n}$ purities,
including the trivial case $\text{Tr}\rho_{\emptyset}^{2}=1$.

###  II.2 Graph States

In order to describe graph states, we first define the Pauli matrices as

$X=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\quad Y=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad Z=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (2)

A graph $G=(V,E)$ is determined by a set of vertices $V$ and a set of edges
$E$ connecting pairs of vertices. Graphs can be used as pictorial
representations of entangled quantum states. Each vertex represents a qubit in
the $\ket{+}$ eigenstate, i.e., the eigenstate of the Pauli $X$ operator
corresponding to the eigenvalue $+1$. Edges denote interactions between qubits
in the form of controlled-$Z$ (CZ) operations. Such quantum states are known
as graph states. Since there is no entanglement between disconnected graphs,
we consider only connected graphs.

The neighbourhood $n(a)$ of a vertex $a\in V$ is the set of vertices connected
to $a$ by a single edge. In a graph state this identifies the qubits that
jointly undergo a CZ operation with $a$.

Each qubit in the graph state $\ket{G}$ can be assigned a Hermitian operator
of the form Raussendorf and Briegel (2001)

$S_{a}=X_{a}\prod_{b\in n(a)}Z_{b}.$ (3)

These $n$ operators form the generators of an Abelian group, known as the
stabilizer $\mathfrak{S}_{G}$ of the state $\ket{G}$. The $2^{n}$ elements in
the stabilizer are the products of all possible subsets of the stabilizer
generators including the identity $I$. The stabilizer is a subgroup of the
Pauli group for $n$ qubits, i.e., $\mathfrak{S}_{G}\subset\mathcal{G}_{n}$.
The state $\ket{G}$ is an eigenstate with eigenvalue $+1$ for any element $S$
of the stabilizer

$S\ket{G}=\ket{G}\quad\forall S\in\mathfrak{S}_{G}.$ (4)

The stabilizer formalism provides an efficient way to fully specify graph
states.

###  II.3 Measurement

When a gate from the Clifford group or a Pauli measurement is applied to a
stabilizer state, the resulting state is also a stabilizer state. This is a
useful property that can be applied when considering measurements on qubits
within graph states.

The simplest measurement to describe on a graph state is that of the Pauli $Z$
operator on a single qubit. As an immediate consequence of such a measurement,
the measured qubit is removed from the graph state. Depending on the
measurement outcome with possible values $+1$ or $-1$, the individual qubit is
in one of the two eigenstates of the $Z$ operator, $\ket{0}$ or $\ket{1}$,
with stabilizer generator $Z$ or $-Z$ respectively.

Though the measured qubit is disconnected from the graph state, it still
affects the remaining state. When qubit $a\in V$ is measured, the graph
becomes $G^{\prime}=G-\\{a\\}$ where all edges to vertex $a$ are removed. The
state of the remaining qubits is $U\ket{G-\\{a\\}}$ where $U$ is a unitary
operator determined by the measurement outcome of $a$. For the outcomes $\pm
1$ the respective unitaries are

$U_{+}=I\quad\text{and}\quad U_{-}=\prod_{b\in n(a)}Z_{b}.$ (5)

In the stabilizer formalism the stabilizer generator $S_{b}^{\prime}$ for each
remaining vertex $b\in V^{\prime}$ will again take the form of Eq. (3). The
unitary operator corresponding to the outcome is applied to each generator as
$US_{b}^{\prime}U^{\dagger}$ to give the generators for the state
$U\ket{G-\\{a\\}}$.

An alternative method is to take the original set of stabilizer generators for
$\ket{G}$ and set the stabilizer generator for the measured qubit to $\pm
Z_{a}$. The generators for qubits that were not within the neighbourhood of
qubit $a$ remain unchanged. However the generators for qubits within the
neighbourhood of $a$, which contain the term $Z_{a}$, are multiplied by the
new generator $\pm Z_{a}$, allowed by the group structure of the new
stabilizer. This produces the new stabilizer generators without explicitly
drawing the graph representing the quantum state.

##  III Purities of Reduced Graph States

Calculating the purities required within the Concentratable Entanglement
relies on the reduced states of different subsets of qubits. Within the
stabilizer formalism, a reduced state can be described as a mixture of several
pure states, each of which has its own set of stabilizer generators. We show
that the number of distinct sets of stabilizer generators corresponds to the
purity of the reduced state that the sets of generators collectively describe.

Consider a bipartition $(A,B)$ of qubits within the graph $G$. The reduced
state of the qubits in $B$ is found by tracing out the qubits in $A$

$\rho_{B}=\text{Tr}_{A}(\ket{G}\bra{G}).$ (6)

This partial trace is equivalent to measuring the qubits in $A$ and discarding
the outcome. These measurements can be performed in the $Z$-direction, where
the outcomes $\pm 1$ each occur with probability $1/2$.

