Hamming weight and tight constraints of multi-qubit entanglement in terms of unified entropy
aa r X i v : . [ qu a n t - ph ] J a n Hamming weight and tight constraints of multi-qubitentanglement in terms of unified entropy
Jeong San Kim ∗ Department of Applied Mathematics and Institute of Natural Sciences,Kyung Hee University, Yongin-si, Gyeonggi-do 446-701, Korea (Dated: October 15, 2018)
Abstract
We establish a characterization of multi-qubit entanglement constraints in terms of non-negativepower of entanglement measures based on unified-( q, s ) entropy. Using the Hamming weight of thebinary vector related with the distribution of subsystems, we establish a class of tight monogamyinequalities of multi-qubit entanglement based on the α th-power of unified-( q, s ) entanglementfor α ≥
1. For 0 ≤ β ≤
1, we establish a class of tight polygamy inequalities of multi-qubitentanglement in terms of the β th-power of unified-( q, s ) entanglement of assistance. Thus ourresults characterize the monogamy and polygamy of multi-qubit entanglement for the full range ofnon-negative power of unified entanglement. PACS numbers: 03.67.Mn, 03.65.Ud ∗ Electronic address: [email protected] . INTRODUCTION Quantum entanglement is a quintessential feature of quantum mechanics revealing thefundamental insights into the nature of quantum correlations. One distinct property of quan-tum entanglement without any classical counterpart is its limited shareability in multi-partyquantum systems, known as the monogamy of entanglement (MoE) [1, 2]. MoE is the fun-damental ingredient for secure quantum cryptography [3, 4], and it also plays an importantrole in condensed-matter physics such as the N -representability problem for fermions [5].Mathematically, MoE is characterized in a quantitative way known as the monogamyinequality ; for a three-qubit quantum state ρ ABC with its two-qubit reduced density matrices ρ AB = tr C ρ ABC and ρ AC = tr B ρ ABC , the first monogamy inequality was established byCoffman-Kundu-Wootters(CKW) as τ (cid:0) ρ A | BC (cid:1) ≥ τ (cid:0) ρ A | B (cid:1) + τ (cid:0) ρ A | C (cid:1) where τ (cid:0) ρ A | BC (cid:1) is the bipartite entanglement between subsystems A and BC , quantifiedby tangle and τ (cid:0) ρ A | B (cid:1) and τ (cid:0) ρ A | C (cid:1) are the tangle between A and B and between A and C ,respectively [6].CKW inequality shows the mutually exclusive relation of two-qubit entanglement be-tween A and each of B and C measured by τ (cid:0) ρ A | B (cid:1) and τ (cid:0) ρ A | C (cid:1) respectively, so that theirsummation cannot exceeds the total entanglement between A and BC , that is, τ (cid:0) ρ A | BC (cid:1) .Later, three-qubit CKW inequality was generalized for arbitrary multi-qubit systems [7] andsome cases of multi-party, higher-dimensional quantum systems more than qubits in termsof various bipartite entanglement measures [8–11].Whereas entanglement monogamy characterizes the limited shareability of entanglementin multi-party quantum systems, the assisted entanglement , which is a dual amount tobipartite entanglement measures, is also known to be dually monogamous, thus polygamousin multi-party quantum systems; for a three-qubit state ρ ABC , a polygamy inequality wasproposed as τ a (cid:0) ρ A | BC (cid:1) ≤ τ a (cid:0) ρ A | B (cid:1) + τ a (cid:0) ρ A | C (cid:1) , where τ a (cid:0) ρ A | BC (cid:1) is the tangle of assistance [12, 13]. Later, the tangle-based polygamyinequality of entanglement was generalized into multi-qubit systems as well as some classof higher-dimensional quantum systems using various entropic entanglement measures [10,24, 15]. General polygamy inequalities of entanglement were also established in arbitrarydimensional multi-party quantum systems [16, 17].Recently, a new class of monogamy inequalities using the α th-power of entanglementmeasures were proposed; it was shown that the α th-power of entanglement of formation andconcurrence can be used to establish multi-qubit monogamy inequalities for α ≥ √ α ≥
2, respectively [18]. Later, tighter monogamy and polygamy inequalities of entanglementusing non-negative power of concurrence and squar of convex-roof extended negativity werealso proposed for multi-qubit systems [19, 20].Here, we provide a full characterization of multi-qubit entanglement monogamy andpolygamy constraints in terms of non-negative power of entanglement measures based onunified entropy [21, 22]. Using the Hamming weight of the binary vector related with the dis-tribution of subsystems, we establish a class of tight monogamy inequalities of multi-qubitentanglement based on the α th-power of unified-( q, s ) entanglement [11] for α ≥
1. For0 ≤ β ≤
1, we establish a class of tight polygamy inequalities of multi-qubit entanglementin terms of the β th-power of unified-( q, s ) entanglement of assistance [15].This paper is organized as follows. In Section II, we review the definitions of unifiedentropy, unified-( q, s ) entanglement and unified-( q, s ) entanglement of assistance as well asmulti-qubit monogamy and polygamy inequalities in terms of unified entanglements. InSection III, we establish a class of tight monogamy inequalities in multi-qubit system basedon the α th-power of unified-( q, s ) entanglement for α ≥
1. In Section IV, we establish aclass of tight polygamy inequalities of multi-qubit entanglement in terms of the β th-powerof unified-( q, s ) entanglement of assistance for 0 ≤ β ≤
