Monogamy of \alphath Power Entanglement Measurement in Qubit Systems

# Monogamy of αth Power Entanglement Measurement in Qubit Systems

Yu Luo    Yongming Li College of Computer Science, Shaanxi Normal University, Xi’an, 710062, China
July 17, 2019
###### Abstract

In this paper, we study the th power monogamy properties related to the entanglement measure in bipartite states. The monogamy relations related to the th power of negativity and the Convex-Roof Extended Negativity are obtained for -qubit states. We also give a tighter bound of hierarchical monogamy inequality for the entanglement of formation. We find that the GHZ state and W state can be used to distinguish both the th power of the concurrence for and the th power of the entanglement of formation for . Furthermore, we compare concurrence with negativity in terms of monogamy property and investigate the difference between them.

###### pacs:
03.67.a, 03.65.Ud, 03.65.Ta

identifier

## I Introduction

Multipartite entanglement is an important physical resource in quantum mechanics, which can be used in quantum computation, quantum communication and quantum cryptography. One of the most surprising phenomenon for multipartite entanglement is the monogamy property, which may be as fundamental as the no-cloning theorem Bruss99 (); Coffman00 (); Osborne06 (); Kay09 (). The monogamy property can be interpreted as the amount of entanglement between and , plus the amount of entanglement between and , cannot be greater than the amount of entanglement between and the pair . Monogamy property have been considered in many areas of physics: one can estimate the quantity of information captured by an eavesdropper about the secret key to be extracted in quantum cryptography Osborne06 (); Barrett05 (), the frustration effects observed in condensed matter physics Ma11 (), even in black-hole physics Susskind13 (); Lloyd14 ().

Historically, monogamy property of various entanglement measure have been discovered. Coffman first considered three qubits , and which may be entangled with each other Coffman00 (), who showed that the squared concurrence follows this monogamy inequality. Osborne proved the squared concurrence follows a general monogamy inequality for -qubit system Osborne06 (). Analogous to the Coffman-Kundu-Wootters (CKW) inequality, Ou proposed the monogamy inequality holds in terms of squared negativity  Ou07 (). Kim showed that the squared convex-roof extended negativity follows monogamy inequality Kim09 (). Oliveira and Bai investigated entanglement of formation(EoF) and showed that the squared EoF follows the monogamy inequality  Oliveira14 (); Bai1401 (). A natural question is why those monogamy property above are squared entanglement measure? In fact, Zhu showed that the th power of concurrence () and the th power of entanglement of formation () follow the general monogamy inequalities Zhu14 (). Sometimes, we can view the coefficient as a kind a of assigned weight to regulate the monogamy property Regula14 (); Salini14 ().

In this paper, we study the monogamy relations related to th power of some entanglement measures. We show that the th power of negativity and the th power of convex-roof extended negativity (CREN) follows the hierarchical monogamy inequality for  Bai1402 (). From the hierarchical monogamy inequality, the general monogamy inequalities related to and are obtained for -qubit states. We find that the GHZ state and W state can be used to distinguish the for , which situation was not clear in Zhu ’s paper Zhu14 (). We also prove the GHZ state and W state can be used to distinguish both the th power of EoF for . The hierarchical monogamy inequality for is also discussed, which improved Bai ’s result Bai1402 (); Bai1401 ().

This paper is organized as follows. In Sec. II,we study the monogamy property of th power of negativity. In Sec. III, we discuss the monogamy property of th power of CREN. In Sec. IV, we study the monogamy property of th power of EoF. In Sec. V, we compare the monogamy property of concurrence with negativity. We summarize our results in Sec. VI.

## Ii Monogamy of αth power of Negativity

Given a bipartite state in the Hilbert space . Negativity is defined as Vidal02 ():

 N(ρAB)=∥ρTAAB∥−12, (1)

where is the partial transpose with respect to the subsystem , denotes the trace norm of , i.e Negativity is a measure of entanglement, and which is a convex function of . if and only if is separable for the and systems Horodecki98 (). For the purposes of discussion , we use following definition of negativity:

 N(ρAB)=∥ρTAAB∥−1. (2)

For any maximally entangled state in two-qubit system, this definition of negativity is equal to 1.