Through repeated application of the $Z$ measurement rule and unitaries given
in Eq. (5), and considering all possible outcomes, the density operator for
the reduced state of the qubits in $B$ is

$\rho_{B}=\frac{1}{2^{|A|}}\sum_{\textbf{z}\in\mathbb{F}_{2}^{A}}U(\textbf{z})\ket{G-A}\bra{G-A}U(\textbf{z})^{\dagger},$
(7)

where $\mathbb{F}_{2}$ is the finite field of two elements $\\{0,1\\}$. An
element of the finite field $\mathbb{F}_{2}^{A}$ is a bitstring z of length
$|A|$. In $\mathbb{F}_{2}^{A}$ addition and subtraction are equivalent and
performed modulo 2 without carry. The local unitaries are Hein  _et al._
(2004)

$U(\textbf{z})=\prod_{a\in A}\Bigg{(}\prod_{b\in n(a)\cap
B}Z_{b}\Bigg{)}^{z_{a}},$ (8)

where $z_{a}=0$ represents the outcome $+1$ and $z_{a}=1$ for $-1$. Within the
unitaries, only qubits in the neighbourhood of measured qubits which are also
in $B$ need to be considered. This is because the individual qubits in $A$ can
be measured in any order, making it possible for neighbourhood qubits to
already be disconnected from the graph. Further, $Z_{b}$ for $b\in A$ has no
effect on the state $\ket{G-A}$, which does not contain qubits in $A$.

For completeness, we prove the following properties of these unitary operators
that have previously been stated and applied Hein  _et al._ (2004), as they
provide the basis for proving our own result regarding the purities of reduced
states.

Lemma. (Hein-Eisert-Briegel) 1. For different measurement outcomes z and
$\textbf{z}^{\prime}$ to give the same unitary operator, i.e.
$U(\textbf{z})=U(\textbf{z}^{\prime})$, the bitstrings must satisfy Hein  _et
al._ (2004)

$U(\textbf{z}-\textbf{z}^{\prime})=I.$ (9)

2\. If $U(\textbf{z})\neq U(\textbf{z}^{\prime})$ then the states
$U(\textbf{z})\ket{G-A}$ and $U(\textbf{z}^{\prime})\ket{G-A}$ are orthogonal.

Proof. 1. Using the definition of the unitary operator in Eq. (8),

$\begin{split}U(\textbf{z}-\textbf{z}^{\prime})&=\prod_{a\in
A}\Bigg{(}\prod_{b\in n(a)\cap B}Z_{b}\Bigg{)}^{z_{a}-z^{\prime}_{a}}\\\
&=\prod_{a\in A}\Bigg{(}\prod_{b\in n(a)\cap
B}Z_{b}\Bigg{)}^{z_{a}}\prod_{a^{\prime}\in A}\Bigg{(}\prod_{b^{\prime}\in
n(a^{\prime})\cap B}Z_{b^{\prime}}\Bigg{)}^{-z^{\prime}_{a^{\prime}}}\\\
&=\prod_{a\in A}\Bigg{(}\prod_{b\in n(a)\cap
B}Z_{b}\Bigg{)}^{z_{a}}\prod_{a^{\prime}\in A}\Bigg{(}\prod_{b^{\prime}\in
n(a^{\prime})\cap B}Z_{b^{\prime}}^{-1}\Bigg{)}^{z^{\prime}_{a^{\prime}}}\\\
&=U(\textbf{z})U(\textbf{z}^{\prime})^{\dagger}.\end{split}$ (10)

This is equal to the identity

$U(\textbf{z}-\textbf{z}^{\prime})=U(\textbf{z})U(\textbf{z}^{\prime})^{\dagger}=I\\\
$ (11)

and by right multiplying by $U(\textbf{z}^{\prime})$, this gives
$U(\textbf{z})=U(\textbf{z}^{\prime})$.

2\. If $U(\textbf{z})\neq U(\textbf{z}^{\prime})$ then
$U(\textbf{z})U(\textbf{z}^{\prime})^{\dagger}\neq I$ using the above proof.
$U(\textbf{z})U(\textbf{z}^{\prime})^{\dagger}$ is a product of Pauli $Z$
operators, as shown in Eq. (10), which in the simplest case consists of a
single operator $Z_{j}$ where $j\in B$. Qubit $j$ in the remaining graph
$G^{\prime}=G-A$ is associated with a stabilizer generator $S^{\prime}_{j}$ in
the form of Eq. (3) that satisfies
$S^{\prime}_{j}\ket{G^{\prime}}=\ket{G^{\prime}}$. The inner product of the
states $U(\textbf{z})\ket{G-A}$ and $U(\textbf{z}^{\prime})\ket{G-A}$ is

$\begin{split}\bra{G^{\prime}}U(\textbf{z}^{\prime})^{\dagger}U(\textbf{z})\ket{G^{\prime}}&=\bra{G^{\prime}}Z_{j}\ket{G^{\prime}}\\\
&=\bra{G^{\prime}}Z_{j}S^{\prime}_{j}\ket{G^{\prime}}\\\
&=\bra{G^{\prime}}Z_{j}X_{j}\textstyle{\prod_{b\in
n(j)}}Z_{b}\ket{G^{\prime}}\\\
&=\bra{G^{\prime}}(-X_{j}Z_{j})\textstyle{\prod_{b\in
n(j)}}Z_{b}\ket{G^{\prime}}\\\ &=-\bra{G^{\prime}}(X_{j}\textstyle{\prod_{b\in
n(j)}}Z_{b})Z_{j}\ket{G^{\prime}}\\\
&=-\bra{G^{\prime}}S_{j}^{\prime\dagger}Z_{j}\ket{G^{\prime}}\\\
&=-\bra{G^{\prime}}Z_{j}\ket{G^{\prime}}\\\
&=-\bra{G^{\prime}}U(\textbf{z}^{\prime})^{\dagger}U(\textbf{z})\ket{G^{\prime}}.\end{split}$
(12)

This is satisfied only when
$\bra{G^{\prime}}U(\textbf{z}^{\prime})^{\dagger}U(\textbf{z})\ket{G^{\prime}}=0$
so the states $U(\textbf{z})\ket{G-A}$ and $U(\textbf{z}^{\prime})\ket{G-A}$
are orthogonal.