1. Finally, we summarize our resultsin Section V.
II. UNIFIED ENTROPY AND MULTI-QUBIT ENTANGLEMENT CON-STRAINTS
For q, s ≥ q = 1 and s = 0, unified-( q, s ) entropy of a quantum state ρ is definedas [21, 22], S q,s ( ρ ) := 1(1 − q ) s [(tr ρ q ) s − . (1)Although unified-( q, s ) entropy has a singularity at s = 0, it converges to R´enyi- q entropy as s tends to 0 [23, 24]. We also note that unified-( q, s ) entropy converges to Tsallis- q entropy325] when s tends to 1, and for any nonnegative s , unified-( q, s ) entropy converges to vonNeumann entropy as q tends to 1,lim q → S q,s ( ρ ) = − tr ρ log ρ =: S ( ρ ) , (2)and these enable us to denote S ,s ( ρ ) = S ( ρ ) and S q, ( ρ ) = R q ( ρ ).Using unified-( q, s ) entropy in Eq. (1), a two-parameter class of bipartite entangle-ment measures was introduced; for a bipartite pure state | ψ i AB , its unified- ( q, s ) entan-glement (UE) [11] is E q,s (cid:16) | ψ i A | B (cid:17) := S q,s ( ρ A ) , (3)for each q, s ≥ ρ A = tr B | ψ i AB h ψ | is the reduced density matrix of | ψ i AB ontosubsystem A . For a bipartite mixed state ρ AB , its UE is E q,s (cid:0) ρ A | B (cid:1) := min X i p i E q,s ( | ψ i i A | B ) , (4)where the minimum is taken over all possible pure state decompositions of ρ AB = P i p i | ψ i i AB h ψ i | . As a dual concept to UE, unified- ( q, s ) entanglement of assistance (UEoA)was also introduced as E aq,s (cid:0) ρ A | B (cid:1) := max X i p i E q,s ( | ψ i i A | B ) , (5)for q, s ≥ ρ AB [15].Due to the continuity of UE in Eq. (4) with respect to the parameters q and s , UE reducesto the one-parameter class of entanglement measures namely R´enyi- q entanglement (RE) [9]as s tends to 0, and it also reduces to another one-parameter class of bipartite entanglementmeasures called Tsallis- q entanglement (TE) [10] as s tends to 1. For any nonnegative s , UEconverges to entanglement of formation (EoF) as q tends to 1,lim q → E q,s (cid:0) ρ A | B (cid:1) = E f (cid:0) ρ A | B (cid:1) , (6)therefore UE is one of the most general classes of bipartite entanglement measures includingthe classes of R´enyi and Tsallis entanglements and EoF as special cases [11].Similarly, the continuity of UEoA in Eq. (5) with respect to the parameters q and s assures that UEoA reduces to R´enyi- q entanglement of assistance (REoA) [9] and Tsallis- q entanglement of assistance (TEoA) [10] when s tends to 0 or 1 respectively. For any4onnegative s , UEoA reduces to entanglement of assistance (EoA)lim q → E aq,s ( ρ AB ) = E a ( ρ AB ) , (7)when q tends to 1 [15].Using UE as the bipartite entanglement measure, a two-parameter class of monogamyinequalities of multi-qubit entanglement was established [11]; for q ≥
2, 0 ≤ s ≤ qs ≤
3, we have E q,s (cid:0) ρ A | A ··· A N (cid:1) ≥ N X i =2 E q,s (cid:0) ρ A | A i (cid:1) (8)for any multi-qubit state ρ A ··· A N where E q,s (cid:0) ρ A | A ··· A N (cid:1) is the UE of ρ A A ··· A N with re-spect to the bipartition between A and A · · · A N , and E q,s (cid:0) ρ A | A i (cid:1) is the unified-( q, s )entanglement of the reduced density matrix ρ A A i for each i = 2 , · · · , N .Later, it was shown that unified entropy can also be used to establish a class of polygamyinequalities of multi-qubit entanglement [15]; for 1 ≤ q ≤ − q + 4 q − ≤ s ≤
