# Multipartite entanglement measures: a review

Mengru Ma  Key Laboratory of Advanced Optoelectronic Quantum Architecture and
Measurement of Ministry of Education, School of Physics, Beijing Institute of
Technology, Beijing 100081, China  Yinfei Li  Key Laboratory of Advanced
Optoelectronic Quantum Architecture and Measurement of Ministry of Education,
School of Physics, Beijing Institute of Technology, Beijing 100081, China
Jiangwei Shang<EMAIL_ADDRESS>Key Laboratory of Advanced
Optoelectronic Quantum Architecture and Measurement of Ministry of Education,
School of Physics, Beijing Institute of Technology, Beijing 100081, China

###### Abstract

Quantum entanglement, a fundamental aspect of quantum mechanics, has captured
significant attention in the era of quantum information science. In
multipartite quantum systems, entanglement plays a crucial role in
facilitating various quantum information processing tasks, such as quantum
teleportation and dense coding. In this article, we review the theory of
multipartite entanglement measures, with a particular focus on the genuine as
well as the operational meaning of multipartite entanglement measures. By
providing a thorough and valuable insight on this field, we hope that this
review would inspire and guide researchers in their endeavors to further
develop novel approaches for characterizing multipartite entanglement.

##  I Introduction

The concept of entanglement has played an exceptional role in quantum physics
ever since its discovery at the beginning of last century [1, 2]. Entanglement
has been recognized as a key resource in various quantum information
processing tasks, such as quantum teleportation [3], superdense coding [4],
and quantum key distribution [5]. Moreover, as a dividing feature between the
quantum and classical worlds, how to detect the presence of entanglement has
received extensive research interest [6]. However, describing the structure of
quantum entanglement for multipartite systems is still challenging, as there
are already an infinite number of entanglement classes for the simple four-
qubit system [7]. Recognizing the significance of entanglement as a valuable
resource, it becomes natural to explore its quantification through theoretical
descriptions. In the seminal work by Vedral _et al._ [8], the fundamental
condition that the amount of entanglement cannot increase under local
operations and classical communication (LOCC) is established, laying the
foundation for entanglement quantification. Furthermore, Vidal [9] advanced
the field by introducing a family of functions, called entanglement monotones,
which are magnitudes that do not increase, on average, under LOCCs. Examples
of entanglement monotones include the entanglement of formation and
distillable entanglement [10], relative entropy of entanglement [11], and the
robustness of entanglement [12]. In addition, entanglement can be quantified
using the eigenvalue spectra of the density matrices, which provides a
practical means for quantitative entanglement measures [10, 11, 13].

For the bipartite system, various entanglement measures are designed, such as
concurrence [14, 13], negativity [15], entanglement of formation [13], etc.
Since a single Schmidt coefficient governs the two-qubit entanglement, all the
entanglement monotones for two-qubit system are equivalent. Quantifying
entanglement for multipartite systems is, however, a great challenge due to
the much richer mathematical structures involved as compared to the bipartite
case. The generalization of concurrence, namely the generalized multipartite
concurrence, can be found in Ref. [16]. In Ref. [17], an entanglement monotone
for genuine multipartite entanglement is introduced, which extends the
negativity and equals to the negativity in the bipartite case. Besides, other
bipartite entanglement measures such as the relative entropy of entanglement
and the robustness of entanglement can be easily generalized to the
multipartite scenario.

Though multipartite systems have deeper underlying structures, the definition
of multipartite entanglement (ME) can be borrowed straightforwardly from the
bipartite case. Consider an $N$-partite mixed state, the definition of
$k$-separability is defined as [18]

$\rho_{k\text{-SEP}}=\sum_{i}p_{i}|\psi_{k\text{-SEP}}^{i}\rangle\langle\psi_{k\text{-SEP}}^{i}|\,,\quad\sum_{i}p_{i}=1\,,$
(1)

where $p_{i}$s can be interpreted as a probability distribution, and
$|\psi_{k\text{-SEP}}^{i}\rangle\langle\psi_{k\text{-SEP}}^{i}|$s are
$k$-separable pure states. The state is fully separable iff ${k=N}$ and is
called genuine multipartite entanglement (GME) iff it is not biseparable (or
${k=1}$). Being a significant form of entanglement, GME provides notable
advantages in various quantum tasks as compared to other types of
entanglement. Biseparable states may be less suitable for many applications
due to their inherent limitations. Hence, the measures of GME are valuable
within the standard LOCC paradigm. For instance, it has been shown that GME is
in general essential to establish a multipartite secret key [19].

In recent years, there has been a continuous endeavor to qualitatively and
quantitatively characterize the entanglement properties of multipartite
systems. This pursuit involves the formulation of entanglement measures
through two main approaches: the axiomatic ansatz and the operational approach
[20]. Under the axiomatic approach, researchers aim to define measures by
establishing a set of desired properties that the measures should possess. An
essential requirement in this context is entanglement monotonicity [9, 21]. On
the other hand, the operational approach quantifies the utility of a state for
specific protocols that rely on entanglement. This perspective evaluates the
usefulness of entanglement in achieving certain tasks or performing specific
operations. However, computing these measures such as the distillable
entanglement tends to be challenging [10, 22, 23].

This review is organized as follows. In Sec. II, we first present an overview
of entanglement measures, starting from the establishment of the LOCC paradigm
to the entanglement monotonicity. We also introduce a few basic axioms that a
good entanglement measure should possess. In Sec. III, we explore various
common multipartite entanglement measures, and Sec. IV focuses on the GME
measures. Subsequently, in Sec. V, we delve into the applications of the GME
measures in various quantum information tasks and explore the multipartite
entanglement measures with an operational meaning. Finally, we provide a
summary and outlook in Sec. VI.

##  II Overview of entanglement measures

In this section, we commence by establishing the foundational tenets of
quantum entanglement within the scope of the LOCC paradigm. Then, essential
guiding postulates that define entanglement measures as well as the convex-
roof extension technique will be introduced. We spotlight the measure of
concurrence in the end.

###  II.1 The LOCC paradigm

Entanglement is a distinctive quantum property of nonlocality that cannot be
found in classical systems. Before presenting the precise mathematical
definition of entanglement, we first examine a typical scenario in the field
of quantum information science. Imagine that a source prepares a quantum
system consisting of two particles. One of the particles is sent to Alice,
while the other is sent to Bob, who are located in distant laboratories. Alice
and Bob can both apply local operations on their particles, including
arbitrary unitary evolution and measurements. We do not pose any restriction
on local dimensions in each lab, meaning that Alice, for example, can
correlate her particle with an infinite-dimensional system through a unitary
evolution. Next, the source may establish a classical correlation between the
particles by selecting and sending one from a set of prepared quantum systems
based on a classical probability distribution. Finally, Alice and Bob are
equipped with quantum memories and classical information channels, enabling
them to conduct their local operations with the aid of classical
communication.

At this point we have established the paradigm of local operations and
classical communication (LOCC) between Alice and Bob. LOCC is a subset of all
the physically realizable operations on quantum states, the significance of
which is best celebrated by the quantum teleportation protocol [3]. In quantum
teleportation, the source distributes a two-qubit entangled state between
Alice and Bob, then Alice performs a joint measurement on the information
qubit she wishes to teleport and the shared entangled qubit. Subsequently,
Alice sends her measurement outcome to Bob. Upon receiving the outcome, Bob
can transform his entangled qubit into an exact copy of Alice’s information
qubit. More precisely, one entangled two-qubit state and two classical bits
are consumed with LOCC operations to transport one qubit.

We notice that entanglement serves as a resource enabling quantum
teleportation, and LOCC is the technique used to harness its power. Moreover,
it is natural to consider LOCC when discussing entanglement as it does not
involve nonlocal quantum operations that may convey quantum information.
Indeed, the set of separable or non-entangled states can be considered as the
states that result from applying LOCC operations on a pure product state [24].
The set of bipartite separable states is therefore defined by

$\rho_{\text{bi-
SEP}}=\sum_{i}p_{i}\rho^{\text{A}}_{i}\otimes\rho^{\text{B}}_{i}\,,\quad\sum_{i}p_{i}=1\,,$
(2)

where $p_{i}$s refer to a probability distribution and $\rho^{\text{X}}$ is a
density matrix supported on the Hilbert space $\mathcal{H}_{\text{X}}$,
${\text{X}=\\{\text{A},\text{B}\\}}$. Entangled states refer to states that
cannot be expressed in the form of Eq. (2).