More generally $U(\textbf{z})U(\textbf{z}^{\prime})^{\dagger}$ can be written
as

$U(\textbf{z})U(\textbf{z}^{\prime})^{\dagger}=Z_{j}\prod_{c\in C}Z_{c}$ (13)

where $C$ is a subset of $B\setminus j$. $\textstyle{\prod_{c\in C}Z_{c}}$
commutes with $S^{\prime}_{j}$ so replacing $Z_{j}$ by the more general
expression for $U(\textbf{z})U(\textbf{z}^{\prime})^{\dagger}$ in Eq. (12),
the states $U(\textbf{z})\ket{G-A}$ and $U(\textbf{z}^{\prime})\ket{G-A}$
satisfy $\bra{G-A}U(\textbf{z}^{\prime})^{\dagger}U(\textbf{z})\ket{G-A}=0$
and hence are orthogonal. $\square$

The unitary operators $U(\textbf{z})$ can used within the stabilizer formalism
to find the reduced state of qubits in $B$. The set of stabilizer generators
for each outcome must be considered separately. The remaining graph $G-A$ can
be found by removing all edges to qubits in $A$ and the stabilizer generator
$S^{\prime}_{b}$ for each qubit $b\in B$ is given by Eq. (3). Then for each
outcome, the corresponding unitary operator must be applied to each stabilizer
generator according to $U(\textbf{z})S^{\prime}_{b}U(\textbf{z})^{\dagger}$,
giving a new stabilizer for each unitary operator.

An alternative method does not require calculation of the unitary operators
corresponding to each measurement outcome. Instead, in the set of stabilizer
generators for the original graph $G$, the stabilizer generator for each qubit
$a\in A$ can be set to $\pm Z_{a}$. Then the group structure of the stabilizer
can be used to determine the stabilizer generators for the remaining graph
only in terms of qubits in $B$. This process is repeated to give a set of
stabilizer generators for each measurement outcome.

The following theorem provides a link between the sets of stabilizer
generators that describe a reduced state and the purity of that reduced state.
This is the main component in calculating the Concentratable Entanglement.

Theorem 1. Consider a graph state bipartitioned into sets $A$ and $B$. Let
$k\in\mathbb{N}$ denote the number of distinct sets of stabilizer generators
for the different measurement outcomes when measuring the qubits in $A$. The
purity of the reduced state of qubits in $B$ is then

$\text{Tr}\rho_{B}^{2}=\frac{1}{k}.$ (14)

Proof. When measuring the qubits in $A$ there are $2^{|A|}$ possible outcomes.
Each outcome is represented by a bitstring $\textbf{z}\in\mathbb{F}_{2}^{A}$,
which is associated with a unitary operator $U(\textbf{z})$. Therefore there
are $2^{|A|}$ unitary operators. However, these unitaries need not be unique.

For any graph state and subset $A$ of qubits, the outcome represented by the
bitstring made entirely of zeros leads to the identity operator

$U(\textbf{0})=I.$ (15)

If $U(\textbf{0})$ is the sole identity operator, operators $U(\textbf{z})$
and $U(\textbf{z}^{\prime})$ are equal when
$\textbf{z}-\textbf{z}^{\prime}=\textbf{0}$ using the Hein-Eisert-Briegel
Lemma. This is solved trivially by $\textbf{z}=\textbf{z}^{\prime}$ but has no
further solutions. Therefore when one operator is the identity, all remaining
unitaries take distinct values.

Consequently, for any unitary to occur more than once, there must be further
operators equal to the identity. If a second operator is the identity, i.e.,
$U(\textbf{w})=I$ for some $\textbf{w}\neq\textbf{0}$, the remaining elements
in the finite field can be matched into pairs whose difference modulo 2 is w.
Therefore there are $2^{|A|}/2=2^{|A|-1}$ different unitaries, each of which
occurs twice.

If a third unitary is set to the identity, $U(\textbf{v})=I$ for
$\textbf{v}\notin\\{\textbf{0},\textbf{w}\\},$ then it follows that a fourth
unitary, $U(\textbf{v}^{\prime})$ where
$\textbf{v}^{\prime}=\textbf{v}-\textbf{w}$, must also be the identity. Hence

$U(\textbf{0})=U(\textbf{w})=U(\textbf{v})=U(\textbf{v}^{\prime})=I.$ (16)

For any remaining $\textbf{z}\in\mathbb{F}_{2}^{A}$, $U(\textbf{z})$ is equal
to three other unitaries, those for which the bitstring differs from z by w, v
or $\textbf{v}^{\prime}$. There are $2^{|A|}/4=2^{|A|-2}$ different unitary
operators, each occurring 4 times.

More generally, if there are $2^{j}$ unitary operators that are the identity
where $0\leq j\leq|A|$, then there are $k=2^{|A|-j}$ different values for the
unitary operators, each of which has multiplicity $2^{j}$.

