1 Introduction

Nowadays, quantum entanglement [1] has been regarded as an important resource for quantum information and computation [2], which enables tasks like quantum cryptography [3], quantum dense coding [4], quantum teleportation [5] including so called entanglement swapping [6] and so on. A fundamental question in quantum entanglement theory is how to detect the existence of entanglement. For this end many separability criteria have been proposed [7, 8]. Another essential issue is how to quantify quantum entanglement in order that we can realize the control in the above applications. The purpose of quantifying entanglement originally came from quantum communication [9, 10]. In many cases quantum entanglement is used as an resource to accomplish such tasks. For instance, the maximally entangled quantum states can be contributed to faithful teleportation [7].

The quantification of quantum entanglement becomes increasingly important. For this reason the concept of entanglement measure is introduced. A variety of entanglement measures have been defined to quantify the amount of entanglement. For the cases of bipartite system, there are the entanglement of formation [10], relative entropy of entanglement [11], distillable entanglement and entanglement cost [10, 12], concurrence [13, 14] and negativity [15, 16], etc. For the multipartite system, some entanglement measures also have been proposed [17–22]. For a deeper discussion, one can recommend several general reviews on entanglement [7, 8, 12]. At present, existing entanglement measures are defined based on different perspectives. Nevertheless, most of them are difficult to be calculated, or restricted to application.

As mentioned above, maximally entangled states reveal some excellent properties in many applications. However, the meaning of “maximally” entangled states is confusing when the number of qubits are larger than three. A considerable amount of research has recently been devoted to the identification of maximally multipartite entangled state (MMES). Vicente et al [23] define the maximally entangled set of n-partite states based on local operations assisted by classical communication (LOCC). It is the set of states which are maximally useful under LOCC manipulation. Facchi et al [24] consider the degree of mixedness (called purity) of the reduced density matrices associated with every bipartition. They think a perfect MMES would be maximally entangled for every bipartition, but perfect MMESs do not necessarily exist. Hence, they define a entanglement measure with respect to the average bipartite entanglement over all possible balanced bipartition. Based on this approach, Zha et al [25–27] do some research to study maximally multi-qubit entangled state. Besides that, some highly entangled multi-qubit states are obtained in the Refs. [28, 29]. The above discussion is mostly based on the case of pure states.

The paper is organized as follows. A multipartite entanglement measure we have proposed is reviewed in Section 2, then we represent it based on reduced density matrices of multi-qudit pure states. In Section 3, a comparison is made of some multipartite entanglement measures which are introduced based on density matrices. In Section 4, as the main part of this paper, we propose a new multipartite entanglement measure for arbitrary pure states. It is proved to be a good entanglement measure. We apply our entanglement measure for some explicit examples in Section 5. Finally, the conclusion of this paper is drawn in Section 6.

2 Preliminary

In this section, we firstly review a multipartite entanglement measure based on coefficient matrices we proposed in Ref. [30]. Then, we will represent this entanglement measure in a different way.

Let us consider a n-qudit pure state \({\left | \psi \right \rangle _{1 {\ldots } n}} = \sum \nolimits _{{i_{1}}, \ldots ,{i_{n}} = 0}^{\left ({{d_{1}} - 1} \right ), {\ldots } ,\left ({{d_{n}} - 1} \right )} {{c_{{i_{1}}, \ldots ,{i_{n}}}}\left | {{i_{1}}, {\ldots } ,{i_{n}}} \right \rangle }\) with Hilbert space H 1⊗H 2⊗⋯⊗H n , where d i m(H i ) = d i , i = 1,2,⋯ , n. Here, \(\left | {{i_{1}}, {\ldots } ,{i_{n}}} \right \rangle \) represent the standard basis states, and \({c_{{i_{1}}, {\ldots } ,{i_{n}}}}\) are the coefficients which satisfy the normalization condition \(\sum\limits _{{i_{1}}, {\ldots } ,{i_{n}}} {{{\left | {{c_{{i_{1}}, {\ldots } ,{i_{n}}}}} \right |}^{2}}} = 1\). We associate with the state \({\left | \psi \right \rangle _{1...n}}\) a \(\left ({\prod\limits _{t = 1}^{l} {{d_{t}}} } \right ) \times \left ({\prod\limits _{t = l + 1}^{n} {{d_{t}}} } \right )\) coefficient matrix \({C_{1 {\ldots } l,\,\left ({l + 1} \right ) {\ldots } n}}\) whose entries are the coefficients \({c_{{i_{1}}, {\ldots } ,{i_{n}}}}\) arranged in ascending lexicographical order. It is listed as:

$$ \left( \begin{array}{ccc} {{c_{\underbrace {0 {\ldots} 0 }_{l},\underbrace {0 {\ldots} 0 }_{n - l}}}} & {\cdots} & {{c_{\underbrace {0 {\ldots} 0 }_{l},\underbrace {{d_{l}} - 1 {\ldots} {d_{n}} - 1}_{n - l}}}} \\ {{c_{\underbrace {0 {\ldots} 1 }_{l},\underbrace {0 {\ldots} 0 }_{n - l}}}} & {\cdots} & {{c_{\underbrace {0 {\ldots} 1 }_{l},\underbrace {{d_{l}} - 1 {\ldots} {d_{n}} - 1}_{n - l}}}} \\ {\vdots} & {\vdots} & {\vdots} \\ {{c_{\underbrace {{d_{1}} - 1\ldots{d_{l}} - 1}_{l},\underbrace {0{\ldots} 0}_{n - l}}}} &\cdots& {{c_{\underbrace {{d_{1}} - 1 {\ldots} {d_{l}} - 1}_{l},\underbrace {{d_{l}} - 1 {\ldots} {d_{n}} - 1}_{n - l}}}} \end{array} \right) , $$
(1)

similar to the coefficient matrix presented in Refs. [31, 32]. We split the n subsystems into two disjoint subsets \(\left \{ {1,2, {\ldots } ,l} \right \}\) and \(\left \{ {l + 1, {\ldots } ,n} \right \}\). In this paper, we will simply write \({C_{1 {\ldots } l,\,\left ({l + 1} \right ) {\ldots } n}}\) as C A . It means that C A represents a bipartition \(\{A|\bar {A}\}\) of {1,2,…, n}, where \(\bar {A}\) is the complement of A in {1,2,…, n}. Furthermore, there are 2n−2 subsets of n subsystems, but we only need to consider these independent 2n−1−1 coefficient matrices rather than all 2n−2 ones.

Now, we recall that the quantity

$$ {E_{A}}\left( {{C_{A}}\left( {{{\left| \psi \right\rangle }_{1...n}}} \right)} \right) = \sqrt {\sum\limits_{i < j} {{{\left( {2\left\| {{\chi_{i}}} \right\|\left\| {{\chi_{j}}} \right\| {\sin {\gamma_{ij}}} } \right)}^{2}}} } $$
(2)

vanishes if and only if the state \(\left | \psi \right \rangle _{1 {\ldots } n}\) is separable with respect to A and \(\bar {A}\). Where \({\chi _{i}}\left ({{\chi _{j}}} \right )\) represents the \(i\,th\left ({j\,th} \right )\) row vector of coefficient matrix C A , and \(\left \| {{\chi _{i}}} \right \|\) denotes the length of the vector χ i , γ i j is the angle between χ i and χ j . For convenience, the quantity in (2) will be simply written as E A . Next, we define the “average” value

$$ {E_{avg}}\left( {{{\left| \psi \right\rangle }_{1 {\ldots} n}}} \right) = \sqrt {\frac{1}{{{2^{n - 1}} - 1}}\sum\nolimits_{A} {{{\left( {{E_{A}}} \right)}^{2}}} } , $$
(3)

which is taken over all possible bipartition of {1,2,…, n}, as a multipartite entanglement measure of \(\left | \psi \right \rangle _{1 {\ldots } n}\). Similarly, we simply denote this average by E a v g .