LOCC is a versatile resource since quantum teleportation implies that an
arbitrary physical operation can be performed with LOCC and shared
entanglement. Moreover, LOCC is capable of concentrating entanglement from
multiple partially entangled states to a reduced number of more entangled
states [25], as well as purifying a noisy entangled state to a pure state
[26]. However, it is worth noting that there are limitations to the
descriptive power of LOCC in capturing nonlocality. Firstly, there are
different classes of entangled states that cannot be transformed between each
other with LOCC operations only. For bipartite cases, the necessary and
sufficient condition that determines the classes is identified by majorization
[27]. Next, there are nonlocal states that has no entanglement at all [28].
Even though the preparation of separable states does not require
transformation of quantum information, it is possible that a set of mutually
orthogonal separable states cannot be faithfully distinguished by LOCC, no
matter which one of the states is presented to the local parties.

Finally, we briefly introduce the mathematical definition of LOCC [10, 29]. A
general operation, i.e., quantum instrument, on a quantum state is a
completely positive trace-preserving (CPTP) map. For instance,
${\mathcal{A}=\\{\mathcal{A}_{1},\,\mathcal{A}_{2},\,\cdots,\,\mathcal{A}_{n}\\}}$
is a CPTP map if each element $\mathcal{A}_{j}$ is a completely positive map
and $\sum_{j}{\mathcal{A}_{j}}$ is trace preserving. Here the maps of interest
are bounded linear maps on a set of bounded linear operators acting on a
Hilbert space $\mathcal{H}$. Considering a quantum state with $K$ parties,
i.e., ${P_{1},\,P_{2},\,\cdots,\,P_{K}}$, the Hilbert space is the tensor
product of the Hilbert space for each local parties, namely
${\mathcal{H}=\bigotimes_{j=1}^{K}\mathcal{H}_{P_{j}}}$. Henceforth we will
use a superscript on an operator or a map to denote the party that it acts on.
The one-way local protocol with respect to party $P_{k}$ is defined by the
following operations. Firstly, the local operation $\mathcal{B}$ on party
$P_{k}$ produces a readout $j$, and $P_{k}$ broadcasts the outcome to all the
other parties. Then, the other parties perform local operations
$\mathcal{E}_{j}^{X},\,X\neq P_{k}$. Thus, a one-way local instrument
${\mathcal{A}^{(P_{k})}=\\{\mathcal{A}_{1},\,\mathcal{A}_{2},\,\cdots,\,\mathcal{A}_{n}\\}}$
with respect to party $P_{k}$ is given by

$\mathcal{A}_{j}=\mathcal{B}_{j}^{P_{k}}\otimes\bigotimes_{X\neq
P_{k}}\mathcal{E}_{j}^{X}\,.$ (3)

Extending the discussion to one-round LOCC operations requires the concept of
coarse-graining. For two quantum instruments
${\mathcal{A}=\\{\mathcal{A}_{i}|i\in S_{\mathcal{A}}\\}}$ and
${\mathcal{B}=\\{\mathcal{B}_{i}|i\in S_{\mathcal{B}}\\}}$, $\mathcal{A}$ is a
coarse-graining of $\mathcal{B}$ if there exists a partition
${S_{\mathcal{B}}=\cup_{i}T_{i,\mathcal{B}}}$ of the set of labels
$S_{\mathcal{B}}$, s.t. ${\forall i\in
S_{\mathcal{A}},\mathcal{A}_{i}=\sum_{j\in
T_{i,\mathcal{B}}}\mathcal{B}_{j}}$. By definition, a quantum instrument
$\mathcal{A}$ can be implemented with a one-round LOCC _iff_ it is a one-way
local operation followed by coarse-graining.

The definition of $r$-round LOCC requires the concept of operation composition
“$\circ$”. An instrument ${\mathcal{A}=\\{\mathcal{A}_{i}|i\in
S_{\mathcal{A}}\\}}$ is _LOCC linked_ to another instrument
$\mathcal{B}=\\{\mathcal{B}_{i}|i\in S_{\mathcal{B}}\\}$ if there exists a set
of one-way local instruments
${\bigl{\\{}\mathcal{E}^{(P_{j})}=\\{\mathcal{E}_{i,j}|i\in
S_{\mathcal{E}}\\},j=1,2,\cdots\bigr{\\}}}$, such that $\mathcal{A}$ is a
coarse-graining of the instrument
$\\{\mathcal{E}_{i,j}\circ\mathcal{B}_{j}\\}$. This can be interpreted as to
first apply the instrument $\mathcal{B}$, then apply a one-round LOCC which
depends on the outcome of $\mathcal{B}$. Thus, an instrument is $r$-round LOCC
_iff_ it is LOCC linked to a $(r-1)$-round LOCC. An immediate observation is
that a $(r-1)$-round LOCC is a $r$-round LOCC, while the inverse statement is
not always true. For instance, in the task of quantum teleportation between
Alice and Bob, one-round LOCC only allows the teleportation of quantum states
in a single direction, while two-round LOCC enables the teleportation in both
directions.

###  II.2 Entanglement monotonicity

As entanglement is an indispensable resource in quantum information science,
an impending task is to quantify it. If a quantum state $\sigma$ can be
converted to another state $\rho$ with LOCC, then for local parties equipped
with LOCC, any quantum information protocol that requires $\rho$ can also be
achieved with $\sigma$. Thus, a fundamental requirement for the quantification
of entanglement is that entanglement cannot increase under LOCC operations,
i.e., LOCC monotonicity. An important fact is that for bipartite pure states
$\ket{\phi}$ and $\ket{\psi}$, $\ket{\phi}$ can be converted to $\ket{\psi}$
via LOCC _iff_ $\ket{\phi}$ is majorized by $\ket{\psi}$, i.e., both states
have the same Schmidt dimension $d$, and the Schmidt coefficients
$\bigl{\\{}\lambda_{i}^{(\phi)}\bigr{\\}}$ and
$\bigl{\\{}\lambda_{i}^{(\psi)}\bigr{\\}}$ satisfy

$\sum_{i=1}^{k}\lambda_{i}^{(\phi)\downarrow}\leq\sum_{i=1}^{k}\lambda_{i}^{(\psi)\downarrow}\,,\quad\forall
k\in\\{1,2,\cdots,d\\}\,,$ (4)

where the superscript $\downarrow$ indicates that the Schmidt coefficients are
organized in a descending order. This reveals that some quantum states cannot
be related to each other via LOCC. Moreover, two pure states are mutually
convertible via LOCC _iff_ they can be transformed via local unitary
rotations, hence deterministic LOCC is not well suited for the classification
of entangled states.

To establish a stronger condition of entanglement ordering, one may consider
stochastic LOCC (SLOCC), where $\sigma$ is converted to $\rho$ via LOCC with a
non-zero probability [7]. SLOCC provides a coarse-grained classification of
multipartite entanglement, which means that quantum states incomparable via
deterministic LOCC can now be related via SLOCC. For two $N$-partite pure
states $\ket{\phi}$ and $\ket{\psi}$, they can be transformed to each other
via SLOCC or they are SLOCC equivalent _iff_ there exists an invertible local
operator (ILO)
${\mathcal{L}_{1}\otimes\mathcal{L}_{2}\otimes\cdots\otimes\mathcal{L}_{N}}$,
s.t.

$\ket{\phi}=\mathcal{L}_{1}\otimes\mathcal{L}_{2}\otimes\cdots\otimes\mathcal{L}_{N}\ket{\psi}\,.$
(5)

An SLOCC class can be built from a pure state $\ket{\psi}$ by constructing a
convex hull of all pure states that are SLOCC equivalent with $\ket{\psi}$ and
of all pure states that can be approximated arbitrarily close by those SLOCC
equivalent pure states [30, 31, 32].

Strong monotonicity of a function $E$ is defined as follows

$\sum_{i}{p_{i}E\bigl{(}\sigma_{i}\bigr{)}}\leq E(\rho)\,,$ (6)

where $\\{p_{i},\sigma_{i}\\}$ is an ensemble produced by an arbitrary LOCC
channel $\Lambda_{\text{LOCC}}$ acting on the multipartite quantum state
$\rho$, such that

$\Lambda_{\text{LOCC}}(\rho)=\sum_{i}{p_{i}\sigma_{i}}\,.$ (7)

For states in the same SLOCC class, the monotonicity provides the entanglement
ordering. However, there are already two types of genuinely entangled states
for three-qubit systems [7, 33], as is shown in Fig. 1, resulting in the fact
that different entanglement measures may lead to distinct entanglement
ordering for multipartite quantum states.

\begin{overpic}[width=195.12877pt]{Classes}\end{overpic} Figure 1: An
illustration of the SLOCC classes for three-qubit states, including the fully-
separable class (S), biseparable class (B), $W$ class, and GHZ class [30].

###  II.3 Postulates of entanglement measures

An entanglement measure (or entanglement monotone) $E(\rho)$ should have some
desirable properties, while not all of the following properties are fulfilled
by all the discussed entanglement quantifiers.

  1. (1)

Entanglement vanishes for separable states. If $\rho$ is separable, then
$E(\rho)=0$.

