Tighter monogamy relations of quantum entanglement for multiqubit W-class states
aa r X i v : . [ qu a n t - ph ] D ec Tighter monogamy relations of quantum entanglement formultiqubit W-class states
Zhi-Xiang Jin and Shao-Ming Fei , School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
Abstract
Monogamy relations characterize the distributions of entanglement in multipartite systems. Weinvestigate monogamy relations for multiqubit generalized W -class states. We present new analyt-ical monogamy inequalities for the concurrence of assistance, which are shown to be tighter thanthe existing ones. Furthermore, analytical monogamy inequalities are obtained for the negativityof assistance. NTRODUCTION
Quantum entanglement [1–8] is an essential feature of quantum mechanics. As one ofthe fundamental differences between quantum entanglement and classical correlations, a keyproperty of entanglement is that a quantum system entangled with one of other subsystemslimits its entanglement with the remaining ones. The monogamy relations give rise to thedistribution of entanglement in the multipartite setting. Monogamy is also an essentialfeature allowing for security in quantum key distribution [9].For a tripartite system A , B and C , the usual monogamy of an entanglement measure E implies that [10] the entanglement between A and BC satisfies E A | BC ≥ E AB + E AC . In Ref.[11, 12], the monogamy of entanglement for multiqubit W -class states has been investigated,and the monogamy relations for tangle and the squared concurrence have been proved. Itgives the general monogamy relations for the x -power [13] of concurrence of assistance forgeneralized multiqubit W -class states.In this paper, we show that the monogamy inequalities for concurrence of assistanceobtained so far can be made tighter. We establish entanglement monogamy relations forthe x -th ( x ≥
2) and y -th ( y <
0) power of the concurrence of assistance which are tighterthan those in [13], which give rise to finer characterizations of the entanglement distributionsamong the multipartite W -class states. Furthermore, we also present the general monogamyrelations for the x -power of negitivity of assistance for generalized multiqubit W -class states. TIGHTER MONOGAMY RELATIONS FOR CONCURRENCE OF ASSISTANCE
We first consider the monogamy inequalities related to concurrence. Let H X denote adiscrete finite dimensional complex vector space associated with a quantum subsystem X .For a bipartite pure state | ψ i AB in vector space H A ⊗ H B , the concurrence is given by [14–16] C ( | ψ i AB ) = q − Tr( ρ A )] , (1)where ρ A is the reduced density matrix by tracing over the subsystem B , ρ A =Tr B ( | ψ i AB h ψ | ). The concurrence for a bipartite mixed state ρ AB is defined by the convexroof extension C ( ρ AB ) = min { p i , | ψ i i} X i p i C ( | ψ i i ) , ρ AB = P i p i | ψ i ih ψ i | , with p i ≥ P i p i = 1 and | ψ i i ∈ H A ⊗ H B .For a tripartite state | ψ i ABC , the concurrence of assistance is defined by [17, 18] C a ( | ψ i ABC ) ≡ C a ( ρ AB ) = max { p i , | ψ i i} X i p i C ( | ψ i i ) , where the maximum is taken over all possible decompositions of ρ AB = Tr C ( | ψ i ABC h ψ | ) = P i p i | ψ i i AB h ψ i | . When ρ AB = | ψ i AB h ψ | is a pure state, then one has C ( | ψ i AB ) = C a ( ρ AB ).For an N -qubit pure state | ψ i AB ··· B N − ∈ H A ⊗ H B ⊗ · · · ⊗ H B N − , the concurrence C ( | ψ i A | B ··· B N − ) of the state | ψ i A | B ··· B N − , viewed as a bipartite state under the partition A and B , B , · · · , B N − , satisfies [19] C α ( ρ A | B ,B ··· ,B N − ) ≥ C α ( ρ AB ) + C α ( ρ AB ) + · · · + C α ( ρ AB N − ) , for α ≥