In Ref. [30], we have proved that the quantities in (2) and (3) can be regarded as entanglement measures since they satisfy the following properties: (i) vanishing if and only if a state is (fully) separable; (ii) invariant under local unitary transformations; (iii) non-increasing under LOCC. Then, according to Schmidt decomposition, we have indicated that the quantity E A in (2) has another expression form [30]

$$ {E_{A}} = \sqrt {2\left( {1 - \sum\limits_{i} {{\lambda_{i}^{2}}} } \right)}, $$
(4)

where \(\sqrt {\lambda _{i}}\) is exactly the singular values of C A like the matrix (1), with the condition \(\sum \nolimits _{i} {{\lambda _{i}}} = 1\). At the same time, it is easy to know λ i is the eigenvalue of ρ A , where \(\rho _{A}=Tr_{\bar {A}}(|\psi \rangle \langle \psi |)\) is the reduced density matrix of subsystem A (Note that λ i is also the eigenvalue of \(\rho _{\bar {A}}\) with \(\rho _{\bar {A}}=Tr_{A}(|\psi \rangle \langle \psi |)\)). Besides, there is no doubt that \({\lambda _{i}^{2}}\) is the eigenvalue of \({\rho _{A}^{2}}\). At this point, in this paper the quantity in (2) can be represented as

$$ {E_{A}} = \sqrt {2\left( {1 - Tr\left( {{\rho_{A}^{2}}} \right)} \right)} . $$
(5)

In fact, (5) is the concurrence [13] when it reduces to two-qubit system. Consequently, we can reconstruct E a v g in (3) as

$$ {E_{avg}} = \sqrt {2\left( {1 - \frac{{\sum\nolimits_{A} {Tr\left( {{\rho_{A}^{2}}} \right)} }}{{{2^{n - 1}} - 1}}} \right)} . $$
(6)

We find that the computational complexity of the entanglement measure E a v g grows exponentially with respect to the number of qudits, due to it needs to take over all possible bipartition of n qudits. More importantly, it may be inappropriate to only evaluate the average bipartite entanglement of a n-qudit state. Thus it is necessary to improve the entanglement measure we proposed. For this aim, we represent it based on reduced density matrix as (5) and (6), to compare with some other similar entanglement measures.

3 Multipartite Entanglement Measures Based on Density Matrices

In this section, let us introduce some multipartite entanglement measures which are proposed based on density matrices. It is worth mentioning that most of them are proposed for n-qubit states, compared with our entanglement measure for arbitrary n-qudit states.

One of the first practical entanglement measures for n-qubit pure states is the MW measure [33, 34]. It is equivalent to the average of all the single-qubit linear entropies [29]

$$ Q\left( {\left| \psi \right\rangle } \right) = 2\left( {1 - \frac{1}{n}\sum\limits_{k = 1}^{n} {Tr{\rho_{k}^{2}}} } \right) , $$
(7)

where ρ k , k = 1,…, n, denotes the marginal density matrix describing the kth qubit of the system after tracing out the rest. It means that the quantity in (7) describes the average entanglement of every qubit of the system with the remaining (n−1) qubits. This MW measure is an entanglement monotone, namely, invariant under local unitary transformations and non-increasing under LOCC. It is proved to be useful in the study of several applications [29]. The quantity in (6) take over all independent 2n−1−1 bipartition of n qudits to measure the average entanglement of a n-qudit state. While the MW measure describes the average entanglement only with respect to n marginal density matrices for a n-qubit state. Sometimes, it is not adequate to quantify entanglement only based on the single-qubit linear entropies. Facchi et al [24] define a entanglement measure as a minimizer of the following quantity

$$ {\pi_{ME}} = {\left( {\begin{array}{*{20}{c}} n\\ {{n_{A}}} \end{array}} \right)^{- 1}}\sum\limits_{\left| A \right| = {n_{A}}} {{\pi_{A}}} , $$
(8)

where n A = [n/2], \(\pi _{A}=Tr_{A}{\rho _{A}^{2}}\) is the purity of an n A -qubit subsystem A. Note that π M E measures the average bipartite entanglement over all possible “balanced” bipartition, and it is related to degree of mixedness of the subsystem A associated with the bipartition \(\{A|\bar {A}\}\). Although they consider a perfect MMES must be completely mixed in its arbitrary subsystem, it only needs to minimize the averaged purity in (8) without purchasing the completely mixed state in every subsystem. They infer that a state realizing this minimization will be a MMES. It is worthy discussing whether it is appropriate to quantify entanglement only in terms of every “balanced” bipartition.

