Tao Li(李陶), Jing-Yi Zhou(周靜怡), Qi Sun(孫琪),?,Zhi-Xiang Jin(靳志祥), Deng-Feng Liang(梁登峰), and Ting Luo(羅婷)

1School of Mathematics and Statistics,Beijing Technology and Business University,Beijing 100048,China

2School of Computer Science and Techonology,Dongguan University of Technology,Dongguan 523808,China

3People’s Public Security University of China,Academy of Information Network Security,Beijing 100038,China

Keywords: monogamy relation,polygramy relation

1.Introduction

In quantum information theory,entanglement is a vital resource due to its importance in quantum communication and quantum information processing.Although entanglement in bipartite quantum systems has been extensively studied with various applications, entanglement in multipartite quantum systems is much more complicated and the corresponding results are relatively few.One distinguishing feature of quantum entanglement,as compared to classical correlations,is that it cannot be freely distributed among the subsystems.For example, it is not possible to prepare three qubits in a way that any two qubits are maximally entangled.This property is called monogamy of entanglement,[1,2]quantitatively,E(ρA|BC)≥E(ρAB)+E(ρAC),whereEis a bipartite entanglement measure,ρABandρACare the reduced density matrices ofρABC.Futhermore, the monogamy property has emerged as the ingredient in some practical applications such as quantum cryptography, quantum teleportation, quantum computation,quantum key distribution.[3-5]

For the systems of three qubits, the first monogamy inequality was established by using concurrence[6]to quantify the shared entanglement among subsystems.However,it is failed in its generalization for higher-dimensional quantum systems.Later, an interesting observation was made showing that a entanglement measure may not satisfy the monogamy relations in itself,but satisfied after theα-th power of it.For example, it has been proved in Ref.[7] that thexth power of the entanglement of formation and concurrence satisfied the monogamy inequalities forandx ≥2,respectively.Many finer monogamy inequalities of multiqubit systems based on non-negative power are available in Refs.[8-11].Recently,the authors[12]introduced the concept of monogamy relations without inequalities.

Monogamy inequality is about the restricted sharability of multipartite entanglement, providing a lower bound of entanglement, while its dual concept to the sharable entanglement, is known to have a polygamous property, which gives an upper bound in multipartite quantum systems.It is mathematically characterized asEa(ρABC)≤Ea(ρAB)+Ea(ρAC)for a three-party quantum stateρABC,where the customary notion“Ea(·)” stands for the assisted entanglement.The polygamy inequality was first given in three-qubit systems using tangle of assistance[13,14]and generalized into multiqubit systems in terms of various assisted entanglements.[14-18]In recent years,monogamy and polygamy inequalities of multi-qubit entanglement have been further studied and extended in terms of non-negative power of entanglement measures and assisted entanglements.Moreover,the monogamy relations for theα-th(0≤α ≤1) power and the polygamy relations for theβ-th(β ≥1)power of unified-(q,s)entanglement and unified-(q,s)entanglement of assistance are obtained in Ref.[16].

In the present paper, complementary monogamy and polygamy inequalities among multipartite systems are investigated.We provide general monogamy inequalities forγ-th(0≤γ ≤α,α ≥1) power of quantum entanglement based on unified-(q,s) entanglement andδ-th (δ ≥β,0≤β ≤1)power of entanglement of assistance based on unified-(q,s)entanglement of assistance, which turn out to be tighter than the existing ones in literature.As a by-product,we derive the corresponding monogamy relations of specific quantum correlations such as entanglement of formation, Renyi-qentanglement and Tsallis-qentanglement by setting the parametersqands.We take Tsallis-2 entanglement as an example to illustrate in detail.Applying the unified-(q,s) entanglement of assistance to specific quantum correlations,e.g., Renyi-qentanglement of assistance and Tsallis-qentanglement of assistance, the corresponding new class of polygamy relations is obtained,which are complementary to the existing ones[16]with different regions of parameterδ.