2, where ρ AB i = Tr B ··· B i − B i +1 ··· B N − ( | ψ i AB ··· B N − h ψ | ). It is further improved thatfor α ≥
2, one has [20], C α ( ρ A | B B ··· B N − ) ≥ C α ( ρ AB ) + α C α ( ρ AB ) + · · · + (cid:16) α (cid:17) m − C α ( ρ AB m ) (2)+ (cid:16) α (cid:17) m +1 (cid:0) C α ( ρ AB m +1 ) + · · · + C α ( ρ AB N − ) (cid:1) + (cid:16) α (cid:17) m C α ( ρ AB N − )and C α ( ρ A | B B ··· B N − ) < K (cid:0) C α ( ρ AB ) + C α ( ρ AB ) + · · · + C α ( ρ AB N − ) (cid:1) (3)for all α <
0, where K = N − .Dual to the Coffman-Kundu-Wootters inequality, the generalized monogamy relationbased on the concurrence of assistance do not satisfy the monogamy relation. But, for an N -qubit generlized W -class states | ψ i AB ··· B N − ∈ H A ⊗ H B ⊗ · · · ⊗ H B N − , the concurrenceof assistance C a ( | ψ i A | B ··· B N − ) of the state | ψ i AB ··· B N − satisfies the inequality [13], C xa ( ρ A | B ,B ··· ,B N − ) ≥ C xa ( ρ AB ) + C xa ( ρ AB ) + · · · + C xa ( ρ AB N − ) , (4)and C ya ( ρ A | B ,B ··· ,B N − ) < C ya ( ρ AB ) + C ya ( ρ AB ) + · · · + C ya ( ρ AB N − ) , (5)where x ≥ y ≤ N -qubit generalized W -class states | ψ i ∈ H A ⊗ H B ⊗ · · · ⊗ H B N − defined by | ψ i = a | · · · i + b | · · · i + · · · + b N | · · · i , (6)with | a | + P Ni =1 | b i | = 1. For the N -qubit generalized W -class states (6), one has [13], C ( ρ AB i ) = C a ( ρ AB i ) , i = 1 , , ..., N − , (7)where ρ AB i = Tr B ··· B i − B i +1 ··· B N − ( | ψ ih ψ | ).[ Theorem 1] . For the N -qubit generalized W -class states | ψ i ∈ H A ⊗ H B ⊗ · · · ⊗ H B N − , let ρ AB j ··· B jm − denote the m -qubit, 2 ≤ m ≤ N , reduced density matrix of | ψ i .If C ( ρ AB ji ) ≥ C ( ρ AB ji +1 ··· B jm − ) for i = 1 , , · · · t , and C ( ρ AB jk ) ≤ C ( ρ AB jk +1 ··· B jm − ) for k = t + 1 , · · · , m − ∀ ≤ t ≤ m − , m ≥
4, the concurrence of assistance satisfies C xa ( ρ A | B j ··· B jm − ) ≥ C xa ( ρ AB j )+ x C xa ( ρ AB j ) + · · · + (cid:16) x (cid:17) t − C xa ( ρ AB jt )+ (cid:16) x (cid:17) t +1 (cid:16) C xa ( ρ AB jt +1 ) + · · · + C xa ( ρ AB jm − ) (cid:17) + (cid:16) x (cid:17) t C xa ( ρ AB jm − ) (8)for all x ≥ [Proof]. For the N -qubit generalized W -class states | ψ i , according to the definitions of C ( ρ ) and C a ( ρ ), one has C a ( ρ A | B j ··· B jm − ) ≥ C ( ρ A | B j ··· B jm − ). When x ≥
2, we have C xa ( ρ A | B j ··· B jm − ) ≥ C x ( ρ A | B j ··· B jm − ) ≥ C x ( ρ AB j )+ x C x ( ρ AB j ) + · · · + (cid:16) x (cid:17) t − C x ( ρ AB jt )+ (cid:16) x (cid:17) t +1 (cid:16) C x ( ρ AB jt +1 ) + · · · + C x ( ρ AB jm − ) (cid:17) + (cid:16) x (cid:17) t C x ( ρ AB jm − )= C xa ( ρ AB j ) + x C xa ( ρ AB j ) + · · · + (cid:16) x (cid:17) t − C xa ( ρ AB jt )+ (cid:16) x (cid:17) t +1 (cid:16) C xa ( ρ AB jt +1 ) + · · · + C xa ( ρ AB jm − ) (cid:17) + (cid:16) x (cid:17) t C xa ( ρ AB jm − ) , (9)where we have used in the first inequality the relation a x ≥ b x for a ≥ b ≥ , x ≥
