Improved monogamy relations with concurrence of assistance and negativity of assistance for multiqubit W-class states

# Improved monogamy relations with concurrence of assistance and negativity of assistance for multiqubit W-class states

Zhi-Xiang Jin    Shao-Ming Fei    Xianqing Li-Jost School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China
###### Abstract

Monogamy relations characterize the distributions of entanglement in multipartite systems. We investigate the monogamy relations satisfied by the concurrence of assistance and the negativity of assistance for multiqubit generalized -class states. Analytical monogamy inequalities are presented for both concurrence of assistance and negativity of assistance, which are shown to be tighter than the existing ones. Detailed examples have been presented.

## I Introduction

Quantum entanglement MAN ; RPMK ; FMA ; KSS ; HPB ; HPBO ; JIV ; CYSG is an essential feature of quantum mechanics. As one of the fundamental differences between quantum entanglement and classical correlations, a key property of entanglement is that a quantum system entangled with one of the subsystems limits its entanglement with the remaining subsystems. The monogamy relations characterize the distribution of quantum entanglement in multipartite systems. Monogamy property is also an essential feature allowing for security in quantum key distribution MP .

For a tripartite system , and , the authors ckw61052306 show that there is a trade-off between entanglement with and its entanglement with . In kw69022309 , the authors present a simple identity which captures the trade-off between entanglement and classical correlation, which can be used to derive rigorous monogamy relations. They also proved various monogamous trade-off relations for other entanglement measures and correlation measures. In tf96.220503 , the author proved the longstanding conjecture of Coffman, Kundu, and Wootters ckw61052306 that the distribution of bipartite quantum entanglement, measured by the tangle , amongst qubits satisfies a tight inequality: , where denotes the bipartite quantum entanglement measured by the tangle under the bipartition and .

Recently, the monogamy of entanglement for multiqubit -class states has been investigated, and the monogamy relations for tangle and the squared concurrence have been presented in Ref. KSJ ; KSJB . In Ref. ZF , the general monogamy inequalities of the -th power of concurrence and entanglement of formation are presented for -qubit states. However, the concurrence of assistance does not satisfy monogamy relations for general quantum states. Therefore, special classes of quantum states have been taken into account for monogamy relations satisfied by the concurrence of assistance. The monogamy relations for the -power of concurrence of assistance for the generalized multiqubit -class states have been derived in ZXN . In jzx1 , a tighter monogamy relation of quantum entnglement for multiqubit -class states has been presented.

In this paper, we show that the monogamy inequalities for concurrence of assistance obtained so far can be further improved. We present entanglement monogamy relations for the -th power of the concurrence of assistance, which are tighter than those in ZXN ; jzx1 and give rise to finer characterizations of the entanglement distributions among the multipartite -class states. Moreover, we present the general monogamy relations for the -power of negativity of assistance for the generalized multiqubit -class states, which are also better than that in ZXN ; jzx1 .

## Ii improved MONOGAMY RELATIONS FOR CONCURRENCE OF ASSISTANCE

Let denote a discrete finite dimensional complex vector space associated with a quantum subsystem . For a bipartite pure state in vector space , the concurrence is given by AU ; PR ; SA , where is the reduced density matrix, . The concurrence for a bipartite mixed state is defined by the convex roof extension , where the minimum is taken over all possible pure state decompositions of , with and and .

For an -qubit state , the concurrence of the state , viewed as a bipartite state under the bipartition and , satisfies ZF

 Cx(ρA|B1,B2⋯,BN−1)≥Cx(ρAB1)+Cx(ρAB2)+⋯+Cx(ρABN−1),

for , where . The above monogamy relation is improved such that for , if for , and for , , , then JZX ,

 Cx(ρA|B1B2⋯BN−1)≥ Cx(ρAB1)+x2Cx(ρAB2)+⋯+(x2)m−1Cx(ρABm) +(x2)m+1(Cx(ρABm+1)+⋯+Cx(ρABN−2)) +(x2)mCx(ρABN−1). (1)

In jll , (II) is further improved such that for , one has

 Cx(ρA|B1B2⋯BN−1)≥ Cx(ρAB1)+hCx(ρAB2)+⋯+hm−1Cx(ρABm) +hm+1(Cx(ρABm+1)+⋯+Cx(ρABN−2)) +hmCx(ρABN−1) (2)

for all , where .

For a tripartite pure state , the concurrence of assistance is defined by TFS ; YCS

 Ca(|ψ⟩ABC)≡Ca(ρAB)=max{pi,|ψi⟩}∑ipiC(|ψi⟩),

where the maximum is taken over all possible decompositions of When is a pure state, then one has .

