Tighter Constraints of Multi-Qubit Entanglement in Terms of Nonconvex Entanglement Measures LCREN and LCRENoA

The monogamy property of entanglement is an intriguing feature of multipartite quantum entanglement. Most entanglement measures satisfying the monogamy inequality have turned out to be convex. Whether nonconvex entanglement measures obey the monogamy inequalities remains less known at present. As a well-known measure of entanglement, the logarithmic negativity is not convex. We elucidate the constraints of multi-qubit entanglement based on the logarithmic convex-roof extended negativity (LCREN) and the logarithmic convex-roof extended negativity of assistance (LCRENoA). Using the Hamming weight derived from the binary vector associated with the distribution of subsystems, we establish monogamy inequalities for multi-qubit entanglement in terms of the αth-power (α≥4ln2) of LCREN, and polygamy inequalities utilizing the αth-power (0≤α≤2) of LCRENoA. We demonstrate that these inequalities give rise to tighter constraints than the existing ones. Furthermore, our monogamy inequalities are shown to remain valid for the high-dimensional states that violate the CKW monogamy inequality. Detailed examples are presented to illustrate the effectiveness of our results in characterizing the multipartite entanglement distributions.


I. INTRODUCTION
Quantum entanglement, an essential aspect of quantum mechanics, provides deep understanding of the nature of quantum correlations by revealing its foundational principles.One unique characteristic of quantum entanglement, which sets it apart from classical systems, is its limited shareability in multi-party quantum systems, known as the monogamy of entanglement (MoE) [1,2].MoE is the fundamental ingredient for secure quantum cryptography [3,4], and it also plays an important role in condensed-matter physics such as the N -representability problem for fermions [5].
Mathematically, MoE is characterized in a quantitative way known as the monogamy inequality; for a three-qubit quantum state ρ ABC with its two-qubit reduced density matrices ρ AB = T r C ρ ABC and ρ AC = T r B ρ ABC , the first monogamy inequality was established by Coffman-Kundu-Wootters (CKW) as τ ρ A|BC ≥ τ ρ A|B + τ ρ A|C where τ ρ A|BC is the bipartite entanglement between subsystems A and BC, quantified by tangle and τ ρ A|B and τ ρ A|C are the tangle between A and B and between A and C, respectively [6].
The CKW inequality demonstrates the mutually exclusive relationship of two-qubit entanglement between A and each of B and C measured by τ ρ A|B and τ ρ A|C respectively.As a result, the sum of the entanglement of the two-qubit systems cannot exceed the total entanglement between A and BC, that is, τ ρ A|BC .Subsequently, the CKW inequality was generalized for arbitrary multi-qubit systems [7] and extended to encompass multi-party and higher-dimensional quantum systems beyond qubits in some certain cases in terms of various bipartite entanglement measures [8][9][10][11].
Whereas entanglement monogamy characterizes the restricted ability to share entanglement in multi-qubit quantum systems, the assisted entanglement, which is a dual amount to bipartite entanglement measures, is also known to be dually monogamous, thus polygamous in multi-qubit quantum systems; for a three-qubit state ρ ABC , a polygamy inequality was proposed as τ a ρ A|BC ≤ τ a ρ A|B + τ a ρ A|C , where τ a ρ A|BC is the tangle of assistance [12,13].Later, the tangle-based polygamy inequality of entanglement was generalized into multi-qubit systems as well as some class of higher-dimensional quantum systems using various entropic entanglement measures [10,14,15].General polygamy inequalities of entanglement were also formulated for multi-qubit quantum systems in arbitrary dimensions.[16,17].
Recently, a new monogamy inequalities employing entanglement measures raised to the power of α were proposed; it was shown that the αth-powered of entanglement of formation and concurrence can be used to establish multi-qubit monogamy inequalities for α ≥ √ 2 and α ≥ 2, respectively [18].Later, tighter monogamy and polygamy inequalities of entanglement using non-negative power of concurrence and squar of convex-roof extended negativity were also proposed for multi-qubit systems [19,20].
It is widely recognized that entanglement measures with convexity always satisfy monogamy inequalities.Gao et.al in Ref. [21] present a measure of entanglement, logarithmic convex-roof extended negativity (LCREN) satisfying important characteristics of an entanglement measure, and investigate the monogamy relation for logarithmic negativity and LCREN both without convexity.They show exactly that the αth power of logarithmic negativity, and a newly defined good measure of entanglement, LCREN, obey a class of general monogamy inequalities in 2⊗2⊗3 systems and 2 ⊗ 2 ⊗ 2 n systems and multi-qubit systems for α ≥ 4 ln 2. They also provide a class of general polygamy inequalities of multi-qubit systems in terms of logarithmic convex-roof extended negativity of assistance (LCRENoA) for 0 ≤ α ≤ 2.
In this paper, we provide a finer characterization of multi-qubit entanglement in terms of nonconvex entanglement measures.By using the Hamming weight of the binary vectors related to the subsystems, we establish a class of monogamy inequalities for multiqubit entanglement based on the αth power of LCREN for α ≥ 4 ln 2. For 0 ≤ α ≤ 2, we establish a class of polygamy inequalities for multi-qubit entanglement in terms of the αth power of LCRENoA.Even for the case of α < 0, we can also provide tight constraints in terms of LCREN and LCRENoA.Thus, a complete characterization for the full range of the power α is given.We further show that our class of monogamy and polygamy inequalities hold in a tighter way than those provided before [21].Moreover, our monogamy inequality is shown to be more effective for the counterexamples of the CKW monogamy inequality in higher-dimensional systems.