In Ref.[35], it introduces a family of entanglement measures R m with m = 2,3,…, n, to examine if the state is a product state of arbitrary m subsystems. Let \(P=\left \{ {{A_{i}}} \right \}_{i = 1}^{m}\) be a partition of {1,2,…, n}, and |A i | is the number of elements in the subset A i . The quantity

$$ {\eta_{{A_{i}}}}\left( \psi \right) = N\left( {\left| {{A_{i}}} \right|} \right)\left( {1 - Tr\left( {\rho_{{A_{i}}}^{2}} \right)} \right),\quad N\left( {\left| {{A_{i}}} \right|} \right) = \frac{{{2^{\left| {{A_{i}}} \right|}}}}{{{2^{\left| {{A_{i}}} \right|}} - 1}} $$
(9)

vanishes if and only if the state |ψ〉 is separable with respect to A i and \(\bar {A_{i}}\). This quantity is similar to our entanglement measure (5) except for a normalization factor N(|A i |). Next, they evaluate the average value for the partition P by the arithmetic mean \({\xi _{P}} = \frac {1}{m}\sum \nolimits _{i = 1}^{m} {{\eta _{{A_{i}}}}}\). Out of all possible partitions P of the total set, finally they evaluate the geometric mean of the quantities ξ P , namely,

$$ {R_{m}} = {\left( {\prod\limits_{d\left( p \right) = m} {{\xi_{P}}} } \right)^{{1 / {\vphantom {1 {S\left( {n,m} \right)}}} \kern-\nulldelimiterspace {S\left( {n,m} \right)}}}},\quad S\left( {n,m} \right) = \sum\limits_{k = 1}^{m} {\frac{{{{\left( { - 1} \right)}^{m - k}}{k^{n - 1}}}}{{\left( {k - 1} \right)!\left( {m - k} \right)!}}} $$
(10)

where d(P) is the number of subsets of the partition P, and S(n, m) represents the number of partitions P with d(P) = m. The quantities R m = 0 if and only if |ψ〉 is separable with respect to (at least) a specific partition P. In particular, for m = n the measure R m (R n ) is precisely the MW measure. Although this entanglement measure can examine the intermediate separability, clearly, it is not easy to calculate. Furthermore, it is worth recalling that R m is still proposed only for n-qubit system.

There is a multipartite entanglement measure for n-qudit state, which is satisfied all necessary conditions for being a entanglement measure, is called GME concurrence [36, 37],

$$ {C_{GME}}\left( {\left| \psi \right\rangle } \right) = \underset{{\gamma_{i}} \in \gamma}\min \sqrt {2\left[ {1 - Tr\left( {\rho_{{A_{{\gamma_{i}}}}}^{2}} \right)} \right]} , $$
(11)

where γ = {γ i } represents the set of all possible bipartition \(\{A_{i}|\bar {A_{i}}\}\) of {1,2,…, n}. Apparently, this GME concurrence is based on the well-known concurrence. Some lower bounds of this measure are obtained in experiment enabling quantification of multipartite entanglement in a broad variety of cases such as quantum computing or cryptography [36]. We can see that the GME concurrence is very similar to our entanglement measure (5). However, the GME concurrence only search for the minimization of those bipartite entanglement over all possible bipartition.

From the above discussion, it follows that every entanglement measure introduced in this section (including our entanglement measure in Section 2) has some connections with others. These analyzes will help us propose a new good entanglement measure.

4 A Multipartite Entanglement Measure for Arbitrary n-qudit States

Now, we will introduce a new entanglement measure for arbitrary n-qudit states.

Suppose a n-qudit pure state |ψ〉∈H 1⊗H 2⊗⋯⊗H n , with respective dimension d i , i = 1,2,⋯ , n. We consider a bipartition \(\{A|\bar {A}\}\) of the total set {1,2,…, n}, where A represents an non-empty subset of {1,2,…, n} and \(\bar {A}\) is the complement of A in the total set. Consequently, there exist a subsystem (it can also be denoted by A) according to the corresponding subset A. Let |A| denotes the number of elements in the subset A, thus the notation A also means a subsystem constituted by |A| qudits. We define an entanglement measure for n-qudit states as

$$ E_{t}(|\psi\rangle)= \underset{\left| A \right| = t}\min \sqrt {\frac{d}{{d - 1}}\left( {1 - Tr({\rho_{A}^{2}})} \right)}, \quad d = \prod\limits_{i \in A} {{d_{i}}} $$
(12)