Different from the Coffman-Kundu-Wootters inequality satisfied by the concurrence, the concurrence of assistance does not satisfy the monogamy relations in general. However, for -qubit generalized -class state, defined by

 |ψ⟩AB1⋯BN−1=a|10⋯0⟩+b1|01⋯0⟩+⋯+bN−1|00⋯1⟩, (3)

with , one has ZXN ,

 C(ρABi)=Ca(ρABi),    i=1,2,...,N−1, (4)

where , and the concurrence of assistance satisfies the monogamy inequality ZXN ,

 Cxa(|ψ⟩A|B1,B2⋯,BN−1)≥Cxa(ρAB1)+Cxa(ρAB2)+⋯+Cxa(ρABN−1), (5)

for . (5) has been further improved such that for , if for , and for , , , then jzx1 ,

 Cxa(|ψ⟩A|B1B2⋯BN−1)≥ Cxa(ρAB1)+x2Cxa(ρAB2)+⋯+(x2)m−1Cxa(ρABm) +(x2)m+1(Cxa(ρABm+1)+⋯+Cxa(ρABN−2)) +(x2)mCxa(ρABN−1). (6)

In fact, as a kind of characterization of the entanglement distribution among the subsystems, the monogamy inequalities satisfied by the concurrence of assistance can be further refined and become tighter.

[Theorem 1]. Let denote the -qubit reduced density matrix of the -qubit generalized -class state . If for , and for , , the concurrence of assistance satisfies

 Cxa(ρA|Bj1⋯Bjm−1)≥Cxa(ρABj1) +hCxa(ρABj2)+⋯+ht−1Cxa(ρABjt) +ht+1(Cxa(ρABjt+1)+⋯+Cxa(ρABjm−2)) +htCxa(ρABjm−1) (7)

for all , where .

[Proof]. For the -qubit generalized -class states , according to the definitions of and , one has . When , we have

 Cxa(ρA|Bj1⋯Bjm−1) ≥Cx(ρA|Bj1⋯Bjm−1)≥Cx(ρABj1) (8) +hCx(ρABj2)+⋯+ht−1Cx(ρABjt) +ht+1(Cx(ρABjt+1)+⋯+Cx(ρABjm−2)) +htCx(ρABjm−1) =Cxa(ρABj1)+hCxa(ρABj2)+⋯+ht−1Cxa(ρABjt) +ht+1(Cxa(ρABjt+1)+⋯+Cxa(ρABjm−2)) +htCxa(ρABjm−1),

where we have used in the first inequality the relation for . The second inequality is due to (II). The equality is due to (4).

As for , for all , comparing with the monogamy relations for concurrence of assistance (5) and (II), our formula (II) in Theorem 1 gives a tighter monogamy relation with larger lower bound. In Theorem 1 we have assumed that some and some for the -qubit generalized -class states. If all for , then we have the following conclusion:

[Theorem 2]. If for , we have

 Cxa(ρA|Bj1⋯Bjm−1)≥Cxa(ρABj1)+hCxa(ρABj2)+⋯+hm−2Cxa(ρABjm−1) (9)

for all , where .

Example 1. Let us consider the 4-qubit generlized -class state,

 |W⟩AB1B2B3=12(|1000⟩+|0100⟩+|0010⟩+|0001⟩). (10)

We have . From our result (II) we have , from (II) one has , and from (5) one has , . One can see that our result is better than that in ZXN and jzx1 for , see Fig. 1.

## Iii Improved Monogamy Relations for Negativity of Assistance

Another well-known quantifier of bipartite entanglement is the negativity. Given a bipartite state in , the negativity is defined by GRF , , where is the partially transposed with respect to the subsystem , denotes the trace norm of , i.e . Negativity is a computable measure of entanglement, and is a convex function of . It vanishes if and only if is separable for the and systems MPR . For the purpose of discussion, we use the following definition of negativity, . For any bipartite pure state , the negativity is given by , where are the eigenvalues for the reduced density matrix of . For a mixed state , the convex-roof extended negativity (CREN) is defined by

 Nc(ρAB)=min∑ipiN(|ψi⟩AB), (11)

where the minimum is taken over all possible pure state decompositions of . CREN gives a perfect discrimination of positive partial transposed bound entangled states and separable states in any bipartite quantum systems PH ; WJM . For a mixed state , the convex-roof extended negativity of assistance (CRENOA) is defined as JAB

 Na(ρAB)=max∑ipiN(|ψi⟩AB), (12)

where the maximum is taken over all possible pure state decompositions of .

