Parameterized Multipartite Entanglement and Genuine Entanglement Measures Based on q-Concurrence

We study genuine multipartite entanglement (GME) and multipartite k-entanglement based on q-concurrence. Well-defined parameterized GME measures and measures of multipartite k-entanglement are presented for arbitrary dimensional n-partite quantum systems. Our GME measures show that the GHZ state is more entangled than the W state. Moreover, our measures are shown to be inequivalent to the existing measures according to entanglement ordering. Detailed examples show that our measures characterize the multipartite entanglement finer than some existing measures, in the sense that our measures identify the difference of two different states while the latter fail.

Several methods have been developed for quantifying the entanglement of quantum states.For multipartite quantum system, Ma et al. introduced a genuine multipartite entanglement measure, known as the genuine multipartite concurrence (GMC), in which the entanglement of any pure state is quantified by selecting the minimum bipartite concurrence over all possible bipartite splits [9].A new GME measure, the geometric mean of bipartite concurrence (GBC), was introduced in [10].Although it does not involve the minimization in calculating the entanglement of pure states, it can only distinguish between genuine entangled states and nongenuine entangled states.For bipartite states, a series of entanglement measures have been proposed, such as entanglement distillation [11,12], entanglement formation [13], negativity [14], and concurrence [15,16].In [17], Hong et al. proposed a measure of entanglement called k (2 ≤ k ≤ n) multipartite entanglement (k-ME) concurrence of any n-partite states, which satisfies the key properties of a well-defined measure of entanglement, such as strictly greater than zero for all k-nonseparable states, vanishing on k-separable states, invariance under local unitary transformations and convexity.Moreover, it satisfies the entanglement monotonicity, namely, the entanglement does not increase under local operations and classical communication (LOCC).Recently, a class of entanglement measures called k geometric mean (k-GM) concurrence has been presented based on the geometric mean of the entanglement associated with k-partitions of n-partite quantum systems [18].Inspired by general Tsallis entropy [19], by using q-concurrence, the authors in [20] provided the parameterized entanglement measures as the generalizations of the k-ME concurrence.
Given that the multipartite entanglement plays a significant role in quantum information processing, the characterization and quantification of multipartite entanglement have been extensively investigated.Nevertheless, due to the extremely complex structure of entanglement in multipartite quantum states, the studies on GME and multipartite kentanglement are still far from being satisfied.By taking into account that the q-concurrence is nonincreasing under local operations and classical communication, and vanishes for biseparable states, we present both GME measures and k-entanglement measures based on q-concurrence.
In this paper, we construct GME measures and k-entanglement measures for multipartite quantum systems in terms of q-concurrence.The paper is organized as follows.In Section 2, we review some fundamental concepts and give well-defined parameterized GME measures for multipartite quantum systems.By detailed examples, we show that our measures are more efficient than the existing ones in detecting the genuine multipartite entanglement.In Section 3, we present parameterized k-entanglement measures for arbitrary dimensional n-partite systems by using q-concurrence.Through detailed examples, we demonstrate that our measures have different state ordering from other ones.Conclusions are given in Section 4.