II. PRELIMINARIES
We first recall the conceptions of LCREN and LCRENoA, and multi-qubit monogamy and polygamy inequalities.For a quantum state ρ AB on Hilbert space H A ⊗ H B , its negativity, N (ρ AB ) is defined as [22][23][24] where ρ T A AB denotes the partial transpose of ρ AB with respect to the subsystem A, and the trace norm ∥X∥ 1 = tr √ XX † .A more easily interpreted and computable measure of entanglement is the logarithmic negativity, which is defined as [22,23] This quantity is an entanglement monotone both under general LOCC and PPT preserving operations but not convex [23].It is, moreover, additive.Due to its construction, the negativity does not recognize entanglement in PPT states.In order to overcome its lack of separability criterion, one modification of negativity is convex-roof extended negativity (CREN), which gives a perfect discrimination of PPT bound entangled states and separable states in any bipartite quantum system.
For a bipartite state ρ AB , its CREN, N (ρ AB ), is defined by [29] N (ρ AB ) = min while the CREN of assistance (CRENoA), which can be considered to be dual to CREN, is defined as [8] N a (ρ AB ) = max where the minimum and maximum are taken over all possible pure-state decompositions of By definition, both the CREN and CRENoA of a pure state are equal to its negativity.For any bipartite state ρ AB , we define LCREN as Clearly, LCREN is invariant under local unitary transformations.One important property is this: E N (ρ AB ) is nonzero if and only if ρ AB is entangled (and so it equals zero if and only if ρ AB is separable).Besides, it is entanglement monotone under LOCC operations.LCREN is not only nonincreasing under LOCC, but also nonincreasing on average under LOCC, which follow from the entanglement monotonicity of CREN under LOCC, the monotonicity logarithm, and concavity of logarithm.
However, just as logarithmic negativity, LCREN is also not convex.Suppose that ρ AB = k p k ρ k with ρ k = |φ k ⟩ AB ⟨φ k | is the optimal decomposition for ρ AB achieving the minimum of (3).Then N (ρ AB ) = k p k N (|φ k ⟩ AB ) by definition.The concavity of logarithm ensures which implies that LCREN is not convex.
We can also show by concrete examples that it is not convex.Consider the mixed qubit state ρ = 1 2 (ρ 1 + ρ 2 ) with and ρ 2 = |01⟩⟨01|.By definition of N (ρ AB ) (Eq.( 3)), for two qubit state ρ AB , we have because the negativity is the concurrence for the pure states.Here C(ρ AB ) is the concurrence of the mixed qubit state [26].According to Ref. [26] we can obtain So one has from which it easily follows that This implies that LCREN is not convex.For any multiqubit state ρ AB0•••B N −1 , a monogamous inequality has been presented in Ref. [21] for α ≥ 4 ln 2, where Similar to the duality between CREN and CRENoA, we can also define a dual to LCREN, namely LCRENoA, by In addition, a class of polygamy inequalities has been obtained for multi-qubit systems in Ref. [21], In the following we show that these inequalities above can be further improved to be much tighter under certain conditions, which provide tighter constraints on the multiqubit entanglement distribution.

