Unified Monogamy Relations of Multipartite Entanglement

Unified-(q, s) entanglement \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathscr{U}}}_{q,s})$$\end{document}(Uq,s) is a generalized bipartite entanglement measure, which encompasses Tsallis-q entanglement, Rényi-q entanglement, and entanglement of formation as its special cases. We first provide the extended (q; s) region of the generalized analytic formula of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr{U}}}_{q,s}$$\end{document}Uq,s. Then, the monogamy relation based on the squared \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr{U}}}_{q,s}$$\end{document}Uq,s for arbitrary multiqubit mixed states is proved. The monogamy relation proved in this paper enables us to construct an entanglement indicator that can be utilized to identify all genuine multiqubit entangled states even the cases where three tangle of concurrence loses its efficiency. It is shown that this monogamy relation also holds true for the generalized W-class state. The αth power \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr{U}}}_{q,s}$$\end{document}Uq,s based general monogamy and polygamy inequalities are established for tripartite qubit states.

Entanglement is a vital asset in quantum information sciences that can enhance quantum technologies such as communication, cryptography and computing beyond classical limitations 1 . Such quantum technologies mostly rely on the distribution of entanglement in multipartite settings. Quantification and characterization of entanglement distribution for multipartite systems is well explained through monogamy relation. Briefly, the monogamy explains that if two parties are maximally entangled, then the rest of the parties cannot share any entanglement with them. This monogamy property, for example, plays a role in security analysis of quantum key distribution 2 and it can also be used to distinguish quantum channels 3 .
The concept of monogamy of entanglement was first introduced by Coffman, Kundu and Wootters 4 -known as CKW inequality. They established the monogamy property for tripartite (A, B, and C) system via an entanglement measure called the concurrence 5 . Furthermore, the monogamy inequality asserts that the summation of individual entanglement content of subsystem A with subsystem B and with subsystem C is less than or equal to the entanglement of subsystem A with combined subsystem BC. This monogamy relation was then generalized to N-qubit systems 6 . Later on, monogamy relations for various entanglement measures have been proved, e.g., concurrence 4,7-9 , entanglement of formation 6,10,11 , negativity 9,12-15 , Tsallis-q entanglement [16][17][18] , and Rényi-q entanglement 19,20 . The dual of monogamy (polygamy) relation via the concurrence of assistance was proposed to quantify the limitation of distributing bipartite entanglement in multipartite systems 21,22 . Polygamy relations were established using various entanglement measures, e.g., convex-roof extended negativity 13 , and Tsallis-q entanglement 9,16 .
This paper proposes the idea to understand the entanglement distribution in multipartite system via the unified-(q, s) entanglement  ( ) q s , . q s ,  encompasses several measures of entanglement such as concurrence, Tsallis-q entanglement (T q -E), Rényi-q entanglement (R q -E), and entanglement of formation (EOF), as its special cases. However, it does not satisfy the usual monogamy relations and violates monogamy for W-class state 23 . The monogamy relation of EOF has been not reported yet in a unified fashion. Three tangle based on the squared concurrence also has some flaws for entanglement detection 24 . This highly motivates us to introduce a general concept of monogamy relations in multiqubit systems, which can overcome these flaws. We propose new monogamy relations for  q s , . To this end, we first give the analytic formula of  q s , for the region , 0 ≤ s ≤ 1, and ≤ + qs (5 13 )/2. Then, we establish the monogamy relation of multiqubit entangled system based on the squared  q s , (SU-(q,s)-E), which encompasses the monogamy relations of EOF, T q -E, and R q -E, as special cases. Therefore, the results in this paper provide a unifying framework for monogamy relations in multiqubit systems, covering several previous monogamy results 6,[16][17][18][19][20]23 .

Results
First, we revise the definition of q s ,  and present the formula with its extended ranges. Then we investigate the monogamy relations for the squared and α ≥ 2 power of q s ,  . Polygamy relation of  q s , for α ≤ 0 is also obtained. We further construct the multipartite entanglement indicator and present some numerical examples.
Unified-(q,s) entanglement. For any bipartite pure state ψ AB , q s ,  is defined as 23 where the minimization and maximization are obtained over all pure state decompositions , encompasses various entanglement measures depending on the parameters q and s. For example, it converges to R q -E, T q -E, and EOF when s → 0, s → 1, and q → 1, respectively.
Refining the analytical formula for q s ,  . For any two-qubit mixed state ρ AB , concurrence  is given as 5 y y y , and σ y denotes the Pauli-y operator. The analytic relationship between q s ,  and concurrence of a bipartite state ρ AB for 1 ≥ s ≥ 0 and 3/s ≥ q ≥ 1 has been unveiled as follows 23 : The analytic formula (5) holds until the f q,s (x) in (6) is monotonically increasing and convex for any q and s value 23 . The monotonicity and convexity follow from the fact that ∂f q,s (x)/∂x ≥ 0 for all q ≥ 0 and ∂ 2 f q,s (x)/∂x 2 ≥ 0 for 1 ≥ s ≥ 0 and 3/s ≥ q ≥ 1 23 .
In the succeeding theorem, we will establish the monogamy inequity of q s www.nature.com/scientificreports www.nature.com/scientificreports/ Theorem 1. SU-(q,s)-E holds the following monogamy inequality for an arbitrary multi-qubit mixed state ρ AB 1 B 2 …B N−1 : is not a logic qubit. However, We can apply the convex roof extension formula (2) of the pure state entanglement. Let where (a) follows from the pure state formula of the  q s , and takes the ,  is a convex function of concurrence for ∈ q s ( , ) ; and (c) is due to the convexity of concurrence for mixed states. (10); (e) and (f) are due to Propositions 1 and 2, respectively.

