Introduction

Quantum correlations that comprise and go beyond entanglement are not monogamous. Only entanglement can be strictly monogamous1, that is, they obey strong constraints on how they can be shared among multipartite systems. This is one of the most important properties for multipartite quantum systems2. So these monogamy relations can be used to characterize the entanglement structure in multipartite systems3 and concretely the difference between the left- and right-hand side of them can be defined as indicators to detect multipartite entanglement not stored in pairs of the focus particle (e.g., the first particle) and the other subset of particles4.

For the squared concurrence, the indicator named three-tangle3 can be used to detect genuine multipartite entanglement (which are entangled states being not decomposable into convex combinations of states separable across any partition) in three-qubit pure states. However, for three-qubit mixed states, there exist some entangled states that have neither two-qubit concurrence nor three-tangle5. To reveal this critical entanglement structure, some multipartite entanglement indicators based on Bai-Xu-Wang-class monogamy relations for the entanglement of formation (EoF) have been proposed4,6,7. In this paper, we will study which multipartite entanglement indicator for EoF works better. By “work better” we mean that is larger than the other8.

We resolve the above problem in the following ways. Firstly, we prove that the αth power of EoF (αEoF, ) obeys a set of hierarchy k-partite monogamy relations of Eq. (10) in an arbitrary n-qubit state . Here, the k-partition means the partition A1, , Ak−1 and . Based on these monogamy relations, a set of new multipartite entanglement indicators are presented correspondingly, which can work better than the 2 EoF-based indicators in n-qubit symmetric states. However, we find that the 2 EoF-based indicator can work better than the αEoF-based indicators for when n is small enough (e.g., n ≤ 9).

Results

This section is organized as follows. In the first subsection, we review the monogamy relations for 2 EoF in n-qubit systems. We then prove in the second subsection that the αEoF obeys hierarchy k-partite monogamy relations for and any n-qubit states. In the third subsection, we construct the entanglement indicators on n-qubit symmetric states and show their monotonic properties. Two examples are given in the forth subsection to verify these results.

Review of monogamy relations for EoF

Coffman, Kundu and Wootters3 proved the first monogamy relation for the squared concurrence in three-qubit states. Then, Osborne and Verstraete9 proved a set of hierarchy k-partite monogamy relations for the squared concurrence in n-qubit states , which have the form

where A1 is the focus qubit, , is the concurrence of in the bipartition and is a k-partite n-qubit state.

Based on these Osborne-Verstraete-class hierarchical monogamy relations in Eq. (1), a set of multipartite entanglement indicators can be constructed as follows

where the entanglement measure is the squared concurrence. These indicators can detect the entanglement not stored in pairs of A1 and any other k − 1 party (i.e., A2, , Ak−1 and )4. However, there exists a special kind of entangled state10 which has zero entanglement indicator. Moreover, the calculation of multiqubit concurrence is extremely hard due to the convex roof extension. Therefore, it is natural to ask whether other monogamy relations beyond the squared concurrence exist.

Recently, Bai et al.4 and Oliveira et al.11 respectively proved that 2 EoF is monogamous in n-qubit states, as follows

Moreover, Bai et al.6 exactly showed that there are a set of hierarchy k-partite monogamy relations for 2 EoF in an arbitrary n-qubit states, which obey the relation

Generally, Zhu and Fei7 proved that αEoF obeys the following monogamy relation in n-qubit states,

where . (In fact, Eq. (5) obviously satisfies for α > 2 which can be obtained from Eq. (4) and ref. 12.)

Because some bipartite multiqubit EoF of can be calculated via quantum discord13,14, the entanglement indicator from Eqs (3, 4, 5) can be obtained and can characterize multipartite entangled states in some n-qubit states4,6,7. In these entanglement indicators, how to choose a better indicator to detect that there exists multipartite entanglement is a problem. In the following subsections, we will try to resolve the problem.

Hierarchy k-partite monogamy relations for αEoF

In this subsection, we firstly summary of some existing conclusions and then get the hierarchy k-partite monogamy relations for αEoF.

As we know, EoF is a well defined measure of entanglement for bipartite states. For any two-qubit state ρAB, an analytical formula was given by Wootters15 as follows

where is the concurrence with the decreasing nonnegative λi being the eigenvalues of the matrix . Here, and is the binary Shannon entropy. Recently, Bai et al.6 proved that f(x) is a monotonic and concave function of x. Moreover, Zhu and Fei7 proved that f(x) satisfies the following relation

where , x and . They also proved that EoF obeys the following relation

for the bipartite quantum state in systems. Because a pure state is equivalent to a two-qubit state under the Schmidt decomposition16, we have

From Eqs (1) and (6, 7, 8, 9) for n-qubit systems, we can easily obtain that the following hierarchy k-partite monogamy relation holds.