For a bipartite pure state , the concurrence is defined as:

 C(|ψAB⟩)=√2[1−Tr(ρ2A)]=2√detρA, (3)

where is the reduced density matrix of subsystem A. For a mixed state , the concurrence can be defined as:

 C(ρAB)=min∑ipiC(|ψiAB⟩), (4)

where the minimum is taken over all possible pure state decompositions of

The next builds a relationship between negativity and concurrence in a system ().

Lemma 1 . For a pure state in a system (), the negativity of bipartition is equal to its concurrence: , where and

Proof: Based on the Schmidt decomposition, we can write the bipartition as: , where are Schmidt coefficients and . , are orthogonal basis for system and system respectively. The density operator , the partial transpose of with respect to system A is given by: . The negativity of is:

 NA|BC = ∥ρTAABC∥−1 (5) = ∥∑i,j√λiλj|ϕj∗A⟩⟨ϕi∗A|⊗|ϕiBC⟩⟨ϕjBC|∥−1 = ∥∑i,j√λiλj|ϕj∗A⟩⟨ϕjBC|⊗|ϕiBC⟩⟨ϕi∗A|∥−1 = ∥∑j√λj|ϕj∗A⟩⟨ϕjBC|⊗∑i√λi|ϕiBC⟩⟨ϕi∗A|∥−1 = ∥R⊗R†∥−1 = ∥R∥2−1 = (∑i√λi)2−1 = 2√λ0λ1 = 2√detρA = CA|BC,

where , and we have used the property of trace norm:

Now we will study the monogamy property of th power of negativity .

Theorem 1 . For a pure state in a system, the th power of negativity satisfies the monogamy inequality:

 NαA|BC≥NαAB+NαAC, (6)

for and satisfy the polygamy inequality:

 NαA|BC

for

Proof: When , by using , we obtain Combine with the result from Re. Zhu14 ():

 CαA|BC≥CαAB+CαAC, (8)

for , We have

 NαA|BC = CαA|BC (9) ≥ CαAB+CαAC ≥ NαAB+NαAC,

the last inequality is due to for any mixed state in a quantum system, concurrence is an upper bound of negative, i.e.  Chen05 (). When without loss of generality, assuming we have: where we used the property for the second inequality: If or the inequality obviously holds.

If we consider any -qubit pure state in -partite cases with From , a set of hierarchical monogamy inequalities of holds:

 NαA1|A2…AN≥k−1∑i=2NαA1Ai+NαA1|Ak…AN, (10)

for and a set of hierarchical polygamy inequalities of holds:

 NαA1|A2…AN≤k−1∑i=2NαA1Ai+NαA1|Ak…AN, (11)

for

These set of hierarchical relations can be used to detect the multipartite entanglement in these -partite Bai1402 (). We can also obtain the following result:

Corollary 1 . For any -qubit pure state the general monogamous inequality hold:

 NαA1|A2⋯AN≥NαA1A2+...+NαA1AN, (12)

for and the general polygamous inequality holds:

 NαA1|A2⋯AN

for

We can see the result of from Re. Ou07 () is a special case of our monogamy inequality Eq. (12).

## Iii Monogamy of αth power Convex-Roof Extended Negativity

Given a bipartite state in the Hilbert space . CREN is defined as the convex roof extended of negativity on pure states Lee03 ():

 ˜N(ρAB)=min∑ipiN(|ψiAB⟩), (14)

where the minimum is taken over all possible pure state decompositions of Obviously, the CREN of a pure state is equal to its Negativity. CREN gives a perfect discrimination of PPT bound entangled states and separable states in any bipartite quantum systems Horodeki97 (); Dur00 (). We have following result for CREN:

Theorem 2 . For a mixed state in a system, the following monogamy inequality holds:

 ˜NαA|BC≥˜NαAB+˜NαAC, (15)

for and following polygamy inequality holds:

 ˜NαA|BC<˜NαAB+˜NαAC, (16)

for

Proof: We only prove the first monogamy inequality, the proof of second inequality is similar to the proof of . Assuming a mixed state in a system, by using the , the definition of CREN and concurrence, we have:

 ˜NA|BC = min∑ipiN(|ψiA|BC⟩) (17) = min∑ipiC(|ψiA|BC⟩) = CA|BC.