  2. (2)

$E(\rho)$ cannot be increased by LOCC. For any LOCC operation
$\Lambda_{\text{LOCC}}$,

$E\bigl{(}\Lambda_{\text{LOCC}}(\rho)\bigr{)}\leq E(\rho)\,.$ (8)

And there is a stronger condition, if the ensemble $\\{p_{k},\rho_{k}\\}$ is
obtained from the state $\rho$ by applying LOCC operations, then the measure
should not increase on average

$\sum_{k}p_{k}E(\rho_{k})\leq E(\rho)\,.$ (9)

  3. (3)

$E(\rho)$ should be invariant under local unitary transformations

$E(\rho)=E\bigl{(}U_{1}\otimes\cdots\otimes U_{N}\rho
U_{1}^{\dagger}\otimes\cdots\otimes U_{N}^{\dagger}\bigr{)}\,.$ (10)

These operations are particular instances of LOCC operations, which are
reversible.

  4. (4)

Convexity. Most entanglement measures are convex, that is, the entanglement
measure cannot increase under convex combination of two or more states,

$E\Bigl{(}\sum_{k}p_{k}\rho_{k}\Bigr{)}\leq\sum_{k}p_{k}E(\rho_{k})\,.$ (11)

  5. (5)

Additivity. If Alice and Bob share $n$ copies of the same state $\rho$, then

$E\bigl{(}\rho^{\otimes n}\bigr{)}=nE(\rho)\,.$ (12)

If there are two different states, then

$E\bigl{(}\rho_{1}\otimes\rho_{2}\bigr{)}=E\bigl{(}\rho_{1}\bigr{)}+E\bigl{(}\rho_{2}\bigr{)}\,.$
(13)

And if the measure satisfies

$E\bigl{(}\rho^{\otimes n}\bigr{)}\leq nE(\rho)\,,$ (14)

it is subadditive.

  6. (6)

Asymptotic continuity. For states $\rho_{n}$ and $\sigma_{n}$ acting on the
Hilbert space
${\mathcal{H}_{n}=\mathcal{H}_{n}^{A}\otimes\mathcal{H}_{n}^{B},n\in\mathbb{N}}$,
we have [9, 34, 21]

$\|\rho_{n}-\sigma_{n}\|_{1}\to
0\Rightarrow\frac{|E(\rho_{n})-E(\sigma_{n})|}{\log_{2}{\text{dim}\mathcal{H}_{n}}}\to
0\,,$ (15)

where $\|\cdot\|_{1}$ is the trace norm. The measures that satisfy the
postulate of asymptotic continuity are valuable for estimating the distillable
entanglement, which will be discussed in Sec. V.2.1.

Note that the first two postulates above constitute the essential criteria
that any entanglement measure must satisfy.

###  II.4 Convex-roof extension

The construction of convex roof is commonly used to quantify the entanglement
of mixed states [9]. Given an entanglement measure of a pure state
$E(\ket{\phi})$, it can be extended to a mixed state by means of convex roof,
such that

$E(\rho)=\inf_{\\{p_{k},\ket{\phi_{k}}\\}}\sum_{k}p_{k}E\bigl{(}\ket{\phi_{k}}\bigr{)}\,,$
(16)

where the infimum is taken over all ensembles
$\bigl{\\{}p_{k},\ket{\phi_{k}}\bigr{\\}}$ of
${\rho=\sum_{k}p_{k}\ket{\phi_{k}}\\!\bra{\phi_{k}}}$. The optimal ensemble
that achieves this infimum is called the optimal ensemble for $E$. Although
concurrence of the two-qubit mixed states can be obtained analytically,
typically the convex roof cannot be easily determined. However, there are a
number of effective schemes to compute the convex roof, such as providing a
lower bound of it as an approximation.

The convex-roof construction is a reliable method for quantifying
entanglement. It produces an entanglement measure that is guaranteed to be
convex. Moreover, if a measure is monotonic on pure states, then its convex-
roof extension is also monotonic on mixed states [2]. Thus, to verify the
monotonicity of a convex roof entanglement measure, one only needs to examine
the monotonicity for pure states. It is worth noting that the properties of
the convex roof can be translated to ones for the concave roof.

###  II.5 Concurrence

Concurrence is one of the most celebrated entanglement measures, which is the
first entanglement measure built with the convex-roof extension. In two-qubit
cases, it has been proven that as the entanglement of formation increases,
concurrence monotonically decreases [13]. Moreover, it has been successfully
extended to higher-dimensional bipartite systems as

$\mathcal{C}(\ket{\psi})=\sqrt{1-\Tr\bigl(\rho^{2}\bigr{missing})}\,,$ (17)

where $\rho$ is the reduced density matrix of the subsystem [35]. The
generalized bipartite concurrence Eq. (17) is also an entanglement monotone
under SLOCC [36]. For a bipartition $\\{A|B\\}$ of a pure state
$\ket{\psi}_{AB}$, we denote the corresponding concurrence as
$\mathcal{C}(|\psi\rangle_{A|B})$ or $\mathcal{C}_{A|B}$.

##  III ME measures

In this section we present examples of multipartite entanglement measures and
discuss how some of them can be defined from measures of the bipartite case.

###  III.1 Squashed entanglement

Many of the axiomatic measures of bipartite states can be extended to the
multipartite case. For a given state $\rho_{AB}$, the squashed entanglement is
given by

$E_{sq}(\rho_{AB})=\frac{1}{2}\inf I\bigl{(}A:B|E\bigr{)}\,,$ (18)

where the infimum is taken over all density matrices $\rho_{ABE}$ such that
${\Tr_{E}\rho_{ABE}=\rho_{AB}}$, and the conditional mutual information is
defined as

$I\bigl{(}A:B|E\bigr{)}=S(AE)+S(BE)-S(ABE)-S(E)\,,$ (19)

where ${S(X)=-\Tr(X\log_{2}X)}$ is the von Neumann entropy of the system $X$.
Multipartite $c$-squashed entanglement is defined as follows [37]. For the
$m$-party state $\rho_{A_{1},A_{2},\cdots,A_{m}}$,

$E_{sq}^{c}(\rho_{A_{1},A_{2},\cdots,A_{m}})=\inf
I\bigl{(}A_{1}:A_{2}:\cdots:A_{m}|E\bigr{)}\,,$ (20)

where the infimum is taken over the extension states
${\sigma_{A_{1},A_{2},\cdots,A_{m},E}}$ of the form ${\sum
p_{i}\rho^{i}_{A_{1},A_{2},\cdots,A_{m}}\otimes\ket{i}_{E}\\!\bra{i}}$. The
conditional mutual information is a sum of conditional bipartite mutual
information

$\displaystyle I$
$\displaystyle(A_{1}:A_{2}:\cdots:A_{m}|E)=I\bigl{(}A_{1}:A_{2}|E\bigr{)}+$
(21)
$\displaystyle\,\,\,I\bigl{(}A_{3}:A_{1}A_{2}|E\bigr{)}+\cdots+I\bigl{(}A_{m}:A_{1}\cdots
A_{m-1}|E\bigr{)}\,.$

Thus $E_{sq}^{c}$ is an entanglement measure which is obtained through the
mixed convex roof of the multipartite quantum mutual information function, and
it is indeed a good entanglement measure. In Ref. [37], the authors also
define the multipartite $q$-squashed entanglement as a generalization of the
bipartite squashed entanglement. For the $m$-party state
$\rho_{A_{1},A_{2},\cdots,A_{m}}$,

$E_{sq}^{q}(\rho_{A_{1},A_{2},\cdots,A_{m}})=\inf
I\bigl{(}A_{1}:A_{2}:\cdots:A_{m}|E\bigr{)}\,,$ (22)

where the infimum is taken over the extension states
$\sigma_{A_{1},A_{2},\cdots,A_{m},E}$ of $\rho_{A_{1},A_{2},\cdots,A_{m}}$. It
is proved that the $q$-squashed entanglement satisfies several important
properties, including the monotonicity under LOCC, convexity, additivity, and
asymptotic continuity. Furthermore, it is shown that multipartite $q$-squashed
entanglement is an upper bound on multipartite distillable key; see Sec.
V.2.2.

###  III.2 Three-tangle

The three-tangle (or residual entanglement) $\tau$ is an entanglement measure
for three-qubit states [38]. It is defined as follows

$\tau\equiv\tau_{ABC}=\mathcal{C}^{2}_{A|BC}-\mathcal{C}^{2}_{AB}-\mathcal{C}^{2}_{AC}\,,$
(23)

where $\mathcal{C}^{2}_{A|BC}$ is the squared concurrence between $A$ and the
pair $BC$, $\mathcal{C}^{2}_{AB}$ is the squared concurrence between $A$ and
$B$, and $\mathcal{C}^{2}_{AC}$ is the squared concurrence between $A$ and
$C$. The three-tangle is permutationally invariant of the parties. And $\tau$
is an entanglement monotone, i.e., decreasing on average under LOCC in all the
three parties [7]. It is noted that bipartite measures cannot determine the
inequivalence of the GHZ and $W$ classes, but three-tangle can.
$\tau(\psi_{\textrm{GHZ}})\neq 0$ for any state in the GHZ class, while it
vanishes for any state in the $W$ class.