To find the purity of the reduced state, first the density operator must be
squared. Using Eq. (7), this gives

$\rho_{B}^{2}=\frac{1}{2^{2|A|}}\sum_{\textbf{z},\textbf{z}^{\prime}}U(\textbf{z})\ket{G^{\prime}}\bra{G^{\prime}}U(\textbf{z})^{\dagger}U(\textbf{z}^{\prime})\ket{G^{\prime}}\bra{G^{\prime}}U(\textbf{z}^{\prime})^{\dagger},$
(17)

where $G^{\prime}=G-A$ for brevity. By the Hein-Eisert-Briegel Lemma, if
$U(\textbf{z})\neq U(\textbf{z}^{\prime})$ then
$\bra{G^{\prime}}U(\textbf{z})^{\dagger}U(\textbf{z}^{\prime})\ket{G^{\prime}}=0$
since the states $U(\textbf{z})\ket{G^{\prime}}$ and
$U(\textbf{z}^{\prime})\ket{G^{\prime}}$ are orthogonal. In the $2^{j}$ cases
when $U(\textbf{z})=U(\textbf{z}^{\prime})$, including the case
$\textbf{z}=\textbf{z}^{\prime}$,
$\bra{G^{\prime}}U(\textbf{z})^{\dagger}U(\textbf{z}^{\prime})\ket{G^{\prime}}=1$.
This leads to

$\rho_{B}^{2}=\frac{2^{j}}{2^{2|A|}}\sum_{\textbf{z}\in\mathbb{F}_{2}^{A}}U(\textbf{z})\ket{G^{\prime}}\bra{G^{\prime}}U(\textbf{z})^{\dagger}=\frac{1}{k}\rho_{B}.$
(18)

The trace is then taken, resulting in a purity given by

$\text{Tr}\rho_{B}^{2}=\frac{1}{k},$ (19)

since $\text{Tr}\rho_{B}=1$.

If there are $k$ different unitaries, this leads to $k$ different sets of
stabilizer generators to describe the graph state once the qubits in $A$ have
been removed by measurement. Each of the $k$ unitaries is applied to give a
set of generators according to $US^{\prime}_{b}U^{\dagger}$ for all $b\in B$.
$\square$

For any bipartition, the purities of the reduced states $\rho_{A}$ and
$\rho_{B}$ are related by Nielsen and Chuang (2010)

$\text{Tr}\rho_{A}^{2}=\text{Tr}\rho_{B}^{2},$ (20)

due to the Schmidt decomposition theorem. When calculating purities using
Theorem 1, the smaller partition from $(A,B)$ can be traced out, reducing the
number of bipartitions of an $n$ qubit state to consider by one half.

Theorem 1 also applies when using the alternative method of finding sets of
stabilizer generators without calculating the unitary operators, since this
method results in the same sets of stabilizer generators.

##  IV Examples

The result of Theorem 1 can be applied within the stabilizer formalism to
calculate the Concentratable Entanglement for cuts of qubits of graph states
with different designs and applications. We will consider several examples
here.

1\. Single qubits. The Concentratable Entanglement for a set containing a
single qubit $a$ in an $n$ qubit graph state $\ket{G}$ is

$C_{\ket{G}}(\\{a\\})=\frac{1}{4}.$ (21)

To show this, consider any connected graph state of $n$ qubits. To find the
Concentratable Entanglement for a set $s$ containing a single qubit, the
purity of the reduced state of the single qubit must be calculated. The
reduced state can be found by tracing out the other $n-1$ qubits and
considering all $2^{n-1}$ possible measurement outcomes. However this can be
simplified by instead using Eq. (20) and only tracing out the qubit of
interest. When bipartioning the set $\mathcal{S}=\\{1,\ldots,n\\}$ of all
qubit labels, the set $A=\\{a\\}$ contains a single qubit and
$B=\mathcal{S}\setminus\\{a\\}$ contains the remaining $n-1$ qubits.

The stabilizer generators for the graph take three forms. Firstly, qubit $a$
has the stabilizer generator

$S_{a}=X_{a}\prod_{b\in n(a)}Z_{b}.$ (22)

Second, the stabilizer generator for each qubit $b\in n(a)$ is

$S_{b}=Z_{a}X_{b}\prod_{k\in\tilde{n}(a)}Z_{k},$ (23)

where $\tilde{n}(a)=n(b)\setminus\\{a\\}$. Finally, qubits $c\in B\setminus
n(a)$ have stabilizer generators of the form

$S_{c}=X_{c}\prod_{m\in n(c)}Z_{m},$ (24)

where $n(c)$ cannot contain qubit $a$.

By tracing out qubit $a$, its stabilizer generator must be updated to $\pm
Z_{a}$. In both cases generators in the form of $S_{c}$ remain valid. However
each $S_{b}$ must be multiplied by $\pm Z_{a}$ so the new generator contains
only Pauli operators acting on qubits in $B$. The possible sets of stabilizer
generators are

$Z_{a}$

$\begin{split}S_{b}^{\prime}&=X_{b}\prod_{k\in\tilde{n}(b)}Z_{k}\\\
S_{c}^{\prime}&=X_{c}\prod_{m\in n(c)}Z_{m}\end{split}$

$-Z_{a}$

$\begin{split}S_{b}^{\prime}&=-X_{b}\prod_{k\in\tilde{n}(b)}Z_{k}\\\
S_{c}^{\prime}&=\phantom{-}X_{c}\prod_{m\in n(c)}Z_{m},\end{split}$

where there is a stabilizer generator in both sets for each $b\in n(a)$ and
each $c\in B\setminus n(a)$. The stabilizer generators for qubits in the
neighbourhood of qubit $a$ acquire a minus sign when the measurement outcome
is $-1$.