Thus we have:

 ˜NαA|BC = CαA|BC (18) ≥ CαAB+CαAC ≥ ˜NαAB+˜NαAC,

for , where the second inequality is due to for any mixed state in a quantum system, concurrence is an upper bound of negative.

From , a set of hierarchical monogamy inequalities of holds for any -qubit mixed state in -partite cases with :

 ˜NαA1|A2…AN≥k−1∑i=2˜NαA1Ai+˜NαA1|Ak…AN, (19)

for and a set of hierarchical polygamy inequalities of holds:

 ˜NαA1|A2…AN≤k−1∑i=2˜NαA1Ai+˜NαA1|Ak…AN, (20)

for

We also have the following corollary:

Corollary 2 . For a mixed state in a -qubit system, the th power of CREN satisfies:

 ˜NαA1|A2…AN≥˜NαA1A2+...+˜NαA1AN, (21)

for and

 ˜NαA1|A2…AN<˜NαA1A2+...+˜NαA1AN, (22)

for .

We can see the result of from Re. Kim09 () is a special case of our monogamy inequality Eq. (21).

## Iv Monogamy of αth power of Entanglement of Formation

Given a bipartite state in the Hilbert space , the entanglement of formation (EoF) is defined as Bennett9601 (); Bennett9602 ():

 E(ρAB)=min∑ipiE(|ψiAB⟩), (23)

where is the von Neumann entropy, the minimum is taken over all possible pure state decompositions of In Re. Wootters98 (), Wootters derived an analytical formula for a two-qubit mixed state :

 E(ρAB)=h(1+√1−C2AB2), (24)

where is the binary entropy and is the concurrence of which is given by Eq. (3) and Eq. (4). Bai have proven a set of hierarchical monogamy inequalities holds for the squared EoF in a system Bai1402 ().

 E2A1|A2…AN≥k−1∑i=2E2A1Ai+E2A1|Ak…AN. (25)

We will show that the hierarchical monogamy inequality holds for the th power of EoF, where Our result can be seen an improvement of Bai ’s work.

Theorem 3 . For a mixed state in a system, the following monogamy inequality for the th power of EoF holds:

 EαA|BC≥EαAB+EαAC, (26)

for and the following polygamy inequality holds:

 EαA|BC≤EαAB+EαAC, (27)

for

Proof: Let’s consider a tripartite pure state in a system. Based on the Schmidt decomposition, the -dimensional qubit C can be viewed as an four-dimensional qubit Osborne06 (). Therefore, we can consider the monogamy relationship in a system:

 EαA|BC = Eα(C2A|BC) (28) ≥ Eα(C2AB+C2AC) ≥ Eα(C2AB)+Eα(C2AC) = Eα(ρAB)+Eα(ρAC),

where the first inequality is due to is a monotonic increasing function and holds, the second inequality is due to the fact Zhu14 (): for all , the last equality is due to a mixed state in a system,  Bai1402 (). Thus, we complete our discussion on pure state.

Consider a mixed state in a system. We use an optimal convex decomposition :

 E(ρA|BC)=∑ipiE(|ϕiABC⟩), (29)

we can derive

 E(ρA|BC) = ∑ipiE(|ϕiABC⟩) (30) = ∑ipiE[C2(|ϕiABC⟩)] ≥ E[∑ipiC2(|ϕiABC⟩)] ≥ E[C2(ρA|BC)] ≥ α√Eα(C2AB)+Eα(C2AC) = Eα(ρAB)+Eα(ρAC),

where the first equality is the definition of mixed state, we have used that is a convex function in the first inequality, the second inequality can be derived by Cauchy-Schwarz inequality: with Thus proving the monogamy inequality. On the other hand, it is easy to check the polygamy inequality for