with \(t=1,2,\cdots ,\displaystyle \lfloor {\frac {n}{2}}\rfloor \). Here, \(\displaystyle \lfloor {\frac {n}{2}}\rfloor \) is the integer part of \(\displaystyle \frac {n}{2}\) and \(\rho _{A}=Tr_{\bar {A}}(|\psi \rangle \langle \psi |)\) is the reduced density matrix of subsystem A. With a specific bipartition \(\{A|\bar {A}\}\) of n-qudit system, \(d=\prod\limits _{i \in A} {{d_{i}}}\) is the dimension of subsystem A. The normalization factor \(\displaystyle \frac {d}{d - 1}\) is chosen so that we have \(0\leq \sqrt {\displaystyle \frac {d}{{d - 1}}\left ({1 - Tr({\rho _{A}^{2}})} \right )} \leq 1\). Similar to some other entanglement measures, our quantity (12) is given by the degree of mixedness of the reduced density matrices associated with all possible bipartitions. The state |ψ〉 is separable with respect to A and \(\bar {A}\) if and only if \(\sqrt {\displaystyle \frac {d}{{d - 1}}\left ({1 - Tr({\rho _{A}^{2}})} \right )}=0\). And the reduced state \(\rho _{A} =\displaystyle \frac {1}{d}I\), where I denotes identity operator, is maximally entangled (maximally mixed) when \(\sqrt {\displaystyle \frac {d}{{d - 1}}\left ({1 - Tr({\rho _{A}^{2}})} \right )}=1\).

Note that when the number of elements in A is fixed, we take the minimum over all possible bipartition \(\{A|\bar {A}\}\) in (12). Then, we can obtain E t (|ψ〉) from t = 1 to \(\displaystyle \lfloor {\frac {n}{2}}\rfloor \). The quantity E t (|ψ〉)=0 means that there (at least) exist a particular bipartition so that the amount of entanglement between subsystems consisting of t qudits and the remaining n−t qubits vanishes. If we denote the state of the subsystem consisting of t qudits by |ϕ A 〉, and let the state of the remaining n−t qubits be \(|\varphi _{\bar {A}}\rangle \). Then we have the following property

$$ E_{t}(|\psi\rangle)=0 \Leftrightarrow |\psi\rangle=|\phi_{A}\rangle \otimes |\varphi_{\bar{A}}\rangle, \quad |A|=t $$
(13)

Namely, our entanglement measure (12) equals to zero if and only if the state is separable. In addition, if E t (|ψ〉) ≠ 0 for arbitrary \(1\leq t \leq \displaystyle \lfloor {\frac {n}{2}}\rfloor \), it indicates that there is no any form of separability in the state |ψ〉. That is, the state |ψ〉 is a genuinely n-partite entangled state. Theoretically, a state |ψ〉 can be conjectured as a maximally multipartite entangled state if every E t (|ψ〉) is equal to 1 for \(1\leq t \leq \displaystyle \lfloor {\frac {n}{2}}\rfloor \). Nevertheless, in many cases such states do not necessarily exist. To sum up, our entanglement measure is good at characterizing multipartite entanglement.

We have known, T r(ρ 2) is invariant under local unitary transformations for every reduced density matrix irrespective of the decomposition, and non-increasing under LOCC. In other word, it is an entanglement monotone. Therefore, the quantity E t (|ψ〉), with \(t=1,2,\cdots ,\displaystyle \lfloor {\frac {n}{2}}\rfloor \), can be qualified as an entanglement measure.

For n-partite mixed state ρ, the entanglement measure (12) can be generalized via a convex roof construction as

$$ {E_{t}}\left( \rho \right) = \underset{\left\{ {{p_{j}},\,\left| {{\psi_{j}}} \right\rangle } \right\}}{inf} \sum\limits_{j} {{p_{j}}{E_{t}}\left( {\left| {{\psi_{j}}} \right\rangle } \right)} , $$
(14)

where the infimum is taken over all possible decompositions \(\rho ={\sum }_{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|\). The case of multipartite entanglement for mixed states is not straightforward. Generally, we only consider pure states in this paper.

5 Applications and Examples

Let us apply our entanglement measure for some explicit examples.