2. Thesecond inequality is due to (2). The equality is due to (7).4
FIG. 1: y is the value of C a ( | ψ i A | B B B ). Solid (red) line is the exact value of C a ( | ψ i A | B B B ), dashed (blue) line is the lower bound of C a ( | ψ i A | B B B ) in (8), anddot-dashed (green) line is the lower bound in [13] for x ≥ x ≥
2, ( x/ t ≥ ≤ t ≤ j m − , comparing with the monogamy relationsfor concurrence of assistance (4), our formula (8) in Theorem 1 gives a tighter monogamyrelation with larger lower bounds. In Theorem 1 we have assumed that some C ( ρ AB ji ) ≥ C ( ρ AB ji +1 ··· B jm − ) and some C ( ρ AB k ) ≤ C ( ρ AB k +1 ··· B m − ) for the N -qubit generalized W -classstates. If all C ( ρ AB ji ) ≥ C ( ρ AB ji +1 ··· B jm − ) for i = 1 , , · · · , m −
2, then we have the followingconclusion: [Theorem 2] . If C ( ρ AB ji ) ≥ C ( ρ AB ji +1 ··· B jm − ) for i = 1 , , · · · , m −
2, then we have C xa ( ρ A | B j ··· B jm − ) ≥ C xa ( ρ AB j ) + x C xa ( ρ AB j ) + · · · + (cid:16) x (cid:17) m − C xa ( ρ AB jm − ) (10)for all x ≥ Example 1 . Let us consider the 4-qubit generlized W -class states, | W i AB B B = 12 ( | i + | i + | i + | i ) . (11)We have C xa ( | ψ i A | B B B ) = ( √ ) x . From our result (8) we have C xa ( | ψ i A | B B B ) ≥ (cid:2) x + ( x ) (cid:3) ( ) x , and from (4) one has C xa ( | ψ i A | B B B ) ≥ ) x , x ≥
2. One can seethat our result is better than that in [13] for x ≥
2, see Fig. 1.We can also derive a tighter upper bound of C ya ( ρ A | B ··· B N − ) for y < Theorem 3] . For the N -qubit generalized W -class states | ψ i ∈ H A ⊗ H B ⊗ · · · ⊗ H B N − ,let ρ AB j ··· B jm − be the m -qubit, 2 ≤ m ≤ N , reduced density matrix of | ψ i with C ( ρ AB ji ) = 0for 1 ≤ i ≤ m −
1, we have C ya ( ρ A | B j ··· B jm − ) < ˜ M (cid:16) C ya ( ρ AB j ) + C ya ( ρ AB j ) + · · · + C ya ( ρ AB jm − ) (cid:17) (12)5 - - - - - H y L FIG. 2: f ( y ) is the value of C ya ( | ψ i A | B B B ). Solid (red) line is the exact value of C ya ( | ψ i A | B B B ), dashed (blue) line is the upper bound of C ya ( | ψ i A | B B B ) in (12), anddotdashed (green) line is the upper bound in [13].for all y <
0, where ˜ M = m − . [Proof]. For y <
0, we have C ya ( ρ A | B j ··· B jm − ) ≤ C y ( ρ A | B j ··· B jm − ) < ˜ M (cid:16) C y ( ρ AB j ) + C y ( ρ AB j ) + · · · + C y ( ρ AB jm − ) (cid:17) = ˜ M (cid:16) C ya ( ρ AB j ) + C ya ( ρ AB j ) + · · · + C ya ( ρ AB jm − ) (cid:17) , (13)where we have used in the first inequality the relation a x ≤ b x for a ≥ b ≥ , x ≤
0. Thesecond inequality is due to (3). The equality is due to (7).As the factor ˜ M = m − is less than one, the inequality (12) is tighter than the one in[13]. This factor ˜ M depends on the number of partite N . Namely, for larger multipartitesystems, the inequality (12) gets even tighter than the one in [13]. Example 2 . Let us consider again the 4-qubit generlized W -class states (11). We have C ya ( | ψ i A | B B B ) = ( √ ) y . From our result (12) we have C ya ( | ψ i A | B B B ) ≤ ( ) y , while from(5) one gets C ya ( | ψ i A | B B B ) ≤ ) y . It can be seen that our result is better than that in[13] for y <
0, see Fig. 2.