Theorem 1 For any n-qubit state , EoF satisfies the following monogamy relation

where and .

The αEoF satisfies the hierarchy monogamy inequality (10) for any , while the αth power of concurrence satisfies hierarchy monogamy inequalities for any α ≥ 29,12. This phenomenon shows a difference between the two kinds of entanglement measures. On the other hand, the inequality (10) is a generalization of Eq. (5) in ref. 6 and Eq. (19) in ref. 7. More specifically, Eq. (10) equals to Eq. (4) when α = 2 and is the same as Eq. (5) when k = n.

Properties of hierarchy entanglement indicators

For any n-qubit state and αEoF , we can define a hierarchy entanglement indicator based on the corresponding monogamy relation in Eq. (10) as follows

where

It can be used to detect the entanglement for the k-partite case of an n-qubit system6 not stored in pairs of A1 and any other k − 1 party.

Here it should be noted that, different from the hierarchy entanglement indicator of the concurrence, the indicator of EoF depends on which qubit is chosen to be the focus qubit. Fortunately, the indicators of the concurrence and EoF are all focus-independent in symmetric quantum systems. In the following, we give some properties about the indicators of EoF only for n-qubit symmetric states.

Theorem 2 For any n-qubit symmetric state , the hierarchy entanglement indicator satisfies

and it is a monotonically increasing function of k, where and .

Proof. When is a symmetric state, it is permutation invariant. Then, , and i ≠ j, we have and

Combining with Eq. (11), we have

Moreover, according to Eq. (5), we have

Then we can derive

where the inequality holds because of Eq. (16). Therefore, the entanglement indicator is a monotonically increasing function of k.

In symmetrical quantum systems, the k-partite n-qubit monogamy relations of αEoF in Eq. (10) can be a monogamy equality (e.g., the corresponding results in the next subsection) and thus the corresponding entanglement indicator can not work. However, we can choose an appropriate indicator

to represent a better entanglement indicator which comes from the following result.

Theorem 3 For any n-qubit symmetric state , the entanglement indicator obeys the following relation

where , and . For any n, we have the following results

  1. 1

    When c = 0, is a monotonically decreasing function of α.

  2. 2

    When c > 0 and b < 1, is a monotonically decreasing function of α if and only if

and is a monotonically increasing function of α if and only if

When c > 0 and b = 1, is also a monotonically increasing function of α.

Proof. From Eqs (10), (12) and (15), we have

According to the definition of b and c and the monogamy inequality (5), we get 0 ≤ c < b ≤ 1.

For any n, we will analytically prove the two necessary and sufficient conditions.

  1. 1

    When c = 0, we have . Because 0 < b ≤ 1, is a monotonically decreasing function of α.

  2. 2

    When , we have

The monotonically decreasing property of is satisfied if and only if the first-order partial derivative , which is equivalent to Eq. (20).

Furthermore, the monotonically increasing property of is satisfied if and only if the first-order partial derivative , which is equivalent to Eq. (21).

From Theorem 3, we can obtain that the necessary and sufficient condition for the unit indicator is and . For any n-qubit symmetrical state, we can numerically compute the corresponding bounds to determine which is better, indicator or the 2 EoF, as follows:

After some deduction, we numerically obtain two bounds N1 and N2 with Eqs (20) and (21). When n ≥ N1, the indicator is better than the 2 EoF indicator which comes from Eq. (20). The 2 EoF indicator is better than the indicator when n ≤ N2, which comes from Eq. (21).

These results can be verified via two n-qubit symmetrical states in the next subsection.

Analytical examples

We will investigate the above results on permutationally invariant states, which are the W state, the superposition of the W state and the Greenberger-Horne-Zeilinger (GHZ) state of n qubits respectively.

For the W state

In this part, we analyze the n-qubit W state which has the form

For this quantum state, the n-partite n-qubit monogamy relations of αth power of concurrence as shown in ref. 7 are saturated and thus these concurrence-based entanglement indicators can not work. However, we will show that the αEoF-based indicator can be used to represent the entanglement in the n-partite n-qubit systems.

Using the symmetry of qubit permutations in the W state, and  17, we have

where and . This set of are positive since the αEoF is monogamous as shown in Eqs (5) and (10).

In order to study the properties of , we firstly prove the function M(n), with

in Eqs (20) and (21), is a monotonically decreasing function of n. The details for illustrating the monotonic property are presented in Methods.

Let

After some deduction, we can derive

when . Thus, combining with the monotonically decreasing property of M(n), we prove that α ≥ M(n) when n ≥ 77, while α ≤ M(n) when n ≤ 76. When α = 2, we get

which means α ≥ M(n) when n ≥ 10, while α ≤ M(n) when n ≤ 9. Combining the above two inequations with Eqs (20) and (21), we obtain the two bounds and . And, we know that when n ≥ N1 and when n ≤ N2. Then we complete the proof that obeys these properties.