1, wehave E aq,s (cid:0) ρ A | A ··· A N (cid:1) ≤ N X i =2 E aq,s (cid:0) ρ A | A i (cid:1) (9)for any multi-qubit state ρ A ··· A N where E aq,s (cid:0) ρ A | A ··· A N (cid:1) is the UEoA of ρ A A ··· A N withrespect to the bipartition between A and A · · · A N , and E aq,s ( ρ A | A i ) is the UEoA of ρ A A i for i = 2 , · · · , N . III. TIGHT MONOGAMY CONSTRAINTS OF MULTI-QUBIT ENTANGLE-MENT IN TERMS OF UNIFIED ENTANGLEMENT
Here we establish a class of tight monogamy inequalities of multi-qubit entanglementusing the α ’th power of UE. Before we present our main results, we first provide somenotations, definitions and a lemma, which are useful throughout this paper.For any nonnegative integer j whose binary expansion is j = n − X i =0 j i i (10)where log j ≤ n and j i ∈ { , } for i = 0 , · · · , n −
1, we can always define a unique binaryvector associated with j , which is defined as −→ j = ( j , j , · · · , j n − ) . (11)5or the binary vector −→ j in Eq. (11), its Hamming weight , ω H (cid:16) −→ j (cid:17) , is the number of 1 ′ s inits coordinates [26]. We also provide the following lemma whose proof is easily obtained bysome straightforward calculus. Lemma 1.
For x ∈ [0 , and nonnegative real numbers α, β , we have (1 + x ) α ≥ αx α (12) for α ≥ , and (1 + x ) β ≤ βx β (13) for ≤ β ≤ . Now we provide our first result, which states that a class of tight monogamy inequalitiesof multi-qubit entanglement can be established using the α -powered UE and the Hammingweight of the binary vector related with the distribution of subsystems. Theorem 2.
For α ≥ , q ≥ and ≤ s ≤ , qs ≤ , we have (cid:0) E q,s (cid:0) ρ A | B B ··· B N − (cid:1)(cid:1) α ≥ N − X j =0 α ω H ( −→ j ) (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α , (14) for any multi-qubit state ρ AB ··· B N − where −→ j = ( j , · · · , j n − ) is the vector from the binaryrepresentation of j and ω H (cid:16) −→ j (cid:17) is the Hamming weight of −→ j .Proof. Without loss of generality, we may assume that the ordering of the qubit subsystems B , . . . , B N − satisfies E q,s (cid:0) ρ A | B j (cid:1) ≥ E q,s (cid:0) ρ A | B j +1 (cid:1) ≥ j = 0 , · · · , N − f ( x ) = x α for α ≥ (cid:0) E q,s (cid:0) ρ A | B B ··· B N − (cid:1)(cid:1) α ≥ N − X j =0 E q,s (cid:0) ρ A | B j (cid:1)! α , (16)which makes it feasible to prove the theorem by showing N − X j =0 E q,s (cid:0) ρ A | B j (cid:1)! α ≥ N − X j =0 α ω H ( −→ j ) (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α . (17)6e first prove Inequality (17) for the case that N = 2 n , a power of 2, by using mathematicalinduction on n , and extend the result for any positive integer N .For n = 1 and a three-qubit state ρ AB B with two-qubit rduced density matrices ρ AB and ρ AB , we have (cid:0) E q,s (cid:0) ρ A | B (cid:1) + E q,s (cid:0) ρ A | B (cid:1)(cid:1) α = (cid:0) E q,s (cid:0) ρ A | B (cid:1)(cid:1) α E q,s (cid:0) ρ A | B (cid:1) E q,s (cid:0) ρ A | B (cid:1) ! α , (18)where Inequalities (12) and (15) implies E q,s (cid:0) ρ A | B (cid:1) E q,s (cid:0) ρ A | B (cid:1) ! α ≥ α E q,s (cid:0) ρ A | B (cid:1) E q,s (cid:0) ρ A | B (cid:1) ! α . (19)From Eq. (18) and Inequality (19), we have (cid:0) E q,s (cid:0) ρ A | B (cid:1) + E q,s (cid:0) ρ A | B (cid:1)(cid:1) α ≥ (cid:0) E q,s (cid:0) ρ A | B (cid:1)(cid:1) α + α (cid:0) E q,s (cid:0) ρ A | B (cid:1)(cid:1) α , (20)which recovers Inequality (17) for n = 1.Now let us assume Inequality (17) is true for N = 2 n − with n ≥
