Generalized monogamy inequalities of convex-roof extended negativity in N-qubit systems

# Generalized monogamy inequalities of convex-roof extended negativity in N-qubit systems

Yanmin Yang , Wei Chen , Gang Li, Zhu-Jun Zheng
School of Mathematics, GuangZhou University, Guangzhou 510006, China
School of Computer Science and Network Security,
Dongguan University of Technology, Dongguan ¡¡523808, China
School of Mathematics, South China University of Technology, Guangzhou 510641, China
e-mail: yangym929@gmail.come-mail: auwchen@scut.edu.cne-mail: zhengzj@scut.edu.cn
###### Abstract

In this paper, we present some generalized monogamy inequalities based on negativity and convex-roof extended negativity (CREN). These monogamy relations are satisfied by the negativity of -qubit quantum systems , under the partitions and . Furthermore, the -class states are used to test these generalized monogamy inequalities.

PACS numbers: 03.67.Mn, 03.65.Ud

Key words: Generalized monogamy inequalities, Negativity, Convex-roof extended negativity

One of the most fundamental properties of quantum correlations is that they are not shareable when distributed among many parties. This property distinguishes the quantum correlations from classical correlations. A simple example is a pure maximally entangled state shared between Alice and Bob. This state cannot share any additional correlation (classical or quantum) with other parties. The composite system with a third party, say Carol, can only be a tensor product of the state of Alice and Bob with the state of Carol. This property has been called the monogamy of entanglement and it means that the monogamy relation of entanglement is a way to characterize the different types of entanglement distribution. The monogamy relations give rise to structures of entanglement in multipartite setting and it is important for many tasks in quantum information theory, particularly, in quantum key distribution [1] and quantum correlations [2, 3] like quantum discord[4].

Although it has been shown that quantum correlation measures and entanglement measures cannot satisfy the traditional monogamy relations, it has been shown that it does satisfy the squared concurrence [5, 6, 7, 8] and the squared entanglement of formation [7, 8, 9, 10, 11]. Other useful entanglement measures are negativity [12] and convex-roof extended negativity(CREN) [13]. The authors in [14] showed that the monogamy inequality holds in terms of squared negativity for three-qubit states and the author in [15] showed a monogamy relation conjecture on squared negativity for tripartite systems. Kim et al. showed that the squared CREN follows the monogamy inequality [16].

In this paper, we study the general monogamy inequalities of CREN in multi-qubit systems. We first recall some basic concepts of entanglement measures. Then we find that the generalized monogamy inequalities always hold based on negativity and CREN in -qubit systems under the partitions and . Detailed examples for -class states are given to test the generalized monogamy inequalities.

Given a bipartite pure state in a quantum system, its concurrence, is defined as [20]

 C(|ψ⟩AB)=√2[1−Tr(ρ2A)]=√2[1−Tr(ρ2B)], (1)

where is reduced density matrix by tracing over the subsystem (and analogously for ).

For any mixed state the concurrence is given by the minimum average concurrence taken over all decompositions of the so-called convex roof [21]

 C(ρAB)=min{pi,|ψi⟩}∑ipiC(|ψi⟩). (2)

The concurrence of assistance (COA) of any mixed state is defined as [22]

 Ca(ρAB) = max{pi,|ψi⟩}∑ipiC(|ψi⟩), (3)

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

If be a two-qubit state, then the COA is defined by [22], [23]

 Ca(ρAB) = Tr(√ρAB˜ρAB) (4) = ∑iλi, (5)

where , is Pauli matrix and is complex conjugation of taken in the standard basis, and

 C(ρAB)=max{0,λ1−∑i>1λi}, (6)

is the concurrence of with being the square roots of the eigenvalues of in decreasing order.

Another well-known quantification of bipartite entanglement is negativity [12], which is based on the positive partial transposition (PPT) criterion [24, 25]. For a bipartite state in a quantum system, its negativity is defined as

 N(ρAB)=∥ρTAAB∥−1, (7)

where is the partial transpose with respect to the subsystem and denotes the trace norm of i.e.