There are two possible sets of stabilizer generators to describe the reduced
states of qubits in $B$. Applying Theorem 1 and Eq. (20), the purity is

$\text{Tr}\rho_{B}^{2}=\frac{1}{2}=\text{Tr}\rho_{a}^{2}.$ (25)

Therefore the Concentratable Entanglement for a single qubit within any
connected graph state, where $s=\\{a\\}$, is

$C_{\ket{G}}(\\{a\\})=1-\frac{1}{2}(1+\text{Tr}\rho_{a}^{2})=\frac{1}{4}.$
(26)

2\. A six qubit graph. Fig. 1 shows a graph state with qubit labels
$\mathcal{S}=\\{1,\ldots,6\\}$. This is graph No. 13 according to the standard
numbering of graph state local unitary (LU) equivalence classes Hein  _et al._
(2004); Danielsen and Parker (2006); Cabello  _et al._ (2011). The graph is
described by stabilizer generators

$\begin{split}S_{1}&=X_{1}Z_{2}\\\ S_{2}&=Z_{1}X_{2}Z_{3}\\\
S_{3}&=\phantom{X_{1}}Z_{2}X_{3}Z_{4}\phantom{X_{5}}Z_{6}\\\
S_{4}&=\phantom{X_{1}Z_{2}}Z_{3}X_{4}Z_{5}\\\
S_{5}&=\phantom{X_{1}X_{2}Z_{3}}Z_{4}X_{5}\\\
S_{6}&=\phantom{X_{1}Z_{2}}Z_{3}\phantom{X_{4}Z_{5}}X_{6}.\end{split}$ (27)

Figure 1: A six qubit graph state. The qubit labels can be bipartitioned, with
qubits in set $A$ represented by hollow vertices and qubits in set $B$ by
filled vertices. (a) $A=\\{4,6\\}$, $B=\\{1,2,3,5\\}$. (b) $A=\\{3,4,6\\}$,
$B=\\{1,2,5\\}$.

To find the purity of the reduced state of qubits 1, 2, 3 and 5, qubits 4 and
6 must be traced out. The stabilizer generators for qubits 4 and 6 are set to
$\pm Z_{4}$ and $\pm Z_{6}$ to account for all possible measurement outcomes,
where qubits 4 and 6 collapse into the eigenstates of the $Z$ operator. This
results in the following sets of stabilizer generators to describe the
remaining state

$\\{Z_{4},Z_{6}\\}$

$\begin{split}S_{1}^{\prime}&=X_{1}Z_{2}\phantom{Z_{3}}\\\
S_{2}^{\prime}&=Z_{1}X_{2}Z_{3}\\\
S_{3}^{\prime}&=\phantom{Z_{1}}Z_{2}X_{3}\\\
S_{5}^{\prime}&=\phantom{X_{1}Z_{2}Z_{3}}X_{5}\end{split}$

$\\{Z_{4},-Z_{6}\\}$

$\begin{split}S_{1}^{\prime}&=X_{1}Z_{2}\phantom{Z_{3}}\\\
S_{2}^{\prime}&=Z_{1}X_{2}Z_{3}\\\ S_{3}^{\prime}&=\phantom{z}{-Z_{2}}X_{3}\\\
S_{5}^{\prime}&=\phantom{X_{1}Z_{2}Z_{3}}X_{5}\end{split}$

$\\{-Z_{4},Z_{6}\\}$

$\begin{split}S_{1}^{\prime}&=X_{1}Z_{2}\phantom{Z_{3}}\\\
S_{2}^{\prime}&=Z_{1}X_{2}Z_{3}\\\ S_{3}^{\prime}&=\phantom{z}{-Z_{2}}X_{3}\\\
S_{5}^{\prime}&=\phantom{X_{1}Z_{2}z}{-}X_{5}\end{split}$

$\\{-Z_{4},-Z_{6}\\}$

$\begin{split}S_{1}^{\prime}&=X_{1}Z_{2}\phantom{Z_{3}}\\\
S_{2}^{\prime}&=Z_{1}X_{2}Z_{3}\\\
S_{3}^{\prime}&=\phantom{Z_{1}}Z_{2}X_{3}\\\
S_{5}^{\prime}&=\phantom{X_{1}Z_{2}z}{-}X_{5}.\end{split}$

There are four distinct sets of stabilizer generators and therefore by Theorem
1 the purity of the reduced state of qubits 1, 2, 3 and 5 is

$\text{Tr}\rho_{1235}^{2}=\frac{1}{4}.$ (28)

Instead, if qubits 3, 4 and 6 are traced out, the stabilizer generators for
each of these qubits must be changed to consider all possible outcomes. Since
the stabilizer generators where qubits 4 and 6 are traced out have already
been calculated, only the stabilizer generator for qubit 3 needs to be
updated. The stabilizer generator for qubit 3 is set to $\pm Z_{3}$ within
each set. Therefore the sets of stabilizer generators for the remaining qubits
are

$\\{Z_{3},Z_{4},Z_{6}\\}$, $\\{Z_{3},Z_{4},-Z_{6}\\}$

$\begin{split}S_{1}^{\prime\prime}&=X_{1}Z_{2}\\\
S_{2}^{\prime\prime}&=Z_{1}X_{2}\\\
S_{5}^{\prime\prime}&=\phantom{X_{1}Z_{2}}X_{5}\end{split}$