2, and consider the casethat N = 2 n . For an ( N + 1)-qubit state ρ AB B ··· B N − with its two-qubit reduced densitymatrices ρ AB j with j = 0 , · · · , N −
1, we have N − X j =0 E q,s (cid:0) ρ A | B j (cid:1)! α = n − − X j =0 E q,s (cid:0) ρ A | B j (cid:1)! α P n − j =2 n − E q,s (cid:0) ρ A | B j (cid:1)P n − − j =0 E q,s (cid:0) ρ A | B j (cid:1) ! α . (21)Because the ordering of subsystems in Inequality (15) implies0 ≤ P n − j =2 n − E q,s (cid:0) ρ A | B j (cid:1)P n − − j =0 E q,s (cid:0) ρ A | B j (cid:1) ≤ , (22)thus Eq. (21) and Inequality (12) lead us to N − X j =0 E q,s (cid:0) ρ A | B j (cid:1)! α ≥ n − − X j =0 E q,s (cid:0) ρ A | B j (cid:1)! α + α n − X j =2 n − E q,s (cid:0) ρ A | B j (cid:1) α . (23)From the induction hypothesis, we have n − − X j =0 E q,s (cid:0) ρ A | B j (cid:1)! α ≥ n − − X j =0 α ω H ( −→ j ) (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α . (24)7oreover, the last summation in Inequality (23) is also a summation of 2 n − terms startingfrom j = 2 n − to j = 2 n −
1. Thus, (after possible indexing and reindexing subsystems, ifnecessary) the induction hypothesis also leads us to n − X j =2 n − E q,s (cid:0) ρ A | B j (cid:1) α ≥ n − X j =2 n − α ω H ( −→ j ) − (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α . (25)From Inequalities (23), (24) and (25), we have n − X j =0 E q,s (cid:0) ρ A | B j (cid:1)! α ≥ n − X j =0 α ω H ( −→ j ) (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α , (26)which recovers Inequality (17) for N = 2 n .Now let us consider an arbitrary positive integer N and a ( N +1)-qubit state ρ AB B ··· B N − .We first note that we can always consider a power of 2, which is an upper bound of N , thatis, 0 ≤ N ≤ n for some n . We also consider a (2 n + 1)-qubit stateΓ AB B ··· B n − = ρ AB B ··· B N − ⊗ σ B N ··· B n − , (27)which is a product of ρ AB B ··· B N − and an arbitrary (2 n − N )-qubit state σ B N ··· B n − .Because Γ AB B ··· B n − is a (2 n + 1)-qubit state, Inequality (26) leads us to (cid:0) E q,s (cid:0) Γ A | B B ··· B n − (cid:1)(cid:1) α ≥ n − X j =0 α ω H ( −→ j ) (cid:0) E q,s (cid:0) Γ A | B j (cid:1)(cid:1) α , (28)where Γ AB j is the two-qubit reduced density matric of Γ AB B ··· B n − for each j = 0 , · · · , n −
1. On the other hand, the separability of Γ AB B ··· B n − with respect to the bipartitionbetween AB · · · B N − and B N · · · B n − assures E q,s (cid:0) Γ A | B B ··· B n − (cid:1) = E q,s (cid:0) ρ A | B B ··· B N − (cid:1) , (29)as well as E q,s (cid:0) Γ A | B j (cid:1) = 0 , (30)for j = N, · · · , n −
1. Moreover, we haveΓ AB j = ρ AB j , (31)8or each j = 0 , · · · , N −