Remark 1. SU-(q,s)-E provides us the broad class of monogamy inequalities and recovers the monogamy relations for
squared EOF, T q -E and R q -E for different values of q and s. Specifically, (9) can be reduced to the following monogamy relations: i. Squared EOF 6,10 , for q → 1 The αth power q s ,  monogamy relation. In this subsection, we establish the αth power q s ,  based general monogamy and polygamy inequalities.

q s A A A q s A A q s A A q s A A q s A A q s A A q s A A q s A A
, , where (a) follows from 1 + β γ > (1 + β) γ for β > 0, and γ ≤ 0. 

Remark 2. Theorem 2 and Theorem 3 have established the monogamy and dual monogamy inequalities for the αth power q s
,  for α ≥ 2 and α ≤ 0, respectively in a tripartite scenario. These relations can be generalized for multiqubit systems by using induction and simple algebraic inequalities.

Multipartite entanglement indicators based on the SU-(q, s)-E. From monogamy relation (9) of
SU-(q,s)-E, we build a multipartite entanglement indicator that can be utilized to detect entanglement in the N-qubit state ρ A 1 A 2 … A N . The indicator  q s , is defined as where the minimization is performed over all pure state decompositions of ρ A 1 A 2 … A N . This indicator essentially originates from the convex-roof of the pure state indicator

Example 1. An N-qubit W-state is defined as
The indicator for the N-qubit W-class state can be written as . Via the established monogamy relation of the squared concurrence, the three tangle J C (genuine tripartite entanglement measure) is defined as 4 (2019) 9:16419 | https://doi.org/10.1038/s41598-019-52817-y www.nature.com/scientificreports www.nature.com/scientificreports/ The three tangle cannot detect the tripartite entangled W-state 4 . However, the indicator q s ,  efficiently detects the entanglement in this state. We plot the indicator as a function of (q,s) for the four and five qubit W-state in Fig. 1. The indicator has nonzero values when entanglement is present in the system.  1 3 The three tangle of ψ ABC is J C ψ = − − p p p ( ) (9 8 6 (1 ) )/9 is calculated through the analytic formula of the  q s , for bipartite states. There is no need for convex-roof for the pure state. In Fig. 2, we draw the comparison between the J C and  q s , . We can see that q s ,  is positive for all values of p.

Discussion
Unified-(q,s) entanglement is a two-parameter class of well defined bipartite entanglement measures. The generalized analytic formula of q s ,  has been proved for the region  ∈ q s ( , ) , which encompasses EOF 5 , Tsallis-q entanglement [16][17][18] and Renyi-q entanglement 19,25 as its special cases. We have investigated the monogamy relation for SU-(q,s)-E, which classifies the entanglement distribution in multipartite systems. The monogamy relation of SU-(q,s)-E enables us  www.nature.com/scientificreports www.nature.com/scientificreports/ to construct an indicator, which overcomes all known flaws and detects genuine multipartite entanglement better than previously known indicators. This superior performance in the detection of multiqubit states is exemplified on W-class states and compared with concurrence based entanglement indicator. The established monogamy relation gives the nontrivial and computable lower bound for the  q s , . Furthermore, we also proved the αth power  q s , based general monogamy and polygamy relations. In summary, the results in this paper provide the unified monogamy relations of multipartite entanglement, covering several previous results as its special cases.

Methods
is a convex function of the concurrence . We prove the convexity of f q,s (x) in the extended 3s)), 0 ≤ s ≤ 1, and ≤ + qs (5 13 )/2, which was previously shown for the region 1 ≥ s ≥ 0 and 3/s ≥ q ≥ 1. We consider the second-order derivative of f q,s (x) for 1 > q > 0 and qs ∈ (3, 5), respectively.
For the region 0 < q < 1, we graphically analyze the solution of U C . It can be shown that for fixed s ∈ [0,1], the value of x to keep the second derivative nonnegative increases monotonically with q 18,25 . Therefore, the critical point exists under the limit x → 1. We apply limit x → 1 to obtain the critical point of q. After applying the limit and some simplification, we have with 0 ≤ s ≤ 1 for the region 0 < q < 1. The second-order derivative is always nonnegative when > ⁎ q q . For qs ∈ (3, 5), we select qs ≤ 4.302 because when s → 1, f q,s (x) approaches to the Tsallis entropy for which the second derivative is known to be nonnegative for q ≤ 4.302 18 . For the analytical proof, we define a new range of s on the basis of this constraint, that is, 0 ≤ s ≤ min{4.302/q,1}. We enforce this constraint by substituting s = 4.302/q in the expression for the second derivative of f q,s (x). In the following, we prove that the second derivative is nonnegative for q ≥ 4.302. The second derivative of f q,s (x) after its simplification is q s q q q q q q q q 2 , 2 2 2 1 12 First, we apply the binomial expansion on A q−1 and B q−1 to write Using the inequality of arithmetic and geometric, i.e., + ≥ x y xy 2 , we obtain where AB = Z 2 . Substituting (26) in (23) and after some manipulations, we finally obtain the inequality:  is an increasing monotonic function of the squared concurrence  2 . Note that we can rewrite the Eq. (5) as    is a convex function of the squared concurrence 2  . The SU-(q,s)-E is convex in  2 when the    where A = (F q + E q ) and B = (E q−1 − F q−1 ). The intermediate value theorem states that if a continuous function has values of opposite sign inside a domain, then it has a root in that domain. The function Z q,s (x) is continuous on the domain D. We divide D into two sub domains,