Example 1

Two genuinely three-qubit entangled states, the GHZ and the W states are shown below, respectively

$$\begin{array}{@{}rcl@{}} |GHZ_{3}\rangle&=&\frac{1}{{\sqrt 2 }}\left( {\left| {000} \right\rangle + \left| {111} \right\rangle } \right), \\ |W_{3}\rangle&=&\frac{1}{{\sqrt 3 }}\left( {\left| {100} \right\rangle + \left| {010} \right\rangle + \left| {001} \right\rangle } \right) . \end{array} $$
(15)

It has been found that they are two inequivalent types of genuine tripartite entanglement according to stochastic local operations and classical communication(SLOCC) [38]. Here, we distinguish them by entanglement measures instead of SLOCC.

Since the states |G H Z 3〉 and |W 3〉 are three-qubit states, we only consider the case t = 1 in (12). That is, we calculate entanglement measure based on the three single-qubit marginal density matrices ρ 1, ρ 2, ρ 3 corresponding to three possible bipartition. One can have

$$ \rho_{A}(|GHZ_{3}\rangle)= \frac{1}{2}\left( {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & 1 \end{array}} \right), \quad \rho_{A}(|W_{3}\rangle)= \frac{1}{3}\left( {\begin{array}{*{20}{c}} 2 & 0 \\ 0 & 1 \end{array}} \right) $$
(16)

with A = 1,2,3. By (12) we obtain E 1(|G H Z 3〉)=1 and \(E_{1}(|W_{3}\rangle )=\displaystyle \frac {{2\sqrt 2 }}{3}\). We can see that |G H Z 3〉 is more entangled than |W 3〉 in term of our entanglement measure. According to the linear entropy, von Neumann entropy, Renyi entropy and the negativity denoted by E L , E V N , E R , E N respectively [29], similarly |G H Z 3〉 exhibits more entanglement than |W 3〉. In particular, the state |G H Z 3〉 is a maximally three-qubit entangled state.

Example 2

There are two notable four-qubit entangled state as follows [28]

$$\begin{array}{@{}rcl@{}} |\psi_{HS}\rangle&=&\frac{1}{{\sqrt 6 }}\left( {\left| {0011} \right\rangle + \left| {1100} \right\rangle + \omega \left( {\left| {1010} \right\rangle + \left| {0101} \right\rangle } \right)} \right. \\ & &\left. { + {\omega^{2}}\left( {\left| {1001} \right\rangle + \left| {0110} \right\rangle } \right)} \right), \\ |\psi_{4}\rangle&=&\frac{1}{2}\left( {\left| {0000} \right\rangle + \left| { + 011} \right\rangle + \left| {1101} \right\rangle + \left| { - 110} \right\rangle } \right) \end{array} $$
(17)

where \(\left | + \right \rangle = \displaystyle \frac {1}{{\sqrt 2 }}\left ({\left | 0 \right \rangle + \left | 1 \right \rangle } \right )\,,\,\left | 1 \right \rangle = \displaystyle \frac {1}{{\sqrt 2 }}\left ({\left | 0 \right \rangle - \left | 1 \right \rangle } \right )\), and \(\omega = \displaystyle {e^{\frac {{2\pi i}}{3}}}\). It is inferred that they are highly entangled four-qubit state, and |ψ H S 〉 is conjectured to be maximally entangled [28].

For four-qubit states, we need to consider the cases t = 1 and t = 2 in (12). We calculate our entanglement measure based on 24−1−1=7 reduced density matrices ρ 1, ρ 2, ρ 3, ρ 4 and ρ 12, ρ 13, ρ 14 of the states |ψ H S 〉,|ψ 4〉. Thus according to (12) we can obtain

$$ E_{1}(|\psi_{HS}\rangle)= E_{1}(|\psi_{4}\rangle)= min\{1,1,1,1\}=1 $$
(18)

and

$$\begin{array}{@{}rcl@{}} E_{2}(|\psi_{HS}\rangle)&=& min \{\sqrt {\frac{8}{9}},\sqrt {\frac{8}{9}},\sqrt {\frac{8}{9}}\}=\frac{{2\sqrt 2 }}{3}, \\ E_{2}(|\psi_{4}\rangle)&=& min \{\sqrt{\frac{5}{6}},1,\sqrt{\frac{5}{6}}\}=\sqrt{\frac{5}{6}} \end{array} $$
(19)

In (18) it shows that the single-qubit reduced states of the states |ψ H S 〉 and |ψ 4〉 are all maximally mixed. Although the both states are highly entangled four-qubit state, the state |ψ H S 〉 exhibit higher bipartition entanglement than |ψ 4〉 in (19). According to the NPT measure and von Neumann entropy [28], one has a similar conclusion that the state |ψ 4〉 is less entangled than |ψ H S 〉. However, the state |ψ H S 〉 is not a perfect maximally entangled state using our entanglement measure.

