Multipartite entanglement indicators based on monogamy relations of n-qubit symmetric states

Constructed from Bai-Xu-Wang-class monogamy relations, multipartite entanglement indicators can detect the entanglement not stored in pairs of the focus particle and the other subset of particles. We investigate the k-partite entanglement indicators related to the αth power of entanglement of formation (αEoF) for k ≤ n, αϵ and n-qubit symmetric states. We then show that (1) The indicator based on αEoF is a monotonically increasing function of k. (2) When n is large enough, the indicator based on αEoF is a monotonically decreasing function of α, and then the n-partite indicator based on works best. However, the indicator based on 2 EoF works better when n is small enough.

Quantum correlations that comprise and go beyond entanglement are not monogamous. Only entanglement can be strictly monogamous 1 , 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 systems 2 . So these monogamy relations can be used to characterize the entanglement structure in multipartite systems 3 , 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 particles 4 .
For the squared concurrence, the indicator named three-tangle 3 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-tangle 5 . 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 proposed 4,6,7 . In this paper, we will study which multipartite entanglement indicator for EoF works better. By "work better" we mean that is larger than the other 8 .
We resolve the above problem in the following ways. Firstly, we prove that the αth power of EoF (αEoF, α ≥ 2 ) obeys a set of hierarchy k-partite ( ∈ , ) k n [3 ] monogamy relations of Eq. (10) in an arbitrary n-qubit state ρ . Here, the k-partition means the partition A 1 , , A k−1 and  A A k n . 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 α ∈ , ) [ 2 2 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 ∈ , k n [3 ] 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.
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 A 1 and any other k − 1 party (i.e., A 2 , , A k−1 and  A A k n ) 4 . However, there exists a special kind of entangled state 10 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 Fei 7 proved that αEoF obeys the following monogamy relation in n-qubit states, can be calculated via quantum discord 13,14 , the entan- from Eqs (3)(4)(5) can be obtained and can characterize multipartite entangled states in some n-qubit states 4,6,7 . 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 ρ AB , an analytical formula was given by Wootters 15 as follows 4 is the concurrence with the decreasing nonnegative λ i being the eigenvalues of the matrix ρ σ σ ρ σ σ 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 Fei 7 proved that f(x) satisfies the following relation where α ≥ 2 , x and ∈ , y [0 1]. They also proved that EoF obeys the following relation is equivalent to a two-qubit state under the Schmidt decomposition 16 From Eqs (1) and (6-9) for n-qubit systems, we can easily obtain that the following hierarchy k-partite monogamy relation holds.
Theorem 1 For any n-qubit state ρ , EoF satisfies the following monogamy relation } and α ≥ 2 . The αEoF satisfies the hierarchy monogamy inequality (10) for any α ≥ 2 , while the αth power of concurrence satisfies hierarchy monogamy inequalities for any α ≥ 2 9,12 . 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 α ( ∈ , ) [ 2 2] , we can define a hierarchy entanglement indicator based on the corresponding monogamy relation in Eq. (10) as follows It can be used to detect the entanglement for the k-partite case of an n-qubit system 6 not stored in pairs of A 1 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 ρ , the hierarchy entanglement indicator satisfies and it is a monotonically increasing function of k, where = , , ,  k n {3 4 } and α ∈ , Combining with Eq. (11), we have where the inequality holds because of Eq. (16). Therefore, the entanglement indicator τ ρ ( ) 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 entangle- can not work. However, we can choose an appropriate indicator , the entanglement indicator obeys the following relation . For any n, we have the following results (1) When c = 0, α ( , ) g n is a monotonically decreasing function of α. When c > 0 and b = 1, α ( , ) g n 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) When c = 0, we have α n is a monotonically decreasing function of α. The monotonically decreasing property of α ( , ) g n is satisfied if and only if the first-order partial derivative α α ∂ ( , )/∂ ≤ g n 0, which is equivalent to Eq. (20). Furthermore, the monotonically increasing property of α ( , ) g n is satisfied if and only if the first-order partial derivative α α ∂ ( , )/∂ ≥ g n 0, which is equivalent to Eq. (21). From Theorem 3, we can obtain that the necessary and sufficient condition for the unit indicator is . For any n-qubit symmetrical state, we can numerically compute the corresponding bounds to determine which is better, 2 EoF indicator or the 2 EoF, as follows: After some deduction, we numerically obtain two bounds N 1 and N 2 with Eqs (20) and (21). When n ≥ N 1 , the 2 EoF indicator is better than the 2 EoF indicator which comes from Eq. (20). The 2 EoF indicator is better than the 2 EoF indicator when n ≤ N 2 , 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 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,  where ( ) = ( − )/ p n n n 4 1 2 and ( ) = / q n n 4 2 . This set of are positive since the αEoF is monogamous as shown in Eqs (5) and (10).
In order to study the properties of α ( , ) g n , we firstly prove the function M(n), with 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) when n ≤ N 2 . Then we complete the proof that α ( , ) g n 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 α ( , ) g n is a monotonically decreasing function of n when α ∈ , [ 2 2] and n ≥ 10. How to exactly prove the result is an open problem.
These results still hold for symmetric n-qubit mixed states as shown in the next part. where = ( + ) /   GHZ 00 00 11 11 2 and ∈ ( , ) p 0 1 . For n = 3, Lohmayer et al. 5 found that, when ∈ ( . , . ) p 0 292 0 627 , it is entangled but without two-qubit concurrence and three-tangle. It is still an unsolved problem 4 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 ρ ( ) are key steps. Any reduced two-qubit states of ρ . Then, according to Eq. (6), we obtain . In the following, we will calculate ρ ( ) . Through introducing a system B which has the same state space as the composite system can be purified as According to the Koashi-Winter formula 4,18 , the bipartite multiqubit EoF can be calculated by the purified state Ψ  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 ρ ( ) D B A B 1 is σ z when n ≥ 6 and ∈ ( , ) p p p L R . Then, after some deduction, we get From Eqs (19), (31) and (33), the indicator has the form has been shown in Fig. 2 for α = 2 and α = 2 respectively. Furthermore, have some properties as follows.
(1) For any α, α ( , ) g n is a monotonically decreasing function of n. The monotonically decreasing property of α ( , ) g n holds because the first-order partial derivative satisfies ( ) (2) Combining with Theorem 3 and Eqs (33)  .
From the above two properties, we know that the nonzero τ ρ ( ) can indicate the existence of the n-qubit entanglement. These results can also be understood as the fact that τ ρ ( ) 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 2 EoF indicator work best when n is large enough, while the 2 EoF indicator works better than the 2 EoF indicator for smaller n. (20)  in Eqs (20) and (21), we analyze the sign of the first-order derivative dM(n)/dn.

The monotonic property of the function in Eqs
After some deduction, we can obtain  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.