III. TIGHT CONSTRAINTS OF MULTI-QUBIT ENTANGLEMENT IN TERMS OF LCREN
Here we establish a class of tight monogamy inequalities of multi-qubit entanglement using the αth-powered of LCREN.Before we present our main results, we first provide some notations, definitions and a lemma, which are useful throughout this paper.
In Ref. [27], Kim established a class of tight monogamy inequalities of multiqubit entanglement in terms of Hamming weight.For any nonnegative integer j with binary expansion j = n−1 i=0 j i 2 i , where log 2 j ≤ n and j i ∈ {0, 1} for i = 0, • • • , n − 1, one can always define a unique binary vector associated with j, Hamming weight ω H − → j of the binary vector − → j is defined to be the number of 1 ′ s in its coordinates [28].Moreover, the Hamming weight ω H − → j is bounded above by log 2 j, We also provide the following lemma whose proof is easily obtained by some straightforward calculus.
[Lemma 1].For x ∈ [0, 1] and nonnegative real numbers α, β, we have for α ≥ 1, and for 0 ≤ α ≤ 1.Now we provide our first result, which states that a class of tight monogamy inequalities of multi-qubit entanglement can be established using the αth-powered LCREN and the Hamming weight of the binary vector related with the distribution of subsystems.
[Proof ].From inequality 8, one has E 4 ln 2 Without loss of generality, we assume that the qubit subsystems B 0 , . . ., B N −1 are so labeled such that We first show that the inequality (15) holds for the case of N = 2 n .For n = 1, let ρ AB0 and ρ AB1 be the two-qubit reduced density matrices of a three-qubit pure state ρ AB0B1 .We obtain Combining ( 12) and ( 16), we have From ( 17) and ( 18), we get Therefore, the inequality (15) holds for n = 1.We assume that the inequality (15) holds for N = 2 n−1 with n ≥ 2, and prove the case of N = 2 n .For an (N + 1)-qubit state ρ AB0B1•••B N −1 with its two-qubit reduced density matrices ρ ABj with j = 0, • • • , N − 1, we have Because the ordering of subsystems in Inequality 16 implies 0 ≤ Thus, the Eq. ( 19) and Inequality ( 12) lead us to According to the induction hypothesis, we get By relabeling the subsystems, the induction hypothesis leads to Thus, we have which is the tensor product of ρ AB0B1...B N −1 and an arbitrary ( where Γ A|Bj is the two-qubit reduced density matrix of Γ AB0B1...B 2 n −1 , j = 0, 1, . . ., 2 n − 1.Therefore, Therefore, our inequality (14) in Theorem 1 is always tighter than the inequality (8) in Ref. [21].
[Proof ].From inequality (8), we only need to prove We use mathematical induction on N here.It is obvious that inequality (24) holds for N = 2 from (14).Assume that it also holds for any positive integer less than N .Since where the first inequality is due to Lemma 12 and the second inequality is due to the induction hypothesis.
conditioned that E 4 ln 2 The vertical axis is the the lower bound of the LCREN E N (ρ A|BCD ).The red thin line represents the lower bound from our result (22).The green dotted line represents the lower bound from our result (14).
For the case of α < 0, we can also derive a tight upper bound of for all α < 0.
[Proof ].Similar to the proof in [19], for arbitrary three-qubit states we have where the first inequality is from α < 0, the second inequality is due to 1 + Combining ( 32) and ( 33), we get Thus, we obtain One can get a set of inequalities through the cyclic permutation of the pair indices B 0 , B 1 , . .., B N −1 in (34).Summing up these inequalities, we get (31).
[Remark4].In (31) we have assumed that all E N (ρ ABi ), i = 0, 1, 2, • • • , N − 1, are nonzero.In fact, if one of them is zero, the inequality still holds if one removes this term from the inequality.Namely, if E N (ρ ABi ) = 0, then one has . Similar to the analysis in proving Theorem 4, one gets ), for α < 0.