Example 3

A five-qubit and a six-qubit highly entangled states are presented as follows [28, 29]

$$\left| \psi_{5} \right\rangle = \frac{1}{2}\left( \left| {000} \right\rangle \left| {{{\Phi}_ - }} \right\rangle + \left| {010} \right\rangle \left| {{{\Psi}_ - }} \right\rangle + \left| {100} \right\rangle \left| {{{\Phi}_ + }} \right\rangle + \left| {111} \right\rangle \left| {{{\Psi}_ + }} \right\rangle \right) , $$
$$\begin{array}{@{}rcl@{}} |\psi_{6}\rangle = \frac{1}{{4\sqrt 2 }}&(&|000000\rangle + |111111\rangle + |000011\rangle + |111100\rangle + |000101\rangle + |111010\rangle \\ &&+|000110\rangle +|111001\rangle +|001001\rangle +|110110\rangle +|001111\rangle +|110000\rangle \\ &&+|010001\rangle +|101110\rangle +|010010\rangle +|101101\rangle +|011000\rangle +|100111\rangle \\ &&+|011101\rangle +|100010\rangle +|001010\rangle +|110101\rangle +|001100\rangle +|110011\rangle \\ &&+|010100\rangle +|101011\rangle +|010111\rangle +|101000\rangle +|011011\rangle +|100100\rangle \\ &&+|011110\rangle +|100001\rangle) \end{array} $$
(20)

where \(\left |{{{\Psi }_ \pm }}\right \rangle =\displaystyle \frac {1}{{\sqrt 2 }}\left ({\left | {00} \right \rangle \pm \left | {11} \right \rangle } \right )\) and \(\left |{{{\Phi }_ \pm }}\right \rangle = \displaystyle \frac {1}{{\sqrt 2 }}\left ({\left | {01} \right \rangle \pm \left |{10}\right \rangle }\right )\).

For the five-qubit state |ψ 5〉, there are 5 single-qubit marginal density matrices ρ 1,…, ρ 5 and 10 two-qubit reduced density matrices ρ 12, ρ 13,…, ρ 45, need to be taken into consideration in our entanglement measure (12). One can calculate

$$\begin{array}{@{}rcl@{}} E_{1}(|\psi_{5}\rangle)&=& min\{1,1,\frac{{\sqrt 3 }}{2},1,1\}= \frac{{\sqrt 3 }}{2} \\ E_{2}(|\psi_{5}\rangle)&=& min\{1,\sqrt{\frac{5}{6}},\sqrt{\frac{{11}}{{12}}},\sqrt{\frac{{11}}{{12}}},\sqrt{\frac{5}{6}}, \sqrt{\frac{{11}}{{12}}},\sqrt{\frac{{11}}{{12}}},\sqrt{\frac{{11}}{{12}}},\sqrt{\frac{{11}}{{12}}},1\}\\ &=& \sqrt{\frac{5}{6}} \end{array} $$
(21)

It is known that the state |ψ 5〉 is a five-qubit highly entangled state with respect to the NPT measure and von Neumann entropy [28]. Using our entanglement measure it indicates that |ψ 5〉 has completely mixed single-qubit marginal density matrices except for the the reduced state ρ 3. Furthermore, it exhibits relatively high bipartition entanglement. Overall, the state |ψ 5〉 is a five-qubit highly entangled state.