The generalization of the three-tangle for $N$ qubits, namely, the $N$-tangle
is discussed in Ref. [39].

###  III.3 Global entanglement

There exist measures that can be calculated as straightforward combinations of
bipartite entanglement measures. Global entanglement of tripartite system, for
instance, is the sum of squared concurrences between a single qubit versus all
other qubits [40, 41, 42],

$Q=\mathcal{C}^{2}\bigl{(}\ket{\phi}_{A|BC}\bigr{)}+\mathcal{C}^{2}\bigl{(}\ket{\phi}_{B|AC}\bigr{)}+\mathcal{C}^{2}\bigl{(}\ket{\phi}_{C|AB}\bigr{)}\,.$
(24)

Its monotonicity under LOCC is simply inherited from the bipartite measures.

By using the relations [38]

$\mathcal{C}^{2}(\ket{\phi}_{i|jk})=\tau_{ABC}+\mathcal{C}^{2}(\rho_{ij})+\mathcal{C}^{2}(\rho_{ik})\,,$
(25)

where ${\rho_{ij}=\Tr_{k}\bigl{(}\ket{\phi}_{ijk}\\!\bra{\phi}\bigr{)}}$ and
$i,j,k$ are distinct systems in $\\{A,B,C\\}$. Then the global entanglement
can be written as

$Q=3\tau_{ABC}+2\bigl{[}\mathcal{C}^{2}(\rho_{AB})+\mathcal{C}^{2}(\rho_{AC})+\mathcal{C}^{2}(\rho_{BC})\bigr{]}\,.$
(26)

###  III.4 Schmidt measure

For a pure state
${\ket{\phi}\in\mathcal{H},\mathcal{H}=\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\otimes\cdots\otimes\mathbb{C}^{d_{N}}}$
in the composite system with parties ${A_{1},A_{2},\cdots,A_{N}}$, the Schmidt
measure is defined as [43]

$E_{P}(\ket{\phi}\\!\bra{\phi})=\log_{2}r\,,$ (27)

where

$\ket{\phi}=\sum_{i=1}^{R}\alpha_{i}\ket{\phi_{A_{1}}^{(i)}}\otimes\ket{\phi_{A_{2}}^{(i)}}\otimes\cdots\otimes\ket{\phi_{A_{N}}^{(i)}}\,,$
(28)

and
${\ket{\phi_{A_{j}}^{(i)}}\in\mathbb{C}^{d_{j}},j=1,2,\cdots,N,\alpha_{i}\in\mathbb{C}}$,
and $r$ is the minimal number of product terms $R$ in the decomposition of
$\ket{\phi}$. For a bipartite system with parties $A_{1}$ and $A_{2}$, the
minimal number of product terms $r$ is given by the Schmidt rank of the state.
One can extend the definition to the full state space
$\mathcal{S}(\mathcal{H})$ by using a convex-roof construction. For
$\rho\in\mathcal{S}(\mathcal{H})$,

$E_{P}(\rho)=\inf\sum_{i}\lambda_{i}E_{P}(\ket{\phi_{i}}\\!\bra{\phi_{i}})\,,$
(29)

where the infimum is taken over all possible convex combinations of the form
${\rho=\sum_{i}\lambda_{i}\ket{\phi_{i}}\\!\bra{\phi_{i}}}$ with
$0\leq\lambda_{i}\leq 1$ for all $i$.

The measure is zero iff the state is fully product. Therefore, it cannot
distinguish genuine multipartite entanglement from bipartite entanglement.

##  IV GME measures

Numerous entanglement measures, particularly for the bipartite case, have been
proposed. Yet, the situation is much more complicated for multipartite
systems. How to quantify genuine entanglement contained in multipartite
quantum states is still lacking in research.

If an $N$-partite state is not biseparable then it is called genuinely
$N$-partite entangled. A measure of genuine multipartite entanglement (GME)
$E(\rho)$ should at least satisfy [44]

  1. (1)

$E(\rho)=0,\forall\rho\in\mathcal{S}_{\text{bi-SEP}}$, where
$\mathcal{S}_{\text{bi-SEP}}$ is the set of biseparable states;

  2. (2)

$E(\rho)>0,\forall\rho\in\mathcal{S}_{\text{GME}}$, where
$\mathcal{S}_{\text{GME}}$ is the set of all GME states;

  3. (3)

(Strong monotonicity) ${E(\rho)\geq\sum_{i}p_{i}E(\rho_{i})}$, where
$\\{p_{i},\rho_{i}\\}$ is an ensemble of states produced by applying LOCC
operations on $\rho$;

  4. (4)

$E\bigl{(}U_{\text{local}}\rho U^{\dagger}_{\text{local}}\bigr{)}=E(\rho)$,
where $U^{\dagger}_{\text{local}}$ is an arbitrary local unitary operator.

###  IV.1 GME-concurrence

For an $N$-partite pure state
$\ket{\phi}\in\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\cdots\otimes\mathcal{H}_{N}$,
where dim${(\mathcal{H}_{i})=d_{i},i=1,2,\cdots,N}$, the GME-concurrence is
defined as [45]

$\mathcal{C}_{\text{GME}}(\ket{\phi}):=\min_{\gamma_{i}\in\gamma}\sqrt{2\Bigl{[}1-\Tr(\rho_{A_{\gamma_{i}}}^{2})\Bigr{]}}\,,$
(30)

where $\gamma=\\{\gamma_{i}\\}$ represents the set of all possible
bipartitions $\\{A_{i}|B_{i}\\}$ of $\\{1,2,\cdots,N\\}$. The GME-concurrence
can be generalized to mixed states $\rho$ via a convex-roof construction,
i.e.,

$\mathcal{C}_{\text{GME}}(\rho)=\inf_{\\{p_{i},\ket{\phi_{i}}\\}}\sum_{i}p_{i}\mathcal{C}_{\text{GME}}\bigl{(}\ket{\phi_{i}}\bigr{)}\,,$
(31)

where the infimum is taken over all possible decompositions
${\rho=\sum_{i}p_{i}\ket{\phi_{i}}\\!\bra{\phi_{i}}}$. It is a special case of
$k$-ME concurrence when ${k=2}$. The measure $k$-ME concurrence can detect all
$k$-nonseparable states, and exhibit properties such as being an entanglement
monotone, vanishing on $k$-separable states, convexity, subadditivity,
invariant under local unitary transformations [46].

###  IV.2 Geometric measure of entanglement

Motivated by the geometric measure (GM) of entanglement introduced by Wei and
Goldbart [47], Ref. [48] presents a genuine multipartite entanglement measure
called the generalized geometric measure (GGM). Consider an $N$-partite pure
quantum state $\ket{\phi_{N}}$, and let

$g_{max}(\phi_{N})=\underset{\psi}{\text{max}}|\innerproduct{\psi}{\phi_{N}}|\,,$
(32)

where the maximum is over all pure quantum states $\ket{\psi}$ that are not
genuinely $N$-party entangled. And $g_{max}$ quantifies the closeness of the
state $\ket{\phi_{N}}$ to all pure quantum states that are not genuinely
multipartite entangled. A pure quantum state of $N$ parties is considered to
be genuinely $N$-party entangled if it cannot be expressed as a product state
of any bipartite partition. Then the GGM is defined as

$E_{G}(\ket{\phi_{N}})=1-g^{2}_{max}(\ket{\phi_{N}})\,.$ (33)

It is noted that the maximization in $g_{max}$ is only over pure states that
are a product over all bipartite partitions. The GGM $E_{G}$ is vanishing for
all pure states that are not genuine multipartite entangled and non-vanishing
for others. The measure is computable for a multipartite pure state of an
arbitrary number of parties, and of arbitrary dimensions, and it is
monotonically decreasing under LOCC.

The generalized geometric measure can be written in a computable form for all
multipartite pure quantum states as [48, 49, 50]

$\displaystyle E_{G}(\phi_{N})=1-\max\Bigl{\\{}\lambda_{A|B}^{2}|$
$\displaystyle A\cup B=\\{1,2,\cdots,N\\},$ $\displaystyle A\cap
B=\varnothing\Bigr{\\}}\,,$ (34)

where $\lambda_{A|B}$ is the maximal Schmidt coefficient of $\ket{\phi_{N}}$
in the bipartite split ${A|B}$. For the three-qubit pure states $\ket{\phi}$,
the GGM reduces to

$E_{G}=1-\max\bigl{\\{}\lambda_{A}^{2},\lambda_{B}^{2},\lambda_{C}^{2}\bigr{\\}}\,,$
(35)

where $\lambda_{A}^{2}$ is the maximal eigenvalue of
$\rho_{A}=\Tr_{BC}\bigl{(}\ket{\phi}\\!\bra{\phi}\bigr{)}$, similarly for
$\lambda_{B}^{2}$ and $\lambda_{C}^{2}$.