$\\{-Z_{3},Z_{4},Z_{6}\\}$, $\\{-Z_{3},Z_{4},-Z_{6}\\}$

$\begin{split}S_{1}^{\prime\prime}&=\phantom{-}X_{1}Z_{2}\\\
S_{2}^{\prime\prime}&=-Z_{1}X_{2}\\\
S_{5}^{\prime\prime}&=\phantom{-X_{1}Z_{2}}X_{5}\end{split}$

$\\{Z_{3},-Z_{4},Z_{6}\\}$, $\\{Z_{3},-Z_{4},-Z_{6}\\}$

$\begin{split}S_{1}^{\prime\prime}&=X_{1}Z_{2}\\\
S_{2}^{\prime\prime}&=Z_{1}X_{2}\\\
S_{5}^{\prime\prime}&=\phantom{-Z}{-X_{5}}\end{split}$

$\\{-Z_{3},-Z_{4},Z_{6}\\}$, $\\{-Z_{3},-Z_{4},-Z_{6}\\}$

$\begin{split}S_{1}^{\prime\prime}&=\phantom{-}X_{1}Z_{2}\\\
S_{2}^{\prime\prime}&=-Z_{1}X_{2}\\\
S_{5}^{\prime\prime}&=\phantom{-{-Z}}{-X_{5}}.\end{split}$

Though there are eight possible measurement outcomes, this results in only
four distinct sets of stabilizer generators. Therefore by Theorem 1 the purity
of the reduced state of qubits 1, 2 and 5 is

$\text{Tr}\rho_{125}^{2}=\frac{1}{4}.$ (29)

This is also the purity of the state $\rho_{346}$ due to Eq. (20) and of the
states $\rho_{145}$ and $\rho_{236}$ because of the symmetry of the graph in
Fig. 1.

This method can be used to find the purities of all possible reduced states of
the graph state. For the six bipartitions where the smaller set contains a
single qubit, the purity is $1/2$ from Example 1. When the smaller set in the
bipartition contains two qubits, a purity of $1/4$ occurs twelve times and a
purity of $1/2$ occurs three times. For the ten bipartitions where each set
has three elements, the purity is $1/8$ in four cases, $1/4$ in four cases and
$1/2$ in two cases. These values can be used to calculate the Concentratable
Entanglement of any subset of qubits and the overall Concentratable
Entanglement of the graph state:

$\begin{split}C_{\ket{G_{13}}}(\mathcal{S})&=1-\frac{1}{2^{6}}\bigg{[}1+6\cdot\frac{1}{2}+\bigg{(}12\cdot\frac{1}{4}+3\cdot\frac{1}{2}\bigg{)}+\bigg{(}8\cdot\frac{1}{8}+8\cdot\frac{1}{4}+4\cdot\frac{1}{2}\bigg{)}+\bigg{(}12\cdot\frac{1}{4}+3\cdot\frac{1}{2}\bigg{)}+6\cdot\frac{1}{2}+1\bigg{]}\\\
&=\frac{21}{32}.\end{split}$ (30)

An alternative measure of entanglement with respect to a bipartition $(A,B)$
is the Schmidt rank, defined as Hein  _et al._ (2006)

$\text{SR}_{B}(\psi)=\log_{2}[\text{rank}(\rho_{B})].$ (31)

This measure is related to the purity of the reduced state of $B$ according to
Hein  _et al._ (2006)

$\text{SR}_{B}(\psi)=-\log_{2}[\text{Tr}(\rho_{B}^{2})].$ (32)

The rank index $RI_{m}$ considers all bipartitions where the smaller set
contains $m$ qubits Hein  _et al._ (2004). It lists the number of times the
Schmidt rank $m$, then $m-1$, through to $1$ occurs. For graph No. 13 in Fig
1, $\text{RI}_{2}=(12,3)$ and $\text{RI}_{3}=(4,4,2)$: for smaller sets in the
bipartition of qubit labels of the graph containing two qubits, there are
twelve sets with a Schmidt rank of 2 and three sets with a Schmidt rank of 1.
For smaller sets of three qubits, the Schmidt rank 3 occurs four times, 2
occurs four times and 1 occurs twice. This corresponds to the occurrences of
different purities of reduced states calculated for the Concentratable
Entanglement. Using Eq. (32), the rank index can be used to give the number of
splits where the purity is $(2^{-m},\ldots,1)$ for bipartitions with a smaller
set of $m$ qubits. For any connected graph of seven or fewer vertices, the
Concentratable Entanglement for $s=\mathcal{S}$ can be calculated using the
results for the rank index in Table II of Hein  _et al._ (2004).

3\. Graph states for repeaters. Azuma et al. Azuma  _et al._ (2015) propose a
quantum repeater design based on graph states; repeater stations create
‘snowflake’ graph states as shown in Fig. 2, to perform entanglement swapping
and set up direct entanglement over larger distances. The inner 1st leaf
qubits (grey) form a fully connected graph so there is a direct path between
any two qubits. Each 1st leaf qubit is further connected to a single 2nd leaf
qubit (black).

Figure 2: A sixteen qubit ‘snowflake’ state proposed for use in repeaters
Azuma  _et al._ (2015). Grey circles represent 1st leaf qubits, each of which
is connected to a 2nd leaf qubit, shown in black.