Parameterized GME Measures for Multipartite Pure States
We first focus on genuine multipartite entanglement.A multipartite pure state is genuinely multipartite entangled if it is not biseparable with respect to any bipartition.A well-defined GME measure should satisfy the following conditions: (a) For all product and biseparable states, the measure must be zero.(b) It is strictly positive for all non-biseparable states.(c) It is nonincreasing under local operations and classical communications.
Denote H f as a d f -dimensional Hilbert vector space.The concurrence of a bipartite pure )), where ρ 1 = Tr 2 (|ψ 12 ⟩⟨ψ 12 |).The q-concurrence (q ≥ 2) is defined by [19], C q (|ψ 12 ⟩) = 1 − Tr(ρ q 1 ).For arbitrary n-partite pure state |ψ⟩ ∈ H 1 ⊗ H 2 ⊗ • • • ⊗ H n , the q-concurrence under bipartition J t | Ĵt is given by where Ĵt is the complement of J t .Conditions (a) and (b) of a well-defined GME measure imply that one needs to take over all possible bipartitions in constructing GME measures.Hence, we choose the form of geometric mean of concurrence.Moreover, we add a parameter to ensure that our constructed GME measures satisfy condition (c) for well-defined GME measures.
In terms of the q-concurrence, we have the following parameterized GME measures.
In [21], the authors suggested that a proper GME measure should satisfy an additional criterion: (d) the GME of the GHZ state is larger than that of the W state.Here, for fourqubit pure ) being all equal to 1, while Γ GME (|W⟩) = 0.4242.Obviously, the GHZ state is more entangled than the W state.Thus, Γ GME (|ψ⟩) is also a proper GME measure in this sense.
Namely, our measure Γ GME (|ψ⟩) can distinguish the difference in the entanglement between |ψ A ⟩ and |ψ B ⟩ as well as between |ψ C ⟩ and |ψ D ⟩.In this sense, our measure characterizes the genuine multipartite entanglement in a more fine way.
Furthermore, concerning the entanglement order [21,22], any two entanglement measures should give rise to the same ordering on the set of entangled states if they are actually equivalent [23].Namely, if two entanglement measures E 1 and E 2 are equivalent, then for any pair of ρ 1 and ρ 2 .The following example shows that two entanglement measures are inequivalent when there exists a different entanglement order in certain intervals.We will highlight the advantages of our measure by comparing it with other measures.
Example 2. Consider the following family of four-qubit pure states, . Set ξ = 1 2 and q = 2. Using Theorem 1, we have that |ϕ θ ⟩ is genuine entangled for θ ∈ (0, π 2 ), see Figure 1.With the increasing in θ from θ 1 to θ 2 , Γ GME decreases.Nevertheless, the GMC increases from θ 1 to θ 2 [9].Thus, for any two arbitrary states within this range, the entanglement order for GMC and Γ GME is different.GMC and Γ GME are inequivalent in this sense.Meanwhile, Γ GME is a smooth function of θ, while GMC displays a sharp peak at θ 2 = 1.1071.
Figure 1.The Γ GME (solid red) and GMC (dashed blue) for the four-qubit states given in (4) versus θ.The peak of Γ GME is at θ 1 , while the GMC has a sharp peak at θ = θ 2 .From θ 1 to θ 2 , Γ GME decreases, while the GMC increases.The GMC has a series of paired equal values in the interval θ ∈ [θ 1 , θ 3 ].