###  IV.3 Tripartite negativity

An ideal measure of the full tripartite entanglement of three qubits should
have at least the following characteristics: i) to be zero for any fully
separable or biseparable state and non-zero for any fully entangled state; ii)
to be invariant under local unitary (LU); iii) to be nonincreasing under LOCC,
that is, to be an entanglement monotone. Tripartite negativity is defined
according to these three conditions, which is expressed as [51]

$\mathcal{N}_{ABC}=\bigl{(}\mathcal{N}_{A|BC}\mathcal{N}_{B|AC}\mathcal{N}_{C|AB}\bigr{)}^{\frac{1}{3}}\,,$
(36)

where the bipartite negativities are defined as

$\mathcal{N}_{I|JK}=-2\sum_{i}\varepsilon_{i}\bigl{(}(\rho^{I})^{T}\bigr{)}\,,$
(37)

where $\varepsilon_{i}\bigl{(}(\rho^{I})^{T}\bigr{)}$ denotes the negative
eigenvalues of $(\rho^{I})^{T}$, and $(\rho^{I})^{T}$ is the partial transpose
of $\rho$ with respect to subsystem $I$, namely, $\langle
i_{I},j_{JK}|(\rho^{I})^{T}|k_{I},l_{JK}\rangle=\langle
k_{I},j_{JK}|\rho|i_{I},l_{JK}\rangle$, with $I=A,B,C$ and $JK=BC,AC,AB$,
respectively. The tripartite negativity fulfills the above three conditions
for pure states.

###  IV.4 Concurrence fill

The challenge of establishing a proper entanglement ordering for genuine
three-qubit entanglement is first tackled by the proposal of concurrence fill
[44, 52]. For an arbitrary pure three-qubit state $\ket{\phi}_{ABC}$ shared by
three parties $A,B,C$, the concurrence between the bipartition $i$ and $jk$ is
${\mathcal{C}\bigl{(}\ket{\phi}_{i|jk}\bigr{)}=\sqrt{2\bigl{[}1-\Tr(\rho_{i}^{2})\bigr{]}}}$,
with ${\rho_{i}=\Tr_{jk}\bigl{(}\ket{\phi}_{ijk}\\!\bra{\phi}\bigr{)}}$ and
$i,j,k$ are distinct systems in $\\{A,B,C\\}$. The three one-to-other
bipartite entanglements are not independent, and they follow the relation [53]

$\mathcal{C}^{2}\bigl{(}\ket{\phi}_{i|jk}\bigr{)}\leq\mathcal{C}^{2}\bigl{(}\ket{\phi}_{j|ik})+\mathcal{C}^{2}(\ket{\phi}_{k|ij}\bigr{)}\,.$
(38)

These three squared bipartite concurrences can be geometrically interpreted as
the lengths of three sides of a triangle, which is the so-called concurrence
triangle. Then the concurrence fill is defined as the square-root of the area
of the concurrence triangle [44]

$\displaystyle\mathcal{F}$
$\displaystyle(\ket{\phi}_{ABC})=\Bigl{[}\frac{16{\cal
P}_{ABC}}{3}\bigl{(}{\cal P}_{ABC}-\mathcal{C}^{2}(\ket{\phi}_{A|BC})\bigr{)}$
(39) $\displaystyle\times\bigl{(}{\cal
P}_{ABC}-\mathcal{C}^{2}(\ket{\phi}_{B|AC})\bigr{)}\bigl{(}{\cal
P}_{ABC}-\mathcal{C}^{2}(\ket{\phi}_{C|AB})\bigr{)}\Bigr{]}^{1/4}\,,$

with half-perimeter

${\cal
P}_{ABC}=\frac{1}{2}\Bigl{[}\mathcal{C}^{2}\bigl{(}\ket{\phi}_{A|BC}\bigr{)}+\mathcal{C}^{2}\bigl{(}\ket{\phi}_{B|AC}\bigr{)}+\mathcal{C}^{2}\bigl{(}\ket{\phi}_{C|AB}\bigr{)}\Bigr{]}\,.$
(40)

In addition to the proposed three-qubit concurrence triangle, the authors in
Ref. [54] show that the triangle relation holds for any continuous
entanglement measure and systems of any dimension. The extension of
concurrence fill to four-partite systems, inspired by a richer geometrical
interpretation, is presented in [55], which interprets as a combination of
areas of cyclic quadrilateral and triangle structures, resulting from two
types of bipartition.

The concurrence fill is proposed for GME quantification and satisfies the
property of faithfulness and smoothness, but it is later proven that the
concurrence fill can be increased under some LOCC operations [42].
Nevertheless, numerical evidence supports that if the edges of the triangle
are chosen as bipartite concurrence, then the area is nonincreasing under
LOCC. Subsequently, various genuine multipartite entanglement measures are
proposed, which are based on the bipartite entanglement monotones [56, 42, 57,
58].

###  IV.5 Geometric mean of bipartite concurrences

The geometric mean of bipartite concurrences (GBC) $\mathcal{G}$ satisfies the
figure of merits of concurrence fill and qualifies as an entanglement monotone
[56], while lacks a geometric interpretation. Denote the set of all possible
bipartitions as $\alpha$ and its cardinality ${c(\alpha)}$, we have

$\mathcal{G}(\rho)=\sqrt[c(\alpha)]{\prod\limits_{\alpha_{j}\in\alpha}\mathcal{C}_{A_{\alpha_{j}}B_{\alpha_{j}}}}\,.$
(41)

The strong monotonicity of GBC is proved as follows. Bipartite concurrence
satisfies the property of strong monotonicity [36], i.e.,

$\sum_{i}{p_{i}\mathcal{C}_{AB}\bigl{(}\sigma_{i}\bigr{)}}\leq\mathcal{C}_{AB}(\rho)\,,$
(42)

where $\\{p_{i},\sigma_{i}\\}$ is an ensemble produced by an arbitrary LOCC
channel $\Lambda_{\text{LOCC}}$ acting on the multipartite quantum state
$\rho$, such that

$\Lambda_{\text{LOCC}}(\rho)=\sum_{i}{p_{i}\sigma_{i}}\,.$ (43)

Note that $\Lambda_{\text{LOCC}}$ is also an LOCC channel with respect to the
bipartition $\\{A|B\\}$. Hereby we examine the monotonicity for pure states,
i.e., when $\rho$ and $\sigma_{i}$ are all pure states. Since all the
bipartite concurrences in the definition of GBC satisfy Eq. (42), we have

$\displaystyle\mathcal{G}(\rho)\geq$
$\displaystyle\sqrt[c(\alpha)]{\prod\limits_{\alpha_{j}\in\alpha}\Bigl{(}\sum_{i}{p_{i}\mathcal{C}_{A_{\alpha_{j}}B_{\alpha_{j}}}\bigl{(}\sigma_{i}\bigr{)}\Bigr{)}}}$
(44) $\displaystyle\geq$
$\displaystyle\sum_{i}{p_{i}\Biggl{(}\sqrt[c(\alpha)]{\prod\limits_{\alpha_{j}\in\alpha}\mathcal{C}_{A_{\alpha_{j}}B_{\alpha_{j}}}\bigl{(}\sigma_{i}\bigr{)}}\Biggr{)}}=\sum_{i}{p_{i}\mathcal{G}\bigl{(}\sigma_{i}\bigr{)}}\,,$

where the concavity of the geometric mean function defined on the convex set
with all the variables ranging from 0 to 1 is employed, which can be easily
proven with the Mahler’s inequality. In fact, any multivariable concave
function of bipartite entanglement monotones preserve the monotonicity.

GBC is the product of bipartite entanglement monotones with an exponential
order $c(\alpha)$ specified by the number of possible bipartitions. However,
the exponential order can be further improved. The square root of the product
of all the bipartite concurrences of a three-qubit pure state is also proven
to be an entanglement monotone [42], which improves the exponential order in
GBC from $1/3$ to $1/2$. Also, it can be generalized to any pure $N$-qubit
state as the product of all the bipartite concurrences to the order of
$1/2^{N-2}$.

###  IV.6 GME measures based on teleportation capability

In order to quantify GME as a resource, one needs to consider the usefulness
of GME in quantum information processing tasks. For three-qubit teleportation,
the GHZ state results in better teleportation fidelity than the $W$ state.
Consequently, many existing GME measures, such as the concurrence fill, are
proper in the sense that they are larger for GHZ state than $W$ state. In Ref.
[59], the authors extended this idea by considering the teleportation
capability for general pure states.