Consider a snowflake state of $2n$ qubits where the 1st leaf qubits have
labels in the set $\mathcal{S}_{1\text{st}}=\\{1_{1},2_{1},\ldots,n_{1}\\}$
and the 2nd leaf qubit labels are
$\mathcal{S}_{2\text{nd}}=\\{1_{2},2_{2},\ldots,n_{2}\\}$. Qubit $a_{2}$ is
the 2nd leaf qubit paired with 1st leaf qubit $a_{1}$. The stabilizer
generator for each 1st leaf qubit $a_{1}\in\mathcal{S}_{1}$ is

$S_{a_{1}}=X_{a_{1}}Z_{a_{2}}\prod_{b\in\mathcal{S}_{1}\setminus\\{a_{1}\\}}Z_{b}.$
(33)

Each 2nd leaf qubit $a_{2}\in\mathcal{S}_{2}$ has a stabilizer generator of
the form

$S_{a_{2}}=X_{a_{2}}Z_{a_{1}}.$ (34)

When measuring a 1st leaf qubit in the $Z$ direction, the outcome $-1$
introduces a minus sign to the stabilizer generator for its 2nd leaf qubit
partner and in the stabilizer generators for all other 1st leaf qubits, as
they are all in the neighbourhood. For a 2nd leaf qubit, the measurement
outcome $-1$ introduces a minus sign only to the stabilizer generator of its
paired qubit in the 1st leaf. Therefore, if a pair $\\{a_{1},a_{2}\\}$ is
traced out, there are only two possible sets of stabilizer generators. The 1st
leaf qubit stabilizer generators either all acquire a minus sign or they do
not.

When $m\leq n$ qubits are traced out, where no two qubits are a pair, each
measurement outcome results in a unique set of stabilizer generators for the
remaining qubits. The purity of the reduced state left when tracing out these
$m$ qubits is $2^{-m}$. Using this result to calculate each of the purities
required, the Concentratable Entanglement for sets $s$ of $n$ qubits
containing no pairings, including the sets containing exclusively 1st leaf or
exclusively 2nd leaf qubits, is

$C_{\ket{G}}(s)=1-\frac{1}{2^{n}}\sum_{m=0}^{n}{n\choose
m}\frac{1}{2^{m}}=1-\bigg{(}\frac{3}{4}\bigg{)}^{n}.$ (35)

This is the maximum Concentratable Entanglement possible for a set of $n$
qubits within a graph state of $2n$ qubits. Therefore within a snowflake
state, all of the 1st leaf qubits or all of the 2nd leaf qubits exhibit
maximal entanglement with respect to the rest of the state.

4\. Limits on overall entanglement. Concentratable Entanglement provides a
measure of the overall entanglement of a state when applied to the set
$\mathcal{S}$. There are theoretical bounds on the maximum and minimum values
that the Concentratable Entanglement can take.

The minimum value of Concentratable Entanglement occurs when the purity of
each possible reduced state is maximised. Since a connected graph represents
an entangled pure state, any reduced state cannot be pure, excluding the
reduced state containing no qubits where $\text{Tr}\rho_{\emptyset}^{2}=1$.
The maximum value each of these purities can take according to Theorem 1 is
$1/2$, where there are two sets of stabilizer generators to describe the
remaining qubits within the reduced state. The minimum value of Concentratable
Entanglement for a graph state of $n$ qubits is therefore

$\begin{split}C_{\ket{G}}(\mathcal{S})&=1-\frac{1}{2^{n}}\bigg{(}\text{Tr}\rho_{\emptyset}^{2}+\text{Tr}\rho_{\mathcal{S}}^{2}+\sum_{i=1}^{n-1}{n\choose
i}\frac{1}{2}\bigg{)}\\\ &=\frac{1}{2}-\frac{1}{2^{n}}.\end{split}$ (36)

This is the Concentratable Entanglement of the GHZ state Beckey  _et al._
(2021). The $n$ qubit GHZ state is LU equivalent to the star graph and the
complete graph, which take this value for Concentratable Entanglement.

If instead the purity of each reduced state is minimised, the Concentratable
Entanglement will be maximised. It is not always possible to minimise the
purity of all reduced states due to the structure of entanglement and the
dependent relationships between different subsets of qubits. The purity of the
reduced state of qubits in $A$ or $B$ is minimised when the number of sets of
stabilizer generators is given by

$k=2^{\min(|A|,|B|)}.$ (37)

This gives a new set of stabilizer generators for each measurement outcome
when tracing out qubits, up to the number of measurement outcomes possible for
the smaller set of $A$ and $B$ due to Eq. (20). Therefore the maximum value
that the Concentratable Entanglement can achieve for a graph state of $n$
qubits is

$C_{\ket{G}}(\mathcal{S})=1-\frac{1}{2^{n}}\sum_{j=0}^{n}{n\choose
j}\frac{1}{2^{\min(j,n-j)}}.$ (38)

Figure 3: The LU classes of graph states, numbered according to Hein  _et al._
(2004); Danielsen and Parker (2006); Cabello  _et al._ (2011), where the
overall Concentratable Entanglement achieves the maximum theoretical value.

Graph states that maximise the Concentratable Entanglement exist only for 2,
3, 5 and 6 qubits, and an example from each LU equivalence class is shown in
Fig. 3. Trivially, a graph state of a single qubit satisfies Eq. (38), however
it does not represent an entangled state and subsequently is of little
interest. For graph states of 2 or 3 qubits, the maximum value of
Concentratable Entanglement is equal to the minimum value, as the smaller
(non-empty) bipartition always contains a single qubit, restricting all
purities to the same value of $1/2$.