Parameterized k-Entanglement Measures for n-Partite Quantum Systems
An n-partite pure state |ψ⟩ is separable under k-partition if it can be expressed as This k-partition strictly obeys the following conditions: (i) Similarly, an n-partite mixed state ρ is k-separable if it can be represented as a convex mixture of k-separable pure states, i.e., ρ = ∑ i p i |ψ i ⟩⟨ψ i |, where {|ψ i ⟩} is k-separable with respect to certain k-partitions.Otherwise, We denote the set of all k-separable states by S k (k = 2, 3, • • • , n), with S 1 denoting the set of all quantum states.Clearly, S n ⊂ S n−1 ⊂ • • • ⊂ S 1 .In particular, the complement S 1 \ S 2 is the set containing all genuine multipartite entangled (2-nonseparable) states.
An entanglement measure for k-separability has to satisfy the following conditions: (i) For all k-separable states, the measure must be zero.(ii) For all k-nonseparable states, the measure must be positive.(iii) It is invariant under local unitary transformations.(iv) The measure is nonincreasing under LOCC for any state ρ (monotonicity).(v) The measure never increases under free operations of LOCC for its LOCC-ensemble {p j , σ j } (strong monotonicity).(vi) Convexity (E(∑ Here, the monotonicity means that the measure does not increase under any LOCC, i.e., E(Λ LOCC (ρ)) ≤ E(ρ).The strong monotonicity says that if ρ is transformed into a state σ j with the probability p j under LOCC, the measure is still nonincreasing on average, namely, ∑ j p j E(σ j ) ≤ E(ρ) holds for the LOCC-ensemble {p j , σ j }.
According to the above conditions (i) and (ii) for a well-defined measure of k-entanglement, we need to take into account all possible k-partitions of multipartite states.By using the concavity of the function y = x ξ (0 < ξ ≤ 1) and the fact that the function g = [ 1 n is concave [24], we construct k-entanglement measures satisfying conditions (iii)-(vi) by adding some parameters.To quantify the k-entanglement with respect to the k-separability of n-partite systems, we first present parameterized k-entanglement measures k = 2, 3, • • • , n for any n-partite pure states |ψ⟩, where where where the infimum is taken over all possible pure state decompositions.We have the following conclusion.Since the q-concurrence is invariant under local unitary transformations, we have (iv) Γ k−ME (ρ) is nonincreasing under LOCC (monotonicity).Firstly, we prove that the inequality Γ k−ME (Λ LOCC (|ψ⟩)) ≤ Γ k−ME (|ψ⟩) holds for any LOCC operations on pure state |ψ⟩.Since the q-concurrence decreases under LOCC, we only need to verify that Γ k−ME (|ψ⟩) is an increasing function of C qJ mα l | Ĵmα l (|ψ⟩).By direct calculation, we have where Now, for an arbitrary mixed state ρ with pure state decomposition {p i , |ψ i ⟩}, we obtain where the first inequality is due to the convexity of parameterized k-entanglement measures Γ k−ME (ρ), and the second inequality is derived from the property that Γ k−ME (|ψ⟩) is nonincreasing under LOCC for any pure states.(v) Γ k−ME (ρ) never increases under free operations of LOCC for its LOCC-ensemble {p j , σ j } (strong monotonicity).
We need to prove that the inequality where the state σ j = K j |ψ⟩⟨ψ|K † j is generated with probability p j by applying LOCC on ρ, ∑ j K † j K j = I (unit operator).If ρ = |ψ⟩⟨ψ| is a pure state, we have where the first inequality is due to the strong monotonicity of q-concurrence, that is, C q (ρ) ≥ ∑ j p j C q (σ j ) [19]; the concavity of the function y = x ξ (0 < ξ ≤ 1) leads to the second inequality; and the third inequality holds, as the function g = [ 1 n is concave [24].
For mixed state ρ = ∑ i p i |ψ i ⟩⟨ψ i |, we have where and the state occurs with the probability p j = Tr(K j ρK † j ) under LOCC.The first inequality holds since Γ k−ME (|ψ⟩) obeys the strong monotonicity for any pure states.The second inequality is due to the definition of Γ k−ME (ρ).
) is due to the convexity of the mixed states.
In [17], the k-ME concurrence of an n-partite pure state |ψ⟩ is defined by , where ρ A t is the reduced density matrix of the subsystem A t and the minimum is taken over all possible k-partitions where |A kα i , |T k | represents the cardinality of the elements in the set T k , and C 2 is the qconcurrence with q = 2.
The example below illustrates that our approach is able to detect multipartite entanglement and is inequivalent to the above multipartite entanglement measures.
Moreover, when θ ∈ (0, θ 2 ) or (θ 5 , π), we find that the 3-GM concurrence increases from 0 to θ 1 and decreases from θ 1 to θ 2 .Then, there always exists at least one pair of states whose 3-GM concurrences have the same value.While our measure Γ 3−ME increases from 0 to θ 2 , it always has different values for θ ∈ (0, θ 2 ).This means that Γ 3−ME is able to identify different entanglements, while the 3-GM concurrence fails in this interval.Therefore, our measure Γ 3−ME is not only inequivalent to the 3-GM concurrence but also has a superior performance in characterizing the multipartite entanglement finely in this case.Similar analysis yields that our Γ 3−ME distinguishes the entanglement in θ ∈ (θ 2 , π 2 ) or ( π 2 , θ 5 ), while the 3-ME concurrence fails (Figure 2).

Conclusions
We have presented parameterized GME measures Γ GME and k-entanglement measures Γ k−ME in terms of q-concurrence.They are proved to be well-defined measures and satisfy all the related conditions such as entanglement monotonicity, invariance under local unitary transformations, convexity, and strong entanglement monotonicity.Our measures are not equivalent to the existing ones in the sense that they give rise to different state orderings.Detailed examples have shown that our measure may characterize better the genuine multipartite entanglement and the k-entanglement of arbitrary n-partite systems.
Direct calculation shows that GMC(|ψ A ⟩) = GMC(|ψ B ⟩) = 0.8660 and GMC(|ψ C ⟩) = GMC(|ψ D ⟩) = 0.8000.It is evident that although the GMC detects genuine multipartite entanglement of above four-qubit pure states, it cannot tell the difference in entanglement neither between |ψ A ⟩ and |ψ B ⟩ nor between |ψ C ⟩ and |ψ D ⟩.The fact is due to GMC only depending on the minimum of concurrence, which is the same for both |ψ A ⟩ and |ψ B ⟩ as well as for both |ψ C ⟩ and |ψ D ⟩.
From our Theorem 1, we have tα i | Ĵtα i represents any bipartition of the state |ψ⟩, L k = {α i } stands for the set that encompasses all possible k-partitions {J 1α i |J 2α i | • • • |J kα i }, and |L k | denotes the cardinality of the elements in the set L k .Γ k−ME (|ψ⟩) is generalized to n-partite mixed states by convexroof extension,