In Fig. 1, we plot these indicators as functions of n and then these properties can be verified from the figure. From the Fig. 1, we numerically find that is a monotonically decreasing function of n when and n ≥ 10. How to exactly prove the result is an open problem.

Figure 1
figure 1

The multipartite entanglement indicators for the W state as functions of n, where in (a) and in (b).

Full size image

These results still hold for symmetric n-qubit mixed states as shown in the next part.

For the superpositions of the GHZ state and the W state

When an n-qubit mixed state is a superpositions of the GHZ state and the W state, it has the form

where and . For n = 3, Lohmayer et al.5 found that, when , it is entangled but without two-qubit concurrence and three-tangle. It is still an unsolved problem4 of how to characterize the entanglement structure in this kind of states for large n.

In Eq. (18), the n-partite entanglement indicators have the forms

Then, the calculations of and are key steps.

Any reduced two-qubit states of has the same form

Using the effective method for calculating concurrence in ref. 15 and after some calculations, we have

where n ≥ 6 and . Then, according to Eq. (6), we obtain .

In the following, we will calculate . Through introducing a system B which has the same state space as the composite system , can be purified as

According to the Koashi-Winter formula4,18, the bipartite multiqubit EoF can be calculated by the purified state , with ,

where is the quantum conditional von Neumann entropy and the quantum discord is defined as13

with the minimum running over all the positive operator-valued measures on the subsystem B. The details for proving Eq. (35) are presented in Methods. Chen et al.19 presented an effective method for choosing an optimal measurement over B and then calculating the quantum discord of two-qubit X states, which can be used to quantify the multipartite entanglement indicator in Eq. (19). After some analysis, we can obtain the optimal measurement for the quantum discord is σz when n ≥ 6 and . Then, after some deduction, we get

From Eqs (19), (31) and (33), the indicator has the form

The distribution of has been shown in Fig. 2 for and α = 2 respectively. Furthermore, and have some properties as follows.

Figure 2
figure 2

The multipartite entanglement indicators for the superposition state as functions of n and p, where , α = 2 and respectively.

Full size image

1. For any α, is a monotonically decreasing function of n. The monotonically decreasing property of holds because the first-order partial derivative satisfies

2. Combining with Theorem 3 and Eqs (33) and (38), we have .

From the above two properties, we know that the nonzero can indicate the existence of the n-qubit entanglement. These results can also be understood as the fact that can detect as many as possible n-qubit entangled states for large n.

Conclusion

Entanglement monogamy is a fundamental property of multipartite entangled states. Based on our established monogamy relations Eq. (10), we obtain a set of useful tools for characterizing the multipartite entanglement not stored in pairs of the focus particle and the other subset of particles, which overcome some flaws of the concurrence. For any n-qubit symmetric state, we prove that the indicator work best when n is large enough, while the 2 EoF indicator works better than the indicator for smaller n.

Methods

The monotonic property of the function in Eqs (20) and (21)

In order to determine the monotonic property of M(n), with

in Eqs (20) and (21), we analyze the sign of the first-order derivative dM(n)/dn.

After some deduction, we can obtain

Then, dM(n)/dn < 0 when

and

Eq. (42) holds if and only if

i.e.,

The inequality (45) holds because is a concave function of n with .

Similarly, we have Eq. (43) holds when

where

and then . From ref. 9, we easily get that dt(n)/dn < 0 where .

In the following, we will prove Eq. (46). Let , where . Using the definition of the partial derivative, it is not different to verify that , , and are all continuous functions. Combining with the exchange order theorem of two second-order mixed partial derivative, we have

According to Eq. (47), we get that F(t) is monotonic and concave as a function of t.

Combining with Eq. (19), we have

Here, the first inequality holds because f is a concave function of n and the monotonically increasing property of F(t) in Eq. (48). The second inequality is satisfied because F(t) is a monotonically increasing function in Eq. (48) and ln x is a concave function of x. And the last inequality holds because F(t) is a concave function as proved in Eq. (48).

Then, we complete the proof that M(n) is a monotonically decreasing function of n.

Proof of the Eq. (35) in the Main Text

Purification can be done for any state , because we can introduce a system B which has the same state space as system and define a pure state20 for the combined system

From ref. 21, we know

Combining with , we can find that Eq. (35) is just Eq. (2) in ref. 17. More specifically,

Then, we complete the proof of the Eq. (35) in the Main Text.

Additional Information

How to cite this article: Liu, F. et al. Multipartite entanglement indicators based on monogamy relations of n-qubit symmetric states. Sci. Rep. 6, 20302; doi: 10.1038/srep20302 (2016).