Based on the discussion above, we show that for a mixed state in a system, a set of hierarchical monogamy inequalities holds for the th power of EoF in -partite case with :

 EαA1|A2…AN≥k−1∑i=2EαA1Ai+EαA1|Ak…AN, (31)

for which can be an improvement of Bai ’s work. And a set of hierarchical polygamy inequalities holds:

 EαA1|A2…AN≤k−1∑i=2EαA1Ai+EαA1|Ak…AN, (32)

for When , the general monogamy inequality hold:

 EαA1|A2…AN≥EαA1A2+...+EαA1AN, (33)

for the specific case have been revealed in Re. Zhu14 (). We also have the general polygamy inequality:

 EαA1|A2…AN≤EαA1A2+...+EαA1AN, (34)

for

## V Monogamy of αth power Concurrence VS Monogamy of αth power Negativity

In this section, we will discuss the monogamy property of th power of concurrence and th power of negativity for Finally, we will discuss the monogamy property of th power of EoF for

Based on the monogamy inequality of concurrence Coffman00 (); Osborne06 (), Re. Zhu14 () considered the general monogamy inequalities of th power concurrence in an -qubit mixed state , and claimed the following inequalities holds:

 CαA1|A2⋯AN≥CαA1A2+...+CαA1AN (35)

for . While the polygamy inequalities holds:

 CαA1|A2⋯AN

for all . It’s not clear for .

For convenience, we define the ”residual tangle” of th power of concurrence as:

 τC(|ψA1A2…AN⟩)=CαA1|A2…AN−CαA1A2−⋯−CαA1AN, (37)

and define the ”residual tangle” of th power of concurrence as:

 τN(|ψA1A2…AN⟩)=NαA1|A2…AN−NαA1A2−⋯−NαA1AN. (38)

Interestingly, We find that the -qubit GHZ state

 |GHZ⟩=1√2(|0⟩⨂N+|1⟩⨂N), (39)

and -qubit W state

 |W⟩=1√N(|00⋯01⟩+|00⋯10⟩+⋯+|10⋯00⟩), (40)

can be used to distinguish the monogamous property of for . In other words, -qubit GHZ state is monogamous for the th power concurrence and -qubit W state is polygamous for the th power concurrence, where . For -qubit GHZ state

 |GHZ⟩=1√2(|0⟩⨂N+|1⟩⨂N), (41)

the concurrence Thus, the ”residual tangle” , -qubit GHZ state is monogamous for the th power concurrence. For -qubit W state

 |W⟩=1√N(|00⋯01⟩+|00⋯10⟩+⋯+|10⋯00⟩), (42)

the concurrence Thus, the ”residual tangle” for all . -qubit W state is polygamous for the th power concurrence

For the ”residual tangle” . The negativity of -qubit GHZ state Thus, , it is coincide with . The situation is different when we consider for -qubit W state. One obtain that and It is easy to check that . can be positive and negative, as showed in Fig:1, we have plotted as the function of for , and consider , and respectively. We find is not always negative, which is different than the case of .

Finally, we will discuss the monogamy property of th power of EoF for . We define the ”residual tangle” of th power of EoF as:

 τE(|ψA1A2…AN⟩)=EαA1|A2…AN−EαA1A2−⋯−EαA1AN. (43)

For -qubit GHZ state, the EoF of -qubit GHZ state Thus, the ”residual tangle” for For -qubit W state, the EoF of -qubit W state for where denotes the binary function. Thus, the ”residual tangle” where and We have proved in the appendix for . On the other hand, can be positive and negative for . As showed in Fig:2, we have plotted as the function of for , and consider , and respectively.

## Vi conclusion

In this paper, We studied the monogamy property of th power of entanglement measure in bipartite states. In particular, we investigated the monogamy properties of negativity and CREN in detail. We showed that the th power of negativity, CREN are monogamous for and polygamous for . We improved the hierarchical monogamy inequality for the th power of EoF, and show that the th power of EoF is hierarchical monogamous for . Finally, we discussed the monogamy property of th power of concurrence and the th power of EoF. We found that the -qubit GHZ state and -qubit W state can be used to distinguish both the th power of the concurrence for and the th power of the EoF for in qubit system. We compared concurrence with negativity in terms of monogamy property and showed the difference between them.