The three-qubit teleportation scheme with a resource state
$|\psi\rangle_{ABC}$ can be described as follows: first measure a system $k$
and observe the outcome, then perform two-qubit teleportation on the remaining
systems. When the teleportation is performed with qubits $i$ and $j$, the
maximal average fidelity of teleportation is [60]

$F_{ij}(|\psi\rangle_{ABC})=\frac{\sqrt{\tau_{ABC}+\mathcal{C}_{ij}^{2}}+2}{3},$
(45)

where $\mathcal{C}_{ij}$ is the concurrence for the reduced density operator
$\rho_{ij}$ of $|\psi\rangle_{ABC}$, and $i,j$ are distinct systems in
$\\{A,B,C\\}$. Also, let
$\mathcal{T}_{ij}(|\psi\rangle_{ABC})=3F_{ij}(|\psi\rangle_{ABC})-2$. The
authors define GME measures $\mathcal{T}_{\text{min}}$ and
$\mathcal{T}_{\text{GM}}$ as

$\displaystyle\mathcal{T}_{\text{min}}$
$\displaystyle\equiv\min\\{\mathcal{T}_{ij},\,\mathcal{T}_{jk},\,\mathcal{T}_{ik}\\}\,,$
(46) $\displaystyle\mathcal{T}_{\text{GM}}$
$\displaystyle\equiv\sqrt[3]{\mathcal{T}_{ij}\mathcal{T}_{jk}\mathcal{T}_{ik}}\,,$
(47)

respectively. Both $\mathcal{T}_{\text{min}}$ and $\mathcal{T}_{\text{GM}}$
satisfy the requirements for GME measures. Moreover, the authors prove that if
both $\mathcal{T}_{ij}$ and $\mathcal{T}_{ik}$ are strictly positive, then so
does $\mathcal{T}_{jk}$. Thus, they define two additional GME measures

$\displaystyle\mathcal{T}_{\text{min}}^{(i)}$
$\displaystyle\equiv\min\\{\mathcal{T}_{ij},\,\mathcal{T}_{ik}\\}\,,$ (48)
$\displaystyle\mathcal{T}_{\text{GM}}^{(i)}$
$\displaystyle\equiv\sqrt{\mathcal{T}_{ij}\mathcal{T}_{ik}}\,,$ (49)

where $\mathcal{T}_{\text{min}}^{(i)}$ and $\mathcal{T}_{\text{GM}}^{(i)}$
quantifies the minimal and the average teleportation capability of the system
$i$, respectively. These measures can also be extended to $N$-qubit states.

##  V Relation with quantum information tasks

Multipartite entanglement is an essential resource for quantum communication,
quantum computing, quantum networks, etc. In this section we discuss the
pivotal role of GME in the tasks of quantum teleportation and dense coding,
followed by an introduction to some operational entanglement measures.

###  V.1 Quantum teleportation and dense coding with GME

In Ref. [61], Yeo and Chua presented a protocol for faithful teleportation of
an arbitrary two-qubit state using a genuine four-qubit entangled state
$\ket{\chi^{00}}$, which is compared to other multipartite entangled states
such as GHZ and $W$ states, then introduced a dense coding scheme using the
same four-qubit state as a shared resource. Afterwards, Ref. [62] generalized
the results of Ref. [61] to teleport an arbitrary $N$-qubit state via genuine
$N$-qubit entanglement channels.

By using a special $2N$-qubit entanglement channel
$\ket{G_{1,1,\cdots,1}}_{\mathbf{A};\mathbf{B}}$ (here $\mathbf{A};\mathbf{B}$
represents Alice’s $N$ qubits and Bob’s $N$ qubits of the channel
respectively), the protocol was proposed to teleport an arbitrary $N$-qubit
state

$\ket{\phi}_{\mathbf{a}}=\alpha_{0}\ket{\overline{0}}+\alpha_{1}\ket{\overline{1}}+\cdots+\alpha_{2^{N}-1}\ket{\overline{2^{N}-1}},$
(50)

where $\mathbf{a}$ are the $N$ qubits to be teleported from Alice to Bob,
${\alpha_{0},\alpha_{1},\cdots,\alpha_{2^{N}-1}}$ are complex coefficients
satisfying
${|\alpha_{0}|^{2}+|\alpha_{1}|^{2}+\cdots+|\alpha_{2^{N}-1}|^{2}=1}$. And
${\ket{\overline{0}},\ket{\overline{1}},\cdots,\ket{\overline{2^{N}-1}}}$ are
binary forms of the states $\ket{00\cdots 0},\ket{0\cdots
01},\cdots,\ket{1\cdots 11}$.

Denote the four Bell states as

$\displaystyle\ket{\Phi_{1,2}}_{AB}=\frac{1}{\sqrt{2}}(\ket{00}_{AB}\pm\ket{11}_{AB})\,,$
(51)
$\displaystyle\ket{\Phi_{3,4}}_{AB}=\frac{1}{\sqrt{2}}(\ket{01}_{AB}\pm\ket{10}_{AB})\,.$

The state $\ket{G_{1,1,\cdots,1}}_{\mathbf{A};\mathbf{B}}$ is a tensor product
of $N$ Bell states
$\ket{\Phi_{1}}_{A_{1},B_{1}}\otimes\ket{\Phi_{1}}_{A_{2},B_{2}}\otimes\cdots\otimes\ket{\Phi_{1}}_{A_{N},B_{N}}$,
which is expressed as

$\displaystyle\ket{G_{1,1,\cdots,1}}$
$\displaystyle{}_{\mathbf{A};\mathbf{B}}=\frac{1}{\sqrt{2^{N}}}\Bigl{[}\ket{\overline{0}}_{A}\ket{\overline{0}}_{B}$
(52)
$\displaystyle+\cdots+\ket{\overline{2^{N}-1}}_{A}\ket{\overline{2^{N}-1}}_{B}\Bigr{]}\,.$

Then, consider the $4^{N}$ generalized Bell states

$\displaystyle\ket{G_{i1,i2,\cdots,iN}}$
$\displaystyle{}_{\mathbf{A};\mathbf{B}}=\ket{\Phi_{i1}}_{A_{1},B_{1}}$ (53)
$\displaystyle\otimes\ket{\Phi_{i2}}_{A_{2},B_{2}}\otimes\cdots\otimes\ket{\Phi_{iN}}_{A_{N},B_{N}}\,,$

where $i1,i2,\cdots,iN=1,2,3,4$.

Now consider the teleportation by using the maximally entangled state (MES)
channel

$\ket{MES}_{\mathbf{A};\mathbf{B}}=\frac{1}{\sqrt{2^{N}}}\sum_{J=1}^{2^{N}}\ket{J}_{A}\ket{J^{\prime}}_{B}\,,$
(54)

where $\ket{J}_{A}$s, $\ket{J^{\prime}}_{B}$s are orthogonal bases of the
Hilbert space of Alice’s $N$ qubits and Bob’s $N$ qubits, respectively. It is
noted that $\ket{MES}_{\mathbf{A};\mathbf{B}}$ may be a _genuine_ multipartite
entangled state, which cannot be reduced to the tensor product of $N$ Bell
states in terms of $2N$ partities. Using local unitary operators applied by
Alice and Bob, $\ket{MES}_{\mathbf{A};\mathbf{B}}$ connects with
$\ket{G_{1,1,\cdots,1}}_{\mathbf{A};\mathbf{B}}$ as

$\ket{MES}_{\mathbf{A};\mathbf{B}}=U_{1}(AN)U_{2}(BN)\ket{G_{1,1,\cdots,1}}_{\mathbf{A};\mathbf{B}}\,,$
(55)

where unitary operators $U_{1}(AN)$ and $U_{2}(BN)$ act on $N$ qubits of Alice
and Bob, respectively. Then, The joint state of the initial state of the $N$
qubits to be teleported and the entanglement channel can be expressed as

$\displaystyle\ket{\phi}$
${}_{\mathbf{a}}\otimes\ket{MES}_{\mathbf{A};\mathbf{B}}$ $\displaystyle=$
$\displaystyle\sum_{i1,i2,\cdots,iN=1}^{4}U_{1}(AN)U_{2}(aN)\ket{G_{i1,i2,\cdots,iN}}_{\mathbf{A};\mathbf{a}}$
$\displaystyle\otimes U(i1)^{-1}U(i2)^{-1}\cdots
U(iN)^{-1}\ket{\phi}_{\mathbf{B}}\,,$ (56)

where $\ket{\phi}_{\mathbf{B}}$ is the same as the initial state of qubits
$\mathbf{a}$ in Eq. (50). Here, ${U(1)=I}$, ${U(2)=\sigma_{z}}$,
${U(3)=\sigma_{x}}$, ${U(4)=i\sigma_{y}}$, with $\sigma_{x,y,z}$ being the
three Pauli operators. If Alice performs a complete projective measurement
jointly on the $2N$ qubits and gets a outcome
$U_{1}(AN)U_{2}(aN)\ket{G_{i1,i2,\cdots,iN}}_{\mathbf{A};\mathbf{a}}$, then
Bob will achieve an accurate replication of $\ket{\phi}_{\mathbf{a}}$ after he
performs the unitary operations $U(i1)\otimes U(i2)\otimes\cdots\otimes U(iN)$
on his qubits in accordance with Alice’s result of the measurement. Hence,
teleportation is achieved by Alice and Bob.