It is required to explore 26−1−1=31 reduced states of the six-qubit state |ψ 6〉 in our entanglement measure (12). By calculating, we find that the state |ψ 6〉 has completely mixed single-qubit reduced states,

$$ E_{1}(|\psi_{6}\rangle)= min\{1,1,1,1,1,1\}=1 . $$
(22)

In addition, all of 15 two-qubit reduced states of the state |ψ 6〉 exhibit same amount of entanglement,

$$ E_{2}(|\psi_{6}\rangle)= min\{\sqrt {\frac{2}{3}},\sqrt {\frac{2}{3}},\ldots,\sqrt {\frac{2}{3}}\}=\sqrt {\frac{2}{3}} . $$
(23)

Similarly, 10 independent three-qubit reduced states of the state |ψ 6〉 exhibit same amount of entanglement,

$$ E_{3}(|\psi_{6}\rangle)= min\{\sqrt {\frac{4}{7}},\sqrt {\frac{4}{7}},\ldots,\sqrt {\frac{4}{7}}\}=\sqrt {\frac{4}{7}} . $$
(24)

It indicates that all of the one, two, three-qubit reduced states of |ψ 6〉 exhibit same amount of entanglement, respectively. Originally the state |ψ 6〉 is introduced for exhibiting all the reduce density matrices for states of one, two, three qubits completely mixed [29]. However, it should be stressed that maximal entanglement is relative to the particular measure of entanglement chosen. We infer that the state |ψ 6〉 is not a perfect maximally entangled state according to our entanglement measure.

Example 4

The n-partite and m-dimensional GHZ state is defined as [39]

$$ {\left| {GHZ}_{n,m} \right\rangle} = \frac{1}{{\sqrt m }}\sum\limits_{i = 0}^{m - 1} {\left| {\underbrace {ii \cdots i}_{n}} \right\rangle } . $$
(25)

It can be calculated that all the reduced density matrices have the following form

$$ \rho = \frac{1}{m} \left( {\begin{array}{*{20}{c}} 1&0& {\cdots} &0&0\\ 0& {\ddots} & {\cdots} &0&0\\ {\vdots} & {\vdots} &1& {\vdots} & {\vdots} \\ 0&0& {\cdots} & {\ddots} &0\\ 0&0& {\cdots} &0&1 \end{array}} \right) , $$
(26)

where every reduced density matrix has m nonzero diagonal element, and the rest are all zero. Then, we can obtain the amount of entanglement according to (12)

$$ E_{t}(|GHZ_{n,m}\rangle)= \sqrt {\frac{{{m^{t}}}}{{{m^{t}} - 1}} \cdot \frac{{m - 1}}{m}} $$
(27)

with \(t=1,2,\cdots ,\displaystyle \lfloor {\frac {n}{2}}\rfloor \). We have known that the state |G H Z n, m 〉 is a maximally entangled state in case n = 3, m = 2 as mentioned in Example 1. Anyway, the state |G H Z n, m 〉 always has completely mixed single-qubit marginal density matrices, i.e. E 1(|G H Z n, m 〉)=1. Besides, it is worth noted that all the reduced states share the same amount of entanglement when t is fixed. Finally it is inferred that the state |G H Z n, m 〉 is an approximate maximally entangled state when m tends to infinity.

6 Conclusion

In this paper, we propose a multipartite entanglement measure for arbitrary pure states, which is given by the degree of mixedness of the reduced density matrices associated with all possible bipartitions. Our quantity equals to zero iff a state is separable. Otherwise, there is no any form of separability in the state. In addition, we demonstrate the situation of maximally multipartite entangled state in terms of our entanglement measure. However, sometimes such maximally entangled states do not necessarily exist. It is shown that our quantity can be qualified as an entanglement measure due to it is an entanglement monotone.

We apply our entanglement measure for some explicit examples. For three-qubit state, the state |G H Z 3〉 is more entangled than the state |W 3〉 using our entanglement measure. It is coincide with some known results. In particular, the state |G H Z 3〉 is a maximally three-qubit entangled state. In the case of states of four qubits, we show that the state |ψ H S 〉 exhibits higher bipartition entanglement than the state |ψ 4〉. Both them are four-qubit highly entangled state, however, the state |ψ H S 〉 could not be presumed to be maximally entangled state according to our entanglement measure. Besides that, although the states |ψ 5〉, |ψ 6〉 and the n-partite, m-dimensional GHZ state |G H Z n, m 〉 have been shown to be relatively high entangled states, they are still not perfect maximally entangled state with respect to our entanglement measure. In conclusion, our entanglement measure is good at characterizing multipartite entanglement. And it is practical and convenient for computation.