IV. TIGHT CONSTRAINTS OF MULTI-QUBIT ENTANGLEMENT IN TERMS OF LCRENOA
In this section, we provide a class of tight polygamy inequalities of multi-qubit entanglement in terms of the αth-powered LCRENoA and the Hamming weight of the binary vector related to the distribution of subsystems for 0 ≤ α ≤ 2. For the case of α < 0, we also propose a monogamy relation for LCRENoA.[Theorem 5].For any multi-qubit state ρ AB0...B N −1 and 0 ≤ α ≤ 2, we have where 0 ≤ α ≤ 2, − → j = (j 0 , • • • , j n−1 ) is the vector from the binary representation of j, and ω H − → j is the Hamming weight of − → j .
According to the induction hypothesis, we get By relabeling the subsystems, the induction hypothesis leads to Thus, we have which is the tensor product of ρ AB0B1...B N −1 and an arbitrary ( where Γ A|Bj is the two-qubit reduced density matrix of Γ AB0B1...B 2 n −1 , j = 0, 1, . . ., 2 n − 1.Therefore, Therefore, our inequality (35) in Theorem 5 is always tighter than the inequality (10) in Ref. [21].
[ Example 3].Let us consider the 3-qubit generalized W state We have 3 ) α from our result (35), and 3 ) α from the result given in Ref. [21].One can see that our result is better than the result in Ref. [21] for 0 ≤ α ≤ 2, see Fig. 3.The green line represents the upper bound from our results.The blue line represents the upper bound from the result in [21].
Similar to the improvement from the inequality (14) to the inequality (22), we can also improve the polygamy inequality in Theorem 6.The proof is similar to the Theorem 2.
Therefore the inequality (42) of Theorem 6 is tighter than the inequality (35) of Theorem 5 under certain conditions.
Similarly, for the quantum state in Eq.( 52 In other words, the LCREN-based monogamy inequality in ( 14) is still valid for the counterexamples of tangle-based monogamy inequality.Thus LCREN is a good alternative for monogamy inequality of multi-qubit entanglement even in higher-dimensional quantum systems so far.

FIG. 1 :
FIG.1:The vertical axis is the the lower bound of the LCREN E N (ρ A|BC ).The red line is the exact values of E N (ρ A|BC ).The green line represents the lower bound from our results.The blue line represents the lower bound from the result in[21].
FIG.2:The vertical axis is the the lower bound of the LCREN E N (ρ A|BCD ).The red thin line represents the lower bound from our result(22).The green dotted line represents the lower bound from our result(14).

FIG. 3 :
FIG.3:The vertical axis is the the upper bound of the LCRENOA E Na (ρ A|BC ).The red line is the exact values of E Na (ρ A|BC ).The green line represents the upper bound from our results.The blue line represents the upper bound from the result in[21].

FIG. 4 :
FIG.4:The vertical axis is the the lower bound of the LCREN E N (|Ψ⟩ A|BC ).The red line is the exact values of E N (|Ψ⟩ A|BC ).The green line represents the lower bound from our results(14).