1. Thus, Inequality (28) together with Eqs. (29), (30) and (31) leadsus to (cid:0) E q,s (cid:0) ρ A | B B ··· B N − (cid:1)(cid:1) α = (cid:0) E q,s (cid:0) Γ A | B B ··· B n − (cid:1)(cid:1) α ≥ n − X j =0 α ω H ( −→ j ) (cid:0) E q,s (cid:0) Γ A | B j (cid:1)(cid:1) α = N − X j =0 α ω H ( −→ j ) (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α , (32)and this completes the proof.For any α ≥ ω H (cid:16) −→ j (cid:17) of the binary vector −→ j = ( j , · · · , j n − ), α ω H ( −→ j ) is greater than or equal to 1, therefore (cid:0) E q,s (cid:0) ρ A | B B ··· B N − (cid:1)(cid:1) α ≥ N − X j =0 α ω H ( −→ j ) (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α ≥ N − X j =0 (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α , (33)for any multi-qubit state ρ AB B ··· B N − and α ≥
1. Thus Inequality (14) of Theorem 2 isgenerally tighter than the monogamy inequalities of multi-qubit entanglement, which justuse the α th-power of entanglement measures.Due to the continuity of UE with respect to the parameters q and s , Inequality (14) ofTheorem 2 reduces to the class of R´enyi- q entropy-based monogamy inequalities of multi-qubit entanglement [9] in a tighter way when s tends to 0; (cid:0) R q (cid:0) ρ A | B B ··· B N − (cid:1)(cid:1) α ≥ N − X j =0 α ω H ( −→ j ) (cid:0) R q (cid:0) ρ A | B j (cid:1)(cid:1) α , (34)for any α ≥ q ≥ ρ AB B ··· B N − where R q (cid:0) ρ A | B B ··· B N − (cid:1) is theRE of ρ AB B ··· B N − with respect to the bipartition between A and B B · · · B N − [9]. When s tends to 1, Inequality (14) reduces to another class of monogamy inequalities, namely,Tsallis- q entropy-based monogamy inequalities of multi-qubit entanglement [10] in a tighterway; (cid:0) T q (cid:0) ρ A | B B ··· B N − (cid:1)(cid:1) α ≥ N − X j =0 α ω H ( −→ j ) (cid:0) T q (cid:0) ρ A | B j (cid:1)(cid:1) α , (35)for any α ≥
1, 2 ≤ q ≤ ρ AB B ··· B N − where T q (cid:0) ρ A | B B ··· B N − (cid:1) is theTE of ρ AB B ··· B N − with respect to the bipartition between A and B B · · · B N − [10].We also note that Inequality (14) of Theorem 2 can be even improved to be a tighterinequality with some condition on two-qubit entanglement;9 heorem 3. For α ≥ , q ≥ , ≤ s ≤ , qs ≤ and any multi-qubit state ρ AB ··· B N − , wehave (cid:0) E q,s (cid:0) ρ A | B ··· B N − (cid:1)(cid:1) α ≥ N − X j =0 α j (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α , (36) conditioned that E q,s (cid:0) ρ A | B i (cid:1) ≥ N − X j = i +1 E q,s (cid:0) ρ A | B j (cid:1) , (37) for i = 0 , · · · , N − .Proof. Due to Inequality (16), it is enough to show N − X j =0 E q,s (cid:0) ρ A | B j (cid:1)! α ≥ N − X j =0 α j (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α , (38)and we use mathematical induction on N . We further note that Inequality (20) in the proofof Theorem 2 assures that Inequality (38) is true for N = 2.Now let us assume the validity of Inequality (38) for any positive integer less than N .For a multi-qubit state ρ AB ··· B N − , we have N − X j =0 E q,s (cid:0) ρ A | B j (cid:1)! α = (cid:0) E q,s (cid:0) ρ A | B (cid:1)(cid:1) α P N − j =1 E q,s (cid:0) ρ A | B j (cid:1) E q,s (cid:0) ρ A | B (cid:1) ! α , (39)where Inequality (12) and the condition in Inequality (37) lead Inequality (39) to P N − j =1 E q,s (cid:0) ρ A | B j (cid:1) E q,s (cid:0) ρ A | B (cid:1) ! α ≥ α P N − j =1 E q,s (cid:0) ρ A | B j (cid:1) E q,s (cid:0) ρ A | B (cid:1) ! α . (40)Thus Eq. (39) and Inequality (40) imply N − X j =0 E q,s (cid:0) ρ A | B j (cid:1)! α ≥ (cid:0) E q,s (cid:0) ρ A | B (cid:1)(cid:1) α + α N − X j =1 E q,s (cid:0) ρ A | B j (cid:1)! α . (41)Because the summation in the right-hand side of Inequality (41) is a summation of N − N − X j =1 E q,s (cid:0) ρ A | B j (cid:1)! α ≥ N − X j =1 α j − (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α . (42)Now, Inequality (41) together with Inequality (42) recover Inequality (38), and this com-pletes the proof. 10or any nonnegative integer j and its corresponding binary vector −→ j , the Hammingweight ω H (cid:16) −→ j (cid:17) is bounded above by log j . Thus we have ω H (cid:16) −→ j (cid:17) ≤ log j ≤ j, (43)which implies (cid:0) E q,s (cid:0) ρ A | B ··· B N − (cid:1)(cid:1) α ≥ N − X j =0 α j (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α ≥ N − X j =0 α ω H ( −→ j ) (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) α , (44)for any α ≥
1. In other words, Inequality (14) in Theorem 2 can be made to be eventighter as Inequality (36) of Theorem 3 for any multi-qubit state ρ AB B ··· B N − satisfying thecondition in Inequality (37). IV. TIGHT POLYGAMY CONSTRAINTS OF MULTI-QUBIT ENTANGLE-MENT IN TERMS OF UNIFIED ENTANGLEMENT OF ASSISTANCE
As a dual property to the Inequality (14) of Theorem 2, we provide a class of polygamyinequalities of multi-qubit entanglement in terms of powered UEoA.
Theorem 4.
For ≤ β ≤ , − q + 4 q − ≤ s ≤ on ≤ q ≤ and any multi-qubit state ρ AB ··· B N − , we have (cid:0) E aq,s (cid:0) ρ A | B B ··· B N − (cid:1)(cid:1) β ≤ N − X j =0 β ω H ( −→ j ) (cid:0) E aq,s (cid:0) ρ A | B j (cid:1)(cid:1) β . (45) Proof.
Without loss of generality, we assume the ordering of the qubit subsystems B , · · · , B N − satisfying E aq,s (cid:0) ρ A | B j (cid:1) ≥ E aq,s (cid:0) ρ A | B j +1 (cid:1) ≥ j = 0 , · · · , N −
2. Moreover, due to the monotonicity of the function f ( x ) = x β for0 ≤ β ≤ (cid:0) E aq,s (cid:0) ρ A | B B ··· B N − (cid:1)(cid:1) β ≤ N − X j =0 E aq,s (cid:0) ρ A | B j (cid:1)! β , (47)11hus it is enough to show that N − X j =0 E aq,s (cid:0) ρ A | B j (cid:1)! β ≤ N − X j =0 β ω H ( −→ j ) (cid:0) E q,s (cid:0) ρ A | B j (cid:1)(cid:1) β . (48)The proof method is similar to that of Theorem 2; we first prove Inequality (48) for thecase that N = 2 n by using mathematical induction on n , and generalize the result to anypositive integer N . For n = 1 and a three-qubit state ρ AB B with two-qubit rduced densitymatrices ρ AB and ρ AB , we have (cid:0) E aq,s (cid:0) ρ A | B (cid:1) + E aq,s (cid:0) ρ A | B (cid:1)(cid:1) β = (cid:0) E aq,s (cid:0) ρ A | B (cid:1)(cid:1) β E aq,s (cid:0) ρ A | B (cid:1) E aq,s (cid:0) ρ A | B (cid:1) ! β , (49)which, together with Inequalities (13) and (46) leads us to (cid:0) E aq,s (cid:0) ρ A | B (cid:1) + E aq,s (cid:0) ρ A | B (cid:1)(cid:1) β ≤ (cid:0) E aq,s (cid:0) ρ A | B (cid:1)(cid:1) β + β (cid:0) E aq,s (cid:0) ρ A | B (cid:1)(cid:1) β . (50)Inequality (50) recovers Inequality (48) for n = 1.Now we assume the validity of Inequality (48) for N = 2 n − with n ≥