2.Monogamy of multiqubit relations for unified entanglement

Due to the flexibility in parameter selection, the unified-(q,s)entropy is interesting both in theory and in applications and has been studied in various fields.For any quantum stateρ ∈?,the unified-(q,s)entropy is defined as[19,20]for eachq,s ≥0, where the maximum is taken over all possible pure state decompositions ofρAB=∑i pi|ψ〉A(chǔ)B〈ψ| and∑i pi=1.

Similarly, as UEoA in Eq.(5) is continuous for the parametersqandsassures that UEoA reduces to Renyi-qentanglement of assistance (REoA)[23]and Tsallis-qentanglement of assistance(TEoA)[18]whenstends to 0 or 1, respectively.For any nonnegatives, withqtends to 1, UEoA reduces to entanglement of assistance(EoA)[24]

Using UE in Eq.(3)to quantify bipartite quantum entanglement, the monogamy inequality was established in multiqubit systems; for anyN-qubit stateρA1A2···ANand its twoqubit reduced density matricesρA1Aiwithi=2,...,N,we have

forq ≥2, 0≤s ≤1,andqs ≤3.[17]Furthermore,in Ref.[16]the authors presented monogamy inequality based on UE in multiqubit systems as

for 0≤α ≤1,q ≥2 and 0≤s ≤1,qs ≤3.

It was also shown that UEoA can be used to characterize the polygamy of multiqubit entanglement as[24]

for any multiqubit stateρA1A2···AN.

3.Tighter monogamy relations for multiqubit for unified entropy

In this section, a corresponding new class of polygamy relations is present which are complementary to the existing ones.We start with the following lemma which gives a useful inequality in the proof of the main theorems.

Lemma 1For any real numbersxandt,if 0≤x ≤1 andt ≥k ≥1,then we have

Proof Letk ≥1.First, we can construct a binary functionf(x,y)=(1+y)x-yxwith 0≤x ≤1,0＜y ≤1/k.Then,based on the fact that

(II) Suppose thatkEq,s(ρAB)≤Eq,s(ρAC).WhenEq,s(ρAB)=0,since 0＜K(α,γ)≤1,the inequality(16)follows from expression (17).WhenEq,s(ρAB)＞0, using Lemma 1 fort=(Eq,s(ρAC))α/(Eq,s(ρAB))αandx=γ/αyields expression(16).

We point out that the monogamy inequalities given by Theorem 1 can be regarded as a complement to the previous work[16]in a sense that the value region of the parameterγfor UE is different.Stated briefly, it is well known that some quantum entanglement measuresEsuch as UE[17]satisfies the original monogamy relations, sayE(ρA|BC)≥E(ρAB)+E(ρAC),while others,such as concurrence,does not satisfy the monogamy relations itself, but satisfies after some certain powers of it.For instance, as can be seen in Ref.[9],there exists a real numberαsuch that for any quantum correlation measureQ,Qxis monogamous ifxbelongs to the interval [α,+∞).Different from that results, the monogamy relations (15) holds for the interval [0,α].So, in this sense,Theorem 1 gives monogamy inequalities that are complementary to the existing ones with different regions of the parameterγfor UE.

More specifically, by using Theorem 1 repeatedly, we have the following theorem for multipartite quantum systems.

Theorem 2 Let 0≤γ ≤α,q ≥2, 0≤s ≤1,qs ≤3,K(α,γ)=[(1+k)γ/α-1]/kγ/α, and letρAB1···BN-1be anyNqubit state withkEq,s(ρABi)≤Eq,s(ρA|Bi+1···BN-1) for everyi=1,...,N-2.Then it holds that

Theorem 2 gives a new class of monogamy relations for multiqubit states, which includes inequality (8) as a special case since inequality (18)reduces to inequality (8) whenα= 1 andk= 1.Fork ＞1, the inequality (18) is tighter than the inequality (8), as [(1+k)γ/α-1]/kγ/α ≥2γ/α-1,where the equality holds only forα=γ.Particularly, forEq,s(ρAB)≥Eq,s(ρAC)andα ≥1, takingγ=1/2,k=1, one has

Fig.1.The axis z represents the lower bounds of UE of|ψ〉A(chǔ)BC,which are functions of α,γ.The red surface represents the UE of the state|ψ〉,blue surface represents the lower bound from our result, green surface(just below the green one)represents the lower bound from the result in Ref.[16].