Remark 1 . In (12) we have assumed that all C ( ρ AB ji ), i = 1 , , · · · , m −
1, arenonzero. In fact, if one of them is zero, the inequality still holds by removing thisterm from the inequality. Namely, if C ( ρ AB ji ) = 0 , then one has C ya ( ρ A | B j ··· B jm − ) < C ya ( ρ AB j ) + · · · + (cid:0) (cid:1) i − C ya ( ρ AB ji − ) + (cid:0) (cid:1) i C ya ( ρ AB ji +1 ) + · · · + (cid:0) (cid:1) m − C ya ( ρ AB jm − ) + (cid:0) (cid:1) m − C ya ( ρ AB jm − ). By cyclically permuting the sub-indices in B j · · · B j m − , we canget a set of inequalities. Summing up these inequalities we have C ya ( ρ A | B j ··· B jm − ) < m − (cid:16) C ya ( ρ AB j ) + · · · + C ya ( ρ AB ji − ) + C ya ( ρ AB ji +1 ) + · · · + C ya ( ρ AB jm − ) + C ya ( ρ AB jm − ) (cid:17) for y < MONOGAMY RELATIONS FOR NAGATIVITY OF ASSISTANCE
Another well-known quantifier of bipartite entanglement is the negativity. Given a bi-partite state ρ AB in H A ⊗ H B , the negativity is defined by [21], N ( ρ AB ) = ( || ρ T A AB || − / ρ T A AB is the partial transpose with respect to the subsystem A , || X || denotes the tracenorm of X , i.e || X || = Tr √ XX † . Negativity is a computable measure of entanglement,and is a convex function of ρ AB . It vanishes if and only if ρ AB is separable for the 2 ⊗ ⊗ N ( ρ AB ) = || ρ T A AB || −
1. For any bipartite pure state | ψ i AB , the negativity N ( ρ AB )is given by N ( | ψ i AB ) = 2 P i 1, where λ i are the eigenvalues for thereduced density matrix of | ψ i AB . For a mixed state ρ AB , the convex-roof extended negativity(CREN) is defined as N c ( ρ AB ) = min X i p i N ( | ψ i i AB ) , (14)where the minimum is taken over all possible pure state decompositions { p i , | ψ i i AB } of ρ AB .CREN gives a perfect discrimination of positive partial transposed bound entangled statesand separable states in any bipartite quantum systems [23, 24]. For a mixed state ρ AB , theconvex-roof extended negativity of assistance (CRENOA) is defined as [25] N a ( ρ AB ) = max X i p i N ( | ψ i i AB ) , (15)where the maximum is taken over all possible pure state decompositions { p i , | ψ i i AB } of ρ AB .Let us consider the relation between CREN and concurrence. For any bipartite purestate | ψ i AB in a d ⊗ d quantum system with Schmidt rank 2, | ψ i AB = √ λ | i + √ λ | i ,one has N ( | ψ i AB ) = k | ψ ih ψ | T B k − √ λ λ = p − Tr ρ A ) = C ( | ψ i AB ). In otherwords, negativity is equivalent to concurrence for any pure state with Schmidt rank 2, and7onsequently it follows that for any two-qubit mixed state ρ AB = P p i | ψ i i AB h ψ i | , N c ( ρ AB ) = min X i p i N ( | ψ i i AB ) (16)= min X i p i C ( | ψ i i AB )= C ( ρ AB ) ,N a ( ρ AB ) = max X i p i N ( | ψ i i AB ) (17)= max X i p i C ( | ψ i i AB )= C a ( ρ AB ) , where the minimum and the maximum are taken over all pure state decompositions { p i , | ψ i i AB } of ρ AB .Combing (7), (16) and (17), we can get the following Lemma. [Lemma 1]. For N -qubit generlized W -class states (6), we have N c ( ρ AB i ) = N a ( ρ AB i ) . (18)As is already known, the negativity satisfies the monogamy relation for N-qubit purestate [25]. In fact, for