## Vii acknowledgments

We thank Zheng-Jun Xi and Chen-Ming Bai for their helpful discussions. Y. Li was supported by NSFC (Grants No.11271237 and No.61228305) and the Higher School Doctoral Subject Foundation of Ministry of Education of China (Grant No.20130202110001). Y. Luo was supported by NSFC (Grant No.61303009).

## Viii appendix

For a binary entropy function for . We have following lower bounding and upper bounding for approximation Calabro09 ():

 1−4(x−12)2≤h(x)≤1−(1−2x)22ln2, (44)

for .

The ”residual tangle” is:

 τE(|W⟩) = hα(1−1N)−(N−1)hα(N+√N2−42N) ≤ [1−(1−2N)22ln2]α−(N−1)[1−4(√N2−42N)2]α < 1−(N−1)(2N)2α,

where and . To prove for and , we define:

 f(x)=(x−1)(2x)2α, (46)

where is a real number and . The derived function of is:

 f′(x)=(2x)2α[(1−2α)x+2αx]. (47)

It is easy to check that for , the derived function . Thus is monotonic increasing, which derived is monotonic decreasing for . The maximum of is .

## References

• (1) D. Bru, Phys. Rev. A60, 4344(1999).
• (2) V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A61, 052306(2000).
• (3) T. J. Osborne and F. Verstraete, Phys. Rev. Lett. 96, 220503(2006).
• (4) A. Kay, D. Kaszlikowski, and R. Ramanathan3, Phys. Rev. Lett. 103, 050501(2009).
• (5) J. Barrett, L. Hardy, and A. Kent, Phys. Rev. Lett. 95, 010503(2005).
• (6) X. S. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P. Walther, Nat. Phys. 7, 399(2009).
• (7) L. Suskind, arXiv:1301,4505.
• (8) S. Lloyd anf J. Preskill, J. High Energy Phys. 08 (2014) 126.
• (9) G. Vidal and R. F. Werner, Phys. Rev. A65, 032314 (2002).
• (10) Y. C. Ou, and H. Fan, Phys. Rev. A75, 062308(2007).
• (11) J. S. Kim, A. Das, and B. S. Sanders, Phys. Rev. A79, 012329 (2009).
• (12) T. R. de Oliveira, Phys. Rev. A89, 034304(2014).
• (13) Y. K. Bai, Y. F. Xu, and Z.D. Wang, Phys. Rev. Lett. 113, 1005503(2014).
• (14) X. N. Zhu, and S. M. Fei, Phys. Rev. A90, 024304(2014).
• (15) B. Regula, S. D. Martino, S. Lee, and G. Adesso, Phys. Rev. Lett. 113, 110501 (2014).
• (16) S. K., R. Prabhu, A. S. De, U. Sen, Ann. Phys. 348, 297 (2014).
• (17) Y. K. Bai, Y. F. Xu, and Z.D. Wang, Phys. Rev. A90, 062343(2014).
• (18) K. Chen, S. Albeverio, and S. M. Fei, Phys. Rev. Lett. 95, 040504(2005).
• (19) S. Lee, D. P. Chi, S. D. Oh, and J. Kim, Phys. Rev. A68, 062304 (2003).
• (20) C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A53, 2046 (1996).
• (21) C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A54, 3824 (1996).
• (22) W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
• (23) M. Horodecki, P. Horodecki, and R. Horodecki. Phys. Rev. Lett. 80, 5239 (1998).
• (24) P. Horodeki, Phys. Lett. A 232, 333 (1997).
• (25) W. Dur, J. I. Cirac, M. Lewenstein, and D. Bru, Phys. Rev. A61, 062313 (2000).
• (26) C. Calabro, The Exponential Complexity of Satisfiability Problems (ph.D thesis) (2009).
• (27) M. A. Nielsen and I. L. Chuang Quantum Computation and Quantum Information (Cambridge: Cambridge Univ. Press) (2000).
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