2, and considerthe case that N = 2 n . For an ( N + 1)-qubit state ρ AB B ··· B N − and its two-qubit reduceddensity matrices ρ AB j with j = 0 , · · · , N −
1, we have N − X j =0 E aq,s (cid:0) ρ A | B j (cid:1)! β = n − − X j =0 E aq,s (cid:0) ρ A | B j (cid:1)! β P n − j =2 n − E aq,s (cid:0) ρ A | B j (cid:1)P n − − j =0 E aq,s (cid:0) ρ A | B j (cid:1) ! β , (51)where the ordering of subsystems in Inequality (46) and Inequality (13) together withEq. (51) lead us to N − X j =0 E aq,s (cid:0) ρ A | B j (cid:1)! β ≤ n − − X j =0 E aq,s (cid:0) ρ A | B j (cid:1)! β + β n − X j =2 n − E aq,s (cid:0) ρ A | B j (cid:1) β . (52)Because each summation on the right-hand side of Inequality (52) is a summation of 2 n − terms , the induction hypothesis assures that n − − X j =0 E aq,s (cid:0) ρ A | B j (cid:1)! β ≤ n − − X j =0 β ω H ( −→ j ) (cid:0) E aq,s (cid:0) ρ A | B j (cid:1)(cid:1) β , (53)and n − X j =2 n − E aq,s (cid:0) ρ A | B j (cid:1) β ≤ n − X j =2 n − β ω H ( −→ j ) − (cid:0) E aq,s (cid:0) ρ A | B j (cid:1)(cid:1) β . (54)12Possibly, we may index and reindex subsystems to get Inequality (54), if necessary.) Thus,Inequalities (52), (53) and (54) recover Inequality (48) when N = 2 n .For an arbitrary positive integer N and a ( N +1)-qubit state ρ AB B ··· B N − , leu us considerthe (2 n + 1)-qubit state Γ AB B ··· B n − in Eq. (27). Because Γ AB B ··· B n − is a (2 n + 1)-qubitstate, we have (cid:0) E aq,s (cid:0) Γ A | B B ··· B n − (cid:1)(cid:1) β ≤ n − X j =0 β ω H ( −→ j ) (cid:0) E aq,s (cid:0) Γ A | B j (cid:1)(cid:1) β , (55)where Γ AB j is the two-qubit reduced density matric of Γ AB B ··· B n − for each j = 0 , · · · , n −
1. Moreover, Γ AB B ··· B n − is a product state of ρ AB B ··· B N − and σ B N ··· B n − , which implies E aq,s (cid:0) Γ A | B B ··· B n − (cid:1) = E aq,s (cid:0) ρ A | B B ··· B N − (cid:1) , (56)and E aq,s (cid:0) Γ A | B j (cid:1) = 0 , (57)for j = N, · · · , n −
1. We also note thatΓ AB j = ρ AB j , (58)for each j = 0 , · · · , N −
1. Thus Inequality (55) together with Eqs. (56), (57) and (58)recovers Inequality (45), and this completes the proof.Similarly to the case of monogamy inequalities, Inequality (45) of Theorem 4 reduces toa class of Tsallis- q entropy-based polygamy inequalities of multi-qubit entanglement in atighter way; (cid:0) T aq (cid:0) ρ A | B B ··· B N − (cid:1)(cid:1) β ≤ N − X j =0 β ω H ( −→ j ) (cid:0) T aq (cid:0) ρ A | B j (cid:1)(cid:1) β , (59)for any 0 ≤ β ≤
1, 1 ≤ q ≤ ρ AB B ··· B N − where T aq (cid:0) ρ A | B B ··· B N − (cid:1) isthe TEoA of ρ AB B ··· B N − with respect to the bipartition between A and B B · · · B N − [10].We further note that Inequality (45) of Theorem 4 can be improved to a class of tighterpolygamy inequalities with some condition on two-qubit entanglement of assistance.13 heorem 5. For ≤ β ≤ , − q + 4 q − ≤ s ≤ on ≤ q ≤ and any multi-qubit state ρ AB ··· B N − , we have (cid:0) E aq,s (cid:0) ρ A | B ··· B N − (cid:1)(cid:1) β ≤ N − X j =0 β j (cid:0) E aq,s (cid:0) ρ A | B j (cid:1)(cid:1) β , (60) conditioned that E aq,s (cid:0) ρ A | B i (cid:1) ≥ N − X j = i +1 E aq,s (cid:0) ρ A | B j (cid:1) , (61) for i = 0 , · · · , N − .Proof. Due to Inequality (47), it is enough to show N − X j =0 E aq,s (cid:0) ρ A | B j (cid:1)! β ≤ N − X j =0 β j (cid:0) E aq,s (cid:0) ρ A | B j (cid:1)(cid:1) β , (62)and we use mathematical induction on N , Moreover, Inequality (50) assures the validity ofInequality (38) for N = 2.Now, let us assume Inequality (62) is true for any nonnegative integer less than N , andconsider a multi-qubit state ρ AB ··· B N − . From the equality N − X j =0 E aq,s (cid:0) ρ A | B j (cid:1)! β = (cid:0) E aq,s (cid:0) ρ A | B (cid:1)(cid:1) β P N − j =1 E aq,s (cid:0) ρ A | B j (cid:1) E aq,s (cid:0) ρ A | B (cid:1) ! β , (63)and Inequality (13) together with the condition in Inequality (61), we have N − X j =0 E aq,s (cid:0) ρ A | B j (cid:1)! β ≤ (cid:0) E aq,s (cid:0) ρ A | B (cid:1)(cid:1) β + β N − X j =1 E aq,s (cid:0) ρ A | B j (cid:1)! β . (64)Because the summation of the right-hand side in Inequality (64) is a summation of N − N − X j =1 E aq,s (cid:0) ρ A | B j (cid:1)! β ≤ N − X j =1 β j − (cid:0) E aq,s (cid:0) ρ A | B j (cid:1)(cid:1) β . (65)Now, Inequalities (64) and (65) recover Inequality (62), and this completes the proof.From Inequality (43), we have ω H (cid:16) −→ j (cid:17) ≤ j for any nonnegative integer j and its corre-sponding binary vector −→ j , therefore (cid:0) E aq,s (cid:0) ρ A | B ··· B N − (cid:1)(cid:1) β ≤ N − X j =0 β j (cid:0) E aq,s (cid:0) ρ A | B j (cid:1)(cid:1) β ≤ N − X j =0 β ω H ( −→ j ) (cid:0) E aq,s (cid:0) ρ A | B j (cid:1)(cid:1) β , (66)14or 0 ≤ β ≤
1. Thus, Inequality (60) of Theorem 5 is tighter than Inequality (45) ofTheorem 4 for 0 ≤ β ≤ ρ AB B ··· B N − satisfying the condition inInequality (61). V. CONCLUSIONS
We have provided a characterization of multi-qubit entanglement monogamy andpolygamy constraints in terms of non-negative power of entanglement measures based onunified entropy. Using the Hamming weight of the binary vector related with the distribu-tion of subsystems, we have established a class of tight monogamy inequalities of multi-qubitentanglement based on the α th-power of UE for α ≥
1. We have further established a classof tight polygamy inequalities of multi-qubit entanglement in terms of the β th-power ofUEoA for 0 ≤ β ≤ q, s ) entropy to establishthe monogamy and polygamy inequalities of multi-qubit entanglement so that our resultsencapsulate the results of R´enyi and Tsallis entanglement-based multi-qubit entanglementconstraints as special cases. Furthermore, our class of monogamy and polygamy inequalitieshold in a tighter way, which can also provide finer characterizations of the entanglementshareability and distribution among the multi-qubit systems. Noting the importance of thestudy on multi-party quantum entanglement, our result can provide a useful methodologyto understand the monogamy and polygamy nature of multi-party quantum entanglement. Acknowledgments
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