any N-qubit state, the monogamy relation of the negativity alwaysholds. Therefore, we can get the following Lemma. [Lemma 2]. For any N-qubit state ρ ∈ H A ⊗ H B ⊗ · · · ⊗ H B N − , we have N xc ( ρ A | B ··· B N − ) ≥ N − X i =1 N xc ( ρ AB i ) , x ≥ . (19) [Proof]. From Ref [25], one has N c ( | ψ i A | B ··· B N − ) ≥ N − X i =1 N c ( ρ AB i ) , (20)for N-qubit pure state. Applying the similar approach in Ref [19], one can get N xc ( | ψ i A | B ··· B N − ) ≥ N − X i =1 N xc ( ρ AB i ) , (21)for N-qubit pure state with x ≥ 2. 8et ρ = P i p i | ψ i i AB ··· B N − h ψ i | be the optimal decomposition of N c ( ρ A | B ··· B N − ) for theN-qubit mixed state, we have N xc ( ρ A | B ··· B N − ) = X i =1 p i N c ( | ψ i A | B ··· B N − ) ! x (22) ≥ X i =1 p i vuut N − X k =1 N c ( ρ AB k ) x ≥ X k X i p i N c ( ρ AB k ) ! x ≥ N − X i =1 N xc ( ρ AB i ) , where the first inequality is due to (20). The second inequality is due to Minkowski in-equality: ( P k ( P i x ik )) ≤ P i ( P k x ik ) . The last inequality is due to ( P i a i ) α ≥ P i a αi for a i ≥ , α ≥ Lemma 3] . For any N-qubit state ρ ∈ H A ⊗ H B ⊗ · · · ⊗ H B N − , if N c ( ρ AB i ) ≥ N c ( ρ A | B i +1 ··· B N − ) for i = 1 , , · · · , m , and N c ( ρ AB j ) ≤ N c ( ρ A | B j +1 ··· B N − ) for j = m +1 , · · · , N − ∀ ≤ m ≤ N − N ≥ 4, we have N xc ( ρ A | B B ··· B N − ) ≥ N xc ( ρ AB ) (23)+ x N xc ( ρ AB ) + · · · + (cid:16) x (cid:17) m − N xc ( ρ AB m )+ (cid:16) x (cid:17) m +1 ( N xc ( ρ AB m +1 ) + · · · + N xc ( ρ AB N − ))+ (cid:16) x (cid:17) m N xc ( ρ AB N − )for all x ≥ [Proof]. From (19), one has N c ( ρ A | BC ) ≥ N c ( ρ AB ) + N c ( ρ AC ) . If N c ( ρ AB ) ≥ N c ( ρ AC ), wehave N xc ( ρ A | BC ) ≥ ( N c ( ρ AB ) + N c ( ρ AC )) x = N xc ( ρ AB ) (cid:18) N c ( ρ AC ) N c ( ρ AB ) (cid:19) x (24) ≥ N xc ( ρ AB ) " x (cid:18) N c ( ρ AC ) N c ( ρ AB ) (cid:19) x = N xc ( ρ AB ) + x N xc ( ρ AC ) , where the second inequality is due to the inequality (1 + t ) x ≥ xt ≥ xt x for x ≥ , ≤ t ≤ 1. 9y using the inequality (24) repeatedly, one gets N xc ( ρ A | B B ··· B N − ) ≥ N xc ( ρ AB ) + x N xc ( ρ A | B ··· B N − ) (25) ≥ N xc ( ρ AB ) + x N xc ( ρ AB ) + (cid:16) x (cid:17) N xc ( ρ A | B ··· B N − ) ≥ · · · ≥ N xc ( ρ AB ) + x N xc ( ρ AB ) + · · · + (cid:16) x (cid:17) m − N xc ( ρ AB m )+ (cid:16) x (cid:17) m N xc ( ρ A | B m +1 ··· B N − ) . As N c ( ρ AB j ) ≤ N c ( ρ A | B j +1 ··· B N − ) for j = m + 1 , · · · , N − 2, by (24) we get N xc ( ρ A | B m +1 ··· B N − ) ≥ x N xc ( ρ AB m +1 ) + N xc ( ρ A | B m +2 ··· B N − ) ≥ x N xc ( ρ AB m +1 ) + · · · + N xc ( ρ AB N − )) + N xc ( ρ AB N − ) . (26)Combining (25) and (26), we have Lemma 3.We can also derive a bound of N xc ( ρ A | B B ··· B N − ) for x < [Lemma 4]. For any N-qubit state ρ ∈ H A ⊗ H B ⊗ · · · ⊗ H B N − , we have N xc ( ρ A | B B ··· B N − ) < M ′ (cid:0) N xc ( ρ AB ) + N xc ( ρ AB ) + · · · + N xc ( ρ AB N − ) (cid:1) (27)for all x < 0, where M ′ = N − . [Proof]. For arbitrary tripartite state, from (19) we have N xc ( ρ A | B B ) ≤ (cid:0) N c ( ρ AB ) + N c ( ρ AB ) (cid:1) x (28)= N xc ( ρ AB ) (cid:18) N c ( ρ AB ) N c ( ρ AB ) (cid:19) x < N xc ( ρ AB ) , where the first inequality is due to x < (cid:16) N c ( ρ AB ) N c ( ρ AB ) (cid:17) x < . On the other hand, we have N xc ( ρ A | B B ) ≤ (cid:0) N c ( ρ AB ) + N c ( ρ AB ) (cid:1) x (29)= N xc ( ρ AB ) (cid:18) N c ( ρ AB ) N c ( ρ AB ) (cid:19) x < N xc ( ρ AB ) . From (28) and (29) we obtain N xc ( ρ A | B B ) < 12 ( N xc ( ρ AB ) + N xc ( ρ AB )) . (30)10y using the inequality (30) repeatedly, one gets N xc ( ρ A | B B ··· B N − ) < (cid:0) N xc ( ρ AB ) + N xc ( ρ A | B ··· B N − ) (cid:1) (31) < N xc ( ρ AB ) + (cid:18) (cid:19) N xc ( ρ AB ) + (cid:18) (cid:19) N xc ( ρ A | B ··· B N − ) < · · · < N xc ( ρ AB ) + (cid:18) (cid:19) N xc ( ρ AB ) + · · · + (cid:18) (cid:19) N − N xc ( ρ AB N − ) + (cid:18) (cid:19) N − N xc ( ρ AB N − ) . By cyclically permuting the sub-indices B , B , · · · , B N − in (31) we can get a set ofinequalities. Summing up these inequalities we obtain (27).In the following, we study the monogamy property of the CRENOA for the N -qubitgeneralized W -class states (6). We can obtain the following theorem.[ Theorem 4] . For the N -qubit generalized W -class states | ψ i ∈ H A ⊗ H B ⊗ · · · ⊗ H B N − , with ρ AB j ··· B jm − the m -qubit, 2 ≤ m ≤ N , reduced density matrix of | ψ i . If N c ( ρ AB ji ) ≥ N c ( ρ AB ji +1 ··· B jm − ) for i = 1 , , · · · t , and N c ( ρ ABj k ) ≤ N c ( ρ AB jk +1 ··· B jm − ) for k = t + 1 , · · · , m − ∀ ≤ t ≤ m − , m ≥ 4, then the CRENOA satisfies N xa ( ρ A | B j ··· B jm − ) ≥ N xa ( ρ AB j )+ x N xa ( ρ AB j ) + · · · + (cid:16) x (cid:17) t − N xa ( ρ AB jt )+ (cid:16) x (cid:17) t +1 (cid:16) N xa ( ρ AB jt +1 ) + · · · + N xa ( ρ AB jm − ) (cid:17) + (cid:16) x (cid:17) t N xa ( ρ AB jm − ) (32)for all x ≥ [Proof]. For the N -qubit generalized W -class states | ψ i , according to the definitions of11 c ( ρ ) and N a ( ρ ), one has N a ( ρ A | B j ··· B jm − ) ≥ N c ( ρ A | B j ··· B jm − ). When x ≥ 2, we have N xa ( ρ A | B j ··· B jm − ) ≥ N xc ( ρ A | B j ··· B jm − ) ≥ N xc ( ρ AB j )+ x N xc ( ρ AB j ) + · · · + (cid:16) x (cid:17) t − N xc ( ρ AB jt )+ (cid:16) x (cid:17) t +1 (cid:16) N xc ( ρ AB jt +1 ) + · · · + N xc ( ρ AB jm − ) (cid:17) + (cid:16) x (cid:17) t N xc ( ρ AB jm − )= N xa ( ρ AB j ) + x N xa ( ρ AB j ) + · · · + (cid:16) x (cid:17) t − N xa ( ρ AB jt )+ (cid:16) x (cid:17) t +1 (cid:16) N xa ( ρ AB jt +1 ) + · · · + N xa ( ρ AB jm − ) (cid:17) + (cid:16) x (cid:17) t N xa ( ρ AB jm − ) , (33)where we have used in the first inequality the relation a x ≥ b x for a ≥ b ≥ , x ≥ 2. Usingthe result of Lemma 3, one gets the second inequality. The equality is due to the Lemma 2.In Theorem 4 we have assumed that some N c ( ρ AB ji ) ≥ N c ( ρ AB ji +1 ··· B jm − ) and some N c ( ρ AB jk ) ≤ N c ( ρ AB jk +1 ··· B jm − ) for the N -qubit generalized W -class states. If all N c ( ρ AB ji ) ≥ N c ( ρ AB ji +1 ··· B jm − ) for i = 1 , , · · · , m − 2, then we have the following con-clusion: [Theorem 5] . If N c ( ρ AB ji ) ≥ N c ( ρ AB ji +1 ··· B jm − ) for i = 1 , , · · · , m − 2, we have N xa ( ρ A | B j ··· B jm − ) ≥ N xa ( ρ AB j ) + x N xa ( ρ AB j ) + · · · + (cid:16) x (cid:17) m − N xa ( ρ AB jm − ) (34)for all x ≥ N ya ( ρ AB ··· B N − ) for y < Theorem 6] . For the N -qubit generalized W -class states | ψ i ∈ H A ⊗ H B ⊗ · · · ⊗ H B N − with N c ( ρ AB ji ) = 0 for 1 ≤ i ≤ m − 1, we have N ya ( ρ A | B j ··· B jm − ) < ˜ M (cid:16) N ya ( ρ AB j ) + N ya ( ρ AB j ) + · · · + N ya ( ρ AB jm − ) (cid:17) (35)for all y < 0, where ˜ M = m − . [Proof]. For y < 0, we have N ya ( ρ A | B j ··· B jm − ) ≤ N yc ( ρ A | B j ··· B jm − ) < ˜ M (cid:16) N yc ( ρ AB j ) + N yc ( ρ AB j ) + · · · + N yc ( ρ AB jm − ) (cid:17) = ˜ M (cid:16) N ya ( ρ AB j ) + N ya ( ρ AB j ) + · · · + N ya ( ρ AB jm − ) (cid:17) , (36)12here we have used in the first inequality the relation a x ≤ b x for a ≥ b ≥ , x ≤ 0. Thesecond inequality is based on Lemma 4. The equality is due to the Lemma 2. Remark 2 . In (35) we have assumed that all N c ( ρ AB ji ), i = 1 , , · · · , m − 1, arenonzero. In fact, if one of them is zero, the inequality still holds if one simply removesthis term from the inequality. Namely, if N c ( ρ AB ji ) = 0 , then one has N ya ( ρ A | B j ··· B jm − ) < N ya ( ρ AB j ) + · · · + (cid:0) (cid:1) i − N ya ( ρ AB ji − ) + (cid:0) (cid:1) i N ya ( ρ AB ji +1 ) + · · · + (cid:0) (cid:1) m − N ya ( ρ AB jm − ) + (cid:0) (cid:1) m − N ya ( ρ AB jm − ). By cyclically permuting the sub-indices in B j · · · B j m − , we canget a set of inequalities. Summing up these inequalities we have N ya ( ρ A | B j ··· B jm − ) < m − (cid:16) N ya ( ρ AB j ) + · · · + N ya ( ρ AB ji − ) + N ya ( ρ AB ji +1 ) + · · · + N ya ( ρ AB jm − ) + N ya ( ρ AB jm − ) (cid:17) ,for y < CONCLUSION Entanglement monogamy is a fundamental property of multipartite entangled states. Wehave presented tighter monogamy inequalities for the x -power of concurrence of assistance C xa ( ρ A | B j ··· B jm − ) of the m -qubit reduced density matrices, 2 ≤ m ≤ N , for the N -qubitgeneralized W -class states, when x ≥ 2. A tighter upper bound of y -power of concurrenceof assistance is also derived for y < 0. The monogamy relations for the x -power of negativ-ity of assistance for the N -qubit generalized W -class states have been also investigated for x ≥ x < 0, respectively. These relations give rise to the restrictions of entanglementdistribution among the qubits in generalized W -class states. It should be noted that en-tanglement of assistances like concurrence of assistance and negativity of assistance are notgenuine measures of quantum entanglement. They quantify the maximum average amountof entanglement between two parties, Alice and Bob, which can be extracted given assistancefrom a third party, Charlie, by performing a measurement on his system and reporting themeasurement outcomes to Alice and Bob. Nevertheless, similar to quantum entanglement,we see that the entanglement of assistances also satisfy certain monogamy relations. Acknowledgments This work is supported by the NSF of China under Grant No.11675113. 13 1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cam-bridge: Cambridge University Press, 2000.[2] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).[3] F. Mintert, M. 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