It is shown that the multipartite entanglement channel in Eq. (55) is
necessary and sufficient for the faithful teleportation of an $N$-qubit state
together with $2N$ bits of classical communication. The GHZ states and $W$
states of $2N$ qubits, however, cannot be transferred into a maximally
entangled state through local operations performed by Alice and Bob, thus they
cannot be effectively utilized for faithful teleportation of an arbitrary
$N$-qubit state.

The dense coding scheme using genuine four-partite entangled state
$\ket{\chi^{00}}$ communicate $4$ bits of classical information by sending two
particles perfectly, which is impossible with a four-partite GHZ or $W$ state
[61]. The maximally entangled five-qubit state (exhibits genuine multipartite
entanglement)
$\ket{\psi_{5}}=\frac{1}{2}(\ket{001}\ket{\Phi_{4}}+\ket{010}\ket{\Phi_{2}}+\ket{100}\ket{\Phi_{3}}+\ket{111}\ket{\Phi_{1}})$
as a resource for superdense coding is discussed in Ref. [63]. In this case,
they assume Alice owns the first three qubits and Bob has the other two of
$\ket{\psi_{5}}$. Then Alice applies one of the sets of unitary
transformations on her particle, which is the process of encoding. After
having performed one of the operations described above, Alice sends off her
qubits to Bob using conventional physical medium. Bob can then perform a five-
partite measurement to extract the classical bits, which corresponds to
decoding. It is noted that Alice and Bob have agreed on dense coding rules
before encoding. The capacity of superdense coding for the state
$\ket{\psi_{5}}$ equals to $5$, reaching the Holevo bound of maximal amount of
classical information that can be encoded.

###  V.2 Operational multipartite entanglement measures

In Secs. III and IV, we have reviewed various entanglement measures that are
defined by axiomatic ansatz. Here we move on to the operationally meaningful
measures which can quantify the value of a state for a specific protocol that
requires entanglement.

####  V.2.1 Distillable entanglement and entanglement cost

Non-maximally entangled states are insufficient for faithful teleportation as
entanglement plays a crucial role in the process. However, by having a
sufficient number of copies of a non-maximally entangled state, one can
achieve asymptotically faithful teleportation at a certain rate. Considering a
scenario where a large number $n$ of copies of a state $\rho$ are given, the
question arises whether it is possible to transform $\rho^{\otimes n}$ to an
output state $\sigma^{\otimes n}$ using LOCC operations, where LOCC protocols
are assumed to be trace-preserving. To address this question from a physical
perspective, we introduce the asymptotic limit approach, and the distance
between the state resulting from the distillation protocol and the desired
state is measured using the trace norm. If ${n\to\infty}$, the arbitrarily
good approximation of $\sigma^{\otimes m}$ by $\sigma_{m}$ becomes possible.
In other words, one can transform $\rho^{\otimes n}$ to an output state
$\sigma_{m}$ that approximates $\sigma^{\otimes m}$ very well for sufficiently
large $m$ with the achievable rate $r=m/n$.

The distillable entanglement is the supremum of the rates at which maximally
entangled states can be obtained from an input supply of states in the form of
$\rho$ over all possible distillation protocols. Denote a general trace-
preserving LOCC operation by $\Lambda$, and let $\Phi(d)$ represent the
density operator corresponding to the maximally entangled state vector in
dimension $d$, then the distillable entanglement is defined as

$E_{D}(\rho)=\sup\biggl{\\{}r:\lim_{n\to\infty}\Bigl{[}\inf_{\Lambda}\|\Lambda(\rho^{\otimes
n})-\Phi(2^{rn})\|_{1}\Bigr{]}=0\biggr{\\}}\,.$ (57)

It is noted that the state $\Phi(2^{rn})$ is local unitarily equivalent to
$rn$ copies of the two-qubit maximally entangled state $\Phi(2)^{\otimes rn}$.

Now consider how many qubits we have to communicate in order to create a state
$\rho$. The entanglement cost quantifies the maximal possible rate $r$ at
which one can generate states that are close approximations of many copies of
$\rho$ using a supply of blocks of two-qubit maximally entangled states. It
provides insights into the efficiency of converting entangled resources to
states that approximate the target state $\rho$, which is given by

$E_{C}(\rho)=\inf\biggl{\\{}r:\lim_{n\to\infty}\Bigl{[}\inf_{\Lambda}\|\rho^{\otimes
n}-\Lambda(\Phi(2^{rn}))\|_{1}\Bigr{]}=0\biggr{\\}}\,.$ (58)

Computing $E_{D}(\rho)$ and $E_{C}(\rho)$ is a challenging task, however,
these measures have significant implications in the investigation of channel
capacities, and also closely related to the entanglement of formation and
relative entropy of entanglement [64, 65, 66].

For the multipartite scenario, the definition of distillable entanglement can
be tailored to specific target states, such as GHZ states or cluster states,
capturing different properties of the state being considered. Similar
situations arise when defining the entanglement cost, where singlet states are
commonly used as the resource, but other resources like GHZ or $W$ states can
also be considered. Different experimental settings and resource availability
motivate various definitions of entanglement cost and distillable
entanglement. The operational point of view is crucial in understanding and
interpreting these measures.

\begin{overpic}[width=180.67499pt]{set}\end{overpic} Figure 2: Schematic
representation of the set of all states for bipartite systems, including the
SEP states, PPT entangled states, hypothetical NPT nondistillable states, and
distillable states.

Bound entangled states are closely related to the problem of entanglement
distillation. Generally, states that are entangled yet not distillable are
termed as _bound entangled_ [2, 67], which include positive partial transpose
(PPT) entangled states. Figure 2 illustrates the set of all states for
_bipartite_ systems, including separable states (SEP), PPT entangled states,
_hypothetical_ non-PPT (NPT) nondistillable states (or NPT bound entangled
states), and distillable states [68, 69, 70]. It is clear that separable
states cannot be distilled, and all PPT states are not distillable. Notably,
for $2\otimes 2$ and $2\otimes 3$ systems, it is known that all PPT states are
separable and all NPT states are distillable. However, whether the
hypothetical NPT nondistillable states exist remains an important and open
question.

####  V.2.2 Multipartite distillable key

The distillable (secret) key quantifies the asymptotic rate at which Alice and
Bob may distill secret classical bits from many copies of a shared quantum
state. Rather than focusing on distilling singlets, one can explore the
distillation of private states, which encompasses a broader class of states.
Let’s start with the $m$-partite private state, or $m$-partite _pdit_ [71]

$\Gamma^{(d)}=\sum_{i,j=0}^{d-1}\frac{1}{d}\ket{ii\cdots i}\\!\bra{jj\cdots
j}\otimes
U_{i}\rho_{A_{1}^{\prime},A_{2}^{\prime},\cdots,A_{m}^{\prime}}U_{j}^{\dagger}\,,$
(59)

where $\rho_{A_{1}^{\prime},A_{2}^{\prime},\cdots,A_{m}^{\prime}}$ denotes a
state on the Hilbert space
$\mathcal{H}_{A_{1}^{\prime},A_{2}^{\prime},\cdots,A_{m}^{\prime}}$, and
$U_{i}(i=0,1,\cdots,d-1)$ are some unitary operations. Thus the state
$\Gamma^{(d)}$ is defined on the Hilbert space
$(\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}\otimes\cdots\otimes\mathcal{H}_{A_{m}})\otimes(\mathcal{H}_{A_{1}^{\prime}}\otimes\mathcal{H}_{A_{2}^{\prime}}\otimes\cdots\otimes\mathcal{H}_{A_{m}^{\prime}})$.
The system $A_{1},A_{2},\cdots,A_{m}$ giving secure bits of key is called the
_key part_ of the private state, and the system
$A_{1}^{\prime},A_{2}^{\prime},\cdots,A_{m}^{\prime}$ defends the key in
system $A_{1},A_{2},\cdots,A_{m}$ from eavesdropper is called the _shield
part_.