For an -qubit state , we denote the negativity of the state , viewed as a bipartite state under the partition and . If for , and for , , , then jzx1

 Nxc(ρA|B1B2⋯BN−1) ≥Nxc(ρAB1)+x2Nxc(ρAB2)+⋯+(x2)m−1Nxc(ρABm) +(x2)m+1(Nxc(ρABm+1)+⋯+Nxc(ρABN−2)) +(x2)mNxc(ρABN−1), (13)

for all . The inequality (III) is further improved that for jll

 Nxc(ρA|B1B2⋯BN−1) ≥Nxc(ρAB1)+hNxc(ρAB2)+⋯+hm−1Nxc(ρABm) +hm+1(Nxc(ρABm+1)+⋯+Nxc(ρABN−2)) +hmNxc(ρABN−1) (14)

where .

The negativity of assistance does not satisfy a monogamy relation in general. However, for an -qubit generlized -class state , if for , and for , , , then the negativity of assistance of the state satisfies the inequality jzx1 ,

 Nxa(|ψ⟩A|B1B2⋯BN−1)≥ Nxa(ρAB1)+x2Nxa(ρAB2)+⋯+(x2)m−1Nxa(ρABm) +(x2)m+1(Nxa(ρABm+1)+⋯+Nxa(ρABN−2)) +(x2)mNxa(ρABN−1), (15)

for all .

In fact, to have a better characterization of the entanglement distribution among the subsystems, the monogamy inequalities satisfied by the negativity of assistance can be further refined and become tighter. Taking into account the fact that for -qubit generlized -class states (3) jzx1 ,

 Nc(ρABi)=Na(ρABi),  i=1,2,⋯,N−1 (16)

where , we have the following relations:

[Theorem 3]. For the -qubit generalized -class states , if for , and for , , then the CRENOA satisfies

 Nxa(ρA|Bj1⋯Bjm−1)≥Nxa(ρABj1) +hNxa(ρABj2)+⋯+ht−1Nxa(ρABjt) +ht+1(Nxa(ρABjt+1)+⋯+Nxa(ρABjm−2)) +htNxa(ρABjm−1) (17)

for all , where .

[Proof]. For the -qubit generalized -class states , according to the definitions of and , one has . When , we have

 Nxa(ρA|Bj1⋯Bjm−1) ≥Nxc(ρA|Bj1⋯Bjm−1)≥Nxc(ρABj1) (18) +hNxc(ρABj2)+⋯+ht−1Nxc(ρABjt) +ht+1(Nxc(ρABjt+1)+⋯+Nxc(ρABjm−2)) +htNxc(ρABjm−1) =Nxa(ρABj1)+hNxa(ρABj2)+⋯+ht−1Nxa(ρABjt) +ht+1(Nxa(ρABjt+1)+⋯+Nxa(ρABjm−2)) +htNxa(ρABjm−1),

where we have used in the first inequality the relation for . Due to the (III), one gets the second inequality. The equality is due to relation (16).

As for , for all , comparing with the monogamy relations for CRENOA in (III), our formula (III) in Theorem 3 gives a tighter monogamy relation with larger lower bounds. In Theorem 3 we have assumed that some and some for the -qubit generalized -class states. If all for , then we have the following conclusion:

[Theorem 4]. If for , we have

 Nxa(ρA|Bj1⋯Bjm−1)≥Nxa(ρABj1)+hNxa(ρABj2)+⋯+hm−2Nxa(ρABjm−1) (19)

for all , where .

Example 2. Let us consider the -qubit pure -class state,

 |W⟩AB1⋯BN−1=1√N(|10⋯0⟩+|01⋯0⟩+⋯+|00⋯1⟩). (20)

It is straightforword to check: , . Let us choose . Then . From our result (III) we have , while from (III) one has , . Obviously, our result is better than that in jzx1 with , see Fig. 2.

## Iv conclusion

Entanglement monogamy is a fundamental property of multipartite entangled states. We have presented tighter monogamy inequalities for the -power of concurrence of assistance of the -qubit reduced density matrices, , for the -qubit generalized -class states, when . The monogamy relations for the -power of negativity of assistance for the -qubit generalized -class states have been also investigated for . These relations give rise to the restrictions of entanglement distribution among the qubits in generalized -class states. It should be noted that entanglement of assistances like concurrence of assistance and negativity of assistance are not genuine measures of quantum entanglement. They quantify the maximum average amount of entanglement between two parties, Alice and Bob, which can be extracted given assistance from a third party, Charlie, by performing a measurement on his system and reporting the measurement 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.

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