Although Theorem 2 gives a new class of monogamy relations for multiqubit states, however, its condition is not always satisfied.To get ride of the strict condition for inequality(18),we give out a universal monogamy inequality as follow.

Fig.2.The red surface represents the difference of the UE between inequalities(18)and(8)on the right side.The blue surface is zero plane of z.

Combining inequalities(21)and(22),we get Theorem 3.

Theorem 3 gives a general monogamy inequality satisfied by theγ-th power of UE for the case of 0＜γ ＜αandα ≥1 with less constriction.Specifically,ifγ=1/2 andk=α=1,we obtain the monogamy inequality established by the UE

which was absent in Ref.[16].

4.Polygamy relations for multiqubit systems

As a dual relation to monogamy inequality,polygamy inequality is one of the hot issues in the study of quantum information theory in recent years.Being an intriguing feature of quantum entanglement, it is also closely related to many quantum information and communication processing tasks.In this section, we will provide a class of polygamy inequalities in multiqubit systems based on UEoA,which are tighter than the existing ones.To this end,we shall first give the following Lemma.

Lemma 2 For any real numbersx,t,andk,ifx ≥1 andt ≥k,then we have

Proof Analogously to Lemma 1,construct a binary functionf(x,y)=(1+y)x-yxwithx ≥1, 0＜y ≤1/k.Obviously,f(x,y) is an increasing function ofydue to the fact that?f/?y=x[(1+y)x-1-yx-1]≥0.Therefore,f(x,y)≤f(x,1/k)=[(1+k)x-1]/kx.Sety=1/twitht ≥k, we obtain

is valid for allt ≥k.Fixedkand lettingtin inequality(26)go to+∞,we get inequality(24).

Using the similar method to the proof of Theorem 1 and Lemma 2,we have

Fig.3.The axis y denotes the upper bound of the UEoA of |ψ〉A(chǔ)BC,which are functions of x-th power of quantum relations.The red solid line represents the UEoA of |ψ〉A(chǔ)BC in Eq.(19), blue dashed line represents the upper bound of our result,green dot-dashed line represents the upper bound given in Ref.[16].

With a similar consideration of Theorem 3, we have the following widespread result with less constriction compared with Theorem 5.

Combining inequalities (34) and (35), we then obtain Theorem 6.

5.Conclusion

Entanglement monogamy and polygamy are two fundamental properties of multipartite entanglement.Based on unified-(q,s) entropy, we provide a characterization of complementary relations for multiqubit states.On the one hand,we derive a class of complementary monogamy inequalities for multiqubit entanglement based on theγ-th(0≤γ ≤α,α ≥1)power of unified-(q,s)entanglement, which turn out to be tighter than the previous results in Ref.[12].On the other hand,we established polygamy relations as a dual property of monogamy in terms of theδ-th(δ ≥β,0≤β ≤1)power of unified-(q,s)entanglement of assistance.

We mention that since unified-(q,s) entropy entanglement is a general bipartite entanglement measure, our work indeed gives a general class of the complementary monogamy and polygamy inequalities.By application of the results above, the corresponding monogamy and polygamy relations for special cases of unified-(q,s) entropy entanglement such as RE and TE can be obtained immediately.Furthermore,monogamy and polygamy relations can be interpreted as one class of distributions of entanglement in multipartite systems,from this point of view,tighter monogamy relations mean better characterizations of the entanglement distribution.Therefore, our results complement and unify the previous results for monogamy relations in literature,which also shed light on the study of the monogamy and polygamy inequalities about quantum correlations.

Acknowledgements

Project supported by the National Natural Science Foundation of China(Grant No.12175147),the Disciplinary Funding of Beijing Technology and Business University, the Fundamental Research Funds for the Central Universities (Grant No.2022JKF02015),and the Research and Development Program of Beijing Municipal Education Commission (Grant No.KM202310011012).