For a given multipartite state $\rho_{A_{1},A_{2},\cdots,A_{m}}$ acting on
${\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}\otimes\cdots\otimes\mathcal{H}_{A_{m}}}$,
consider a sequence $\Lambda_{n}$ of LOCC operations such that
${\Lambda_{n}\bigl{(}\rho_{A_{1},A_{2},\cdots,A_{m}}^{\otimes
n}\bigr{)}=\sigma^{(n)}}$, where $n$ is the number of copies of the state. A
set of operations ${{\cal P}=\cup_{n=1}^{\infty}\\{\Lambda_{n}\\}}$ is called
a pdit distillation protocol of state $\rho_{A_{1},A_{2},\cdots,A_{m}}$ if the
condition

$\lim_{n\to\infty}\|\sigma^{(n)}-\Gamma^{(d_{n})}\|_{1}=0$ (60)

holds. Here $\Gamma^{(d_{n})}$ is a multipartite pdit whose key part is of
dimension $d_{n}\times d_{n}$. For a protocol ${\cal P}$, its rate is given by
taking the limit superior of $(\log_{2}{d_{n}})/n$, i.e.,

$\mathcal{R}({\cal P})=\limsup_{n\to\infty}\frac{\log_{2}{d_{n}}}{n}\,.$ (61)

Then the distillable key of state $\rho_{A_{1},A_{2},\cdots,A_{m}}$ is given
by [72]

$K_{D}^{(m)}(\rho_{A_{1},A_{2},\cdots,A_{m}})=\sup_{{\cal P}}\mathcal{R}({\cal
P})\,.$ (62)

For the $N$-partite quantum state $\rho$, the normalized multipartite squashed
entanglement is an upper bound on the distillable key [37]

$K_{D}^{(m)}(\rho)\leq\frac{1}{m}E_{sq}^{q}(\rho)\,.$ (63)

Another upper bound of the distillable key is the regularized version of the
relative entropy [72].

When considering the distribution of secret keys, the multipartite private
states, which are the output of a protocol that distills secret key among
trusted parties, are necessarily genuine multipartite entangled [19]. Besides,
the distillation of private states is the goal of quantum key repeater, a
device that extends the distance of quantum key distribution (QKD) while being
capable of tolerating higher noise levels and beyond the limitations of
entanglement distillation [73, 74].

####  V.2.3 Accessible entanglement and source entanglement

If a quantum state $\rho$ can be transformed to another quantum state $\sigma$
by LOCC, then $\rho$ is considered to be at least as useful as $\sigma$. This
is because any application or protocol that can be achieved using $\sigma$ can
also be achieved using $\rho$, but the reverse may not be true. If the parties
are provided with $\rho$, they can transform it to $\sigma$ without incurring
any additional cost and proceed with the desired protocol. The study of LOCC
transformations allows us to determine the relative usefulness of different
quantum states and forms the basis for quantifying entanglement. And LOCC
corresponds to those operations which can be implemented without depleting the
existing entanglement resources, thus entanglement measures must be quantities
that are nonincreasing under these LOCC transformations.

In order to further understand multipatite entanglement, entanglement measures
with an operational meaning for arbitrary multipartite states (pure or mixed)
of any dimension are introduced [75]. These measures, known as _accessible_
and _source_ entanglement, can be computed once the possible LOCC
transformations are characterized. If there exists a deterministic LOCC
protocol that can transform state $\ket{\psi}$ to state $\ket{\phi}$, we say
that state $\ket{\psi}$ can _reach_ state $\ket{\phi}$, and state $\ket{\phi}$
is _accessible_ from state $\ket{\psi}$. For a given state $\ket{\psi}$, the
accessible set $M_{a}(\ket{\psi})$ is defined as the set of states that are
reachable via LOCC from $\ket{\psi}$. And the source set, $M_{s}(\ket{\psi})$
is the set of states that can reach $\ket{\psi}$. Then consider $\mu$ as an
arbitrary measure in the set of local unitarily equivalent classes, the
corresponding accessible volume which measures the amount of states that can
be accessed by $\ket{\psi}$, and the source volume which measures the amount
of states that can be used to reach the state are
${V_{a}(\ket{\psi})=\mu[M_{a}(\psi)]}$ and
${V_{s}(\ket{\psi})=\mu[M_{s}(\psi)]}$, respectively. The intuition is that
the larger the accessible volume is, the more valuable the state is, while the
larger the source volume is, the less powerful the state is. Thus, the
accessible entanglement and the source entanglement are given by

$E_{a}(\ket{\psi})=\frac{V_{a}(\ket{\psi})}{V_{a}^{\sup}}\,,\,\,\,E_{s}(\ket{\psi})=1-\frac{V_{s}(\ket{\psi})}{V_{s}^{\sup}}\,,$
(64)

where $V_{a}^{\sup}$ and $V_{s}^{\sup}$ represent the maximally accessible
volume and the maximally source volume according to the measure $\mu$,
respectively. It is worth noting that the operational properties of these
measures make it easy to prove that they are nonincreasing under LOCC. Also,
${M_{s}(\ket{\psi})=\emptyset}$ iff the state is in the MES because the
maximally entangled states are the only states that cannot be reached by any
other state [76]. For example, the $W$ state achieves the maximum values of
both ${E_{a}(\ket{W})=1}$ and ${E_{s}(\ket{W})=1}$. This highlights the
exceptional nature of the $W$ state, making it the most valuable state within
the $W$ class.

####  V.2.4 Concentratable entanglement

As a computable and operationally meaningful family of entanglement monotones,
the concentratable entanglement is introduced in Ref. [77]. Consider an
$N$-qubit pure quantum state $\ket{\phi}$, $\mathcal{S}=\\{1,2,\cdots,N\\}$ as
the set of labels for each qubit and its power set ${\cal P}(\mathcal{S})$,
for any set of qubit labels ${s\in{\cal
P}(\mathcal{S})\\!\setminus\\!\\{\emptyset\\}}$, the concentratable
entanglement is defined as

$\mathcal{C}_{C}(s)=1-\frac{1}{2^{c(s)}}\sum_{\alpha\in{\cal
P}(s)}\Tr(\rho_{\alpha}^{2})\,,$ (65)

where $c(s)$ is the cardinality, and $\rho_{\alpha}$ represents the joint
reduced state. The purity $\Tr(\rho_{\alpha}^{2})$ can be computed via an
overlap test, for instance to employ the $N$-qubit parallelized swap test to
compute the state overlap ${|\innerproduct{\phi_{1}}{\phi_{2}}|^{2}}$. For the
example of $N$-qubit $\ket{W}$ and $\ket{\textrm{GHZ}}$ state, one finds that
$\mathcal{C}_{C}^{\ket{\textrm{GHZ}}}(s)>\mathcal{C}_{C}^{\ket{W}}(s)$ and the
concentratable entanglement detects more multipartite entanglement in
$\ket{\textrm{GHZ}}$ than in $\ket{W}$.

The $N$-tangle, concurrence, and linear entropy of entanglement can be seen as
special cases of concentratable entanglement. For two-qubit systems,
$\mathcal{C}_{C}(s)=\tau_{(2)}/4=\mathcal{C}^{2}/4$ for all $s\in{\cal
P}(\mathcal{S})$, where $\tau_{(N)}=|\innerproduct{\phi}{\tilde{\phi}}|^{2}$
with $\ket{\tilde{\phi}}=\sigma_{y}^{\otimes N}\ket{\phi^{\ast}}$ and
$\mathcal{C}$ is the concurrence. Concentratable entanglement quantifies the
probability that at least one swap test fails when $N$ of them are applied in
parallel across two copies of $\ket{\phi}$. It is operationally interpreted as
the probability of producing Bell pairs in the parallelized swap test. The
controlled-swap (c-swap) operation is an experimentally accessible tool and a
fundamental component in quantum communication protocols for measuring and
witnessing entanglement.

##  VI Summary and Outlook

In this review, we have surveyed various multipartite entanglement measures
with an emphasis on the genuine and operational multipartite entanglement
measures. The intention behind this review is to provide valuable insights and
serve as a helpful resource for future research in the field of quantum
information processing. In addition, we hope that this review would inspire
and guide researchers in their endeavors to further develop novel approaches
for characterizing multipartite entanglement.

In spite of continuous progress, however, the current status of entanglement
measure theory is still marked by a number of outstanding open problems, some
of which include

  1. 1)

The quest for an effective and universally applicable measure to quantify
genuine multipartite entanglement in systems involving more than three parties
remains as a significant challenge.

  2. 2)

While theoretical entanglement measures provide valuable insights, operational
measures are designed to be practically applicable in experimental scenarios.
Thus, the exploration and advancement of new operational measures for
multipartite entanglement is essential.

  3. 3)

Extending entanglement measures to mixed-state multipartite systems is also a
meaningful direction. Real-world quantum systems often involve mixed states
due to unavoidable interactions with the environment, hence understanding and
quantifying multipartite entanglement in mixed states is critical for the
successful implementation of various quantum information processing tasks.

In summary, we believe that the research on multipartite entanglement measures
is still ongoing and is likely to have a crucial impact on other tasks in the
era of quantum information science.

##  VII Declaration of Competing Interest

The authors declare that they have no conflicts of interest in this work.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grants No. 92265115 and No. 12175014) and the National Key R&D Program of
China (Grant No. 2022YFA1404900).

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