An absolutely maximally entangled (AME) state Helwig  _et al._ (2012) has
maximal entanglement in all bipartitions. If $B$ is the smaller set containing
$j$ qubits in a bipartition $(A,B)$, then the bipartition is maximally
entangled if the reduced state of qubits in $B$ can be written as the mixed
state

$\rho_{B}=\frac{1}{2^{j}}I_{2^{j}}.$ (39)

The purity of this state is $2^{-j}$. An AME state will achieve the maximum
value of Concentratable Entanglement. The states of 2, 3, 5 and 6 qubits that
maximise the Concentratable Entanglement are known AME states.

5\. Graphs up to nine qubits. For graph states consisting of nine qubits or
fewer, we have calculated the overall Concentratable Entanglement of the
state, as shown in Fig. 4. The respective values of 0 and 1/4 for the
Concentratable Entanglement of the unique graph states of 1 and 2 qubits are
not shown. The theoretical maximum and minimum values of Concentratable
Entanglement bound the area shaded in grey. Star states achieving the minimum
Concentratable Entanglement, which are LU equivalent to GHZ states lie on the
lower boundary of this area. The states with the greatest Concentratable
Entanglement of each size are shown, and can be seen to achieve the maximum
only for states of 3, 5 and 6 qubits.

Figure 4: The possible values of overall Concentratable Entanglement of graph
states of 3 to 9 qubits. Theoretical bounds on the Concentratable Entanglement
define the area in grey. We consider special classes of states, which are
given different markers, as shown in the legend.

Figure 4 also highlights the Concentratable Entanglement for linear and ring
graph states. Linear cluster states of photons have been experimentally
realised Schwartz  _et al._ (2016); Istrati  _et al._ (2020): an important
step in developing the two dimensional clusters needed for MBQC and therefore
their overall Concentratable Entanglement is shown. Ring states are similar to
linear states although a further CZ operation is applied between the first and
final qubits in the chain, thus completing a ring. Ring states have higher
Concentratable Entanglement than linear states except for states of up to four
qubits, where a ring and linear state are LU equivalent.

All of the overall Concentratable Entanglement values for graph states of nine
qubits or fewer are shown in Fig. 4. Graph states within the same LU class
will have the same overall Concentratable Entanglement, however using
Concentratable Entanglement as a measure of the overall entanglement in a
state is not sufficient to identify the LU class of a graph state, as the same
value can occur for more than one class. For example, for graph states of
seven qubits, there are 26 LU equivalence classes, but only 16 possible values
of Concentratable Entanglement. If Concentratable Entanglement is calculated
using Eq. (1), the reduced state purities can be studied to identify the LU
class of a state. However, if a quantum computer were used to perform a SWAP
test, further calculations would be required, such as performing SWAP tests to
calculate the Concentratable Entanglement for subsets of qubits within the
state to identify the LU class from the entanglement structure.

##  V Discussion & Further Work

We have presented a method for calculating the purity of reduced states and
hence the Concentratable Entanglement for graph states entirely within the
stabilizer formalism, using only the stabilizer generators of a graph state.
By evaluating the Concentratable Entanglement for different graph states we
have found closed forms when considering a single qubit in any graph state and
either the inner or outer leaf qubits in a snowflake state for use in a
quantum repeater. We have also shown how Concentratable Entanglement relates
to previous results regarding Schmidt rank and AME states. However the overall
Concentratable Entanglement of graph states in different LU classes is shown
to overlap in certain cases, suggesting that calculation of this version of
the measure alone is insufficient to identify the entanglement structure of a
graph state.

Though Concentratable Entanglement was introduced for pure qubit states, it
may be possible to use it as an entanglement measure for pure qudit states.
Further work could then look to extend the definition in terms of purities
given in Eq. (1) to graph states of qudits. Extension of the SWAP test to
qudits Prove  _et al._ (2022) may allow for calculation of Concentratable
Entanglement via test outcomes when using a quantum computer in a similar
manner to the qubit case. Qudit graph states Looi  _et al._ (2008); Keet  _et
al._ (2010); Helwig (2013) are stabilizer states for qudits of prime dimension
$d$ and allow for weighted or multiple edges. Reduced states could be found by
considering all possible measurement outcomes, setting the stabilizer
generators for measured qubits accordingly, and using the group structure of
the stabilizer. The purity of the reduced state could again be calculated
based on the number of unique sets of stabilizer generators.

It is unlikely that the result for calculating purities using stabilizer
generators within the stabilizer formalism can be applied to hypergraphs. Like
qudit graph states, hypergraphs Rossi  _et al._ (2013); Qu  _et al._ (2013);
Steinhoff  _et al._ (2017) allow for weighted or repeated edges between the
same qudits but hyperedges that can connect any number of qudits are
introduced. The stabilizer generator for each qudit is given in terms of CZ
gates on multiple qudits, which are not within the Pauli group. Therefore
hypergraphs with hyperedges connecting more than two qudits are not true
stabilizer states and do not allow for the group properties of a stabilizer to
be used to find updated generators following a measurement.

Currently Concentratable Entanglement is a measure that applies only to pure
states. Even single qubit errors either in the generation or transmission of
qubits within graph states can result in a mixed state. An extension of
Concentratable Entanglement to mixed states could enable the study of effects
such errors have on the entanglement within graph states. This is the subject
of future investigations.

##  VI Acknowledgements

ARC is supported by an EPSRC Doctoral Training Partnership (DTP) Scholarship.
PK is supported by the EPSRC Quantum Communications Hub, Grant No.
EP/M013472/1, and the EPSRC grant Large Baseline Quantum-Enhanced Imaging
Networks, Grant No. EP/V021303/1.

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