In a quantum system, if a bipartite pure state with the Schmidt decomposition,

 |ψ⟩AB=d−1∑i=0√λi|ii⟩,   λi≥0,    d−1∑i=0λi=1, (8)

then [16]

 N(ρAB)=2∑i

To overcome the lack of separability criterion, one modification of negativity is convex-roof extended negativity (CREN), which gives a perfect discrimination of PPT bound entangled states and separable states in any bipartite quantum system. For a bipartite mixed state CREN is defined as

 ˜N(ρAB)=min{pi,|ψi⟩}∑ipiN(|ψi⟩), (10)

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

Similar to the duality between concurrence and COA, we can also define a dual of CREN, namely convex-roof extended negativity of assistance (CRENOA), by taking the maximum value of average negativity over all possible pure state decomposition of mixed state , i.e.

 ˜Na(ρAB)=max{pi,|ψi⟩}∑ipiN(|ψi⟩). (11)

CREN is equivalent to concurrence for any pure state with Schmidt rank two [16]. It follows that for any two-qubit mixed state ,

 C(ρAB)=min{pi,|ψi⟩}∑ipiC(|ψi⟩)=min{pi,|ψi⟩}∑ipi˜N(|ψi⟩)=˜N(ρAB), (12) Ca(ρAB)=max{pi,|ψi⟩}∑ipiC(|ψi⟩)=max{pi,|ψi⟩}∑ipi˜N(|ψi⟩)=˜Na(ρAB). (13)

For any -qubit pure state , it has been shown that the concurrence and COA of satisfy monogamy inequalities [5, 17]:

 N−1∑i=1C2(ρABi)≤C2(|ψ⟩A|B1⋯BN−1)≤N−1∑i=1C2a(ρABi), (14)

where .

Combining with Eq.(12) and Eq.(13), we have

 N−1∑i=1˜N2(ρABi)≤C2(|ψ⟩A|B1⋯BN−1)≤N−1∑i=1˜N2a(ρABi). (15)

The concurrence is related to the linear entropy of a state [18],

 T(ρ)=1−Tr(ρ2). (16)

Given a bipartite state , has the property [19],

 T(ρA)+T(ρB)≥T(ρAB)≥|T(ρA)−T(ρB)|. (17)

From the definition of pure state concurrence together with Eq.(17), we have

 C2(|ψ⟩A|BC1⋯CN−2)+C2(|ψ⟩B|AC1⋯CN−2)≥C2(|ψ⟩AB|C1⋯CN−2), (18) |C2(|ψ⟩A|BC1⋯CN−2)−C2(|ψ⟩B|AC1⋯CN−2)|≤C2(|ψ⟩AB|C1⋯CN−2). (19)

For an -qubit pure state , the negativity of the state , viewed as a bipartite state with partition , satisfies the following monogamy inequalities.

###### Theorem 1.

For any -qubit pure state we have

 N2(|ψ⟩AB|C1⋯CN−2)≥max{N−2∑i=1[˜N2(ρACi)−˜N2a(ρBCi)],N−2∑i=1[˜N2(ρBCi)−˜N2a(ρACi)]}, (20)

where and

###### Proof.

Let be a -qubit pure state, then we have a Schmidt decomposition Then from Eq.(1), we get

 C(|ψ⟩AB|C1⋯CN−2)=√2(1−Trρ2AB) (21)

where

 ρAB=TrC1⋯CN−2(|ψ⟩AB|C1⋯CN−2⟨ψ|)=3∑i=0λi|i⟩⟨i|.

We thus obtain

 C(|ψ⟩AB|C1C2⋯CN−2)=2√∑i

From Eq.(9), we have

 N(|ψ⟩AB|C1C2⋯CN−2)=2∑i

Consequently, we have

 N2(|ψ⟩AB|C1⋯CN−2) ≥ C2(|ψ⟩AB|C1⋯CN−2) ≥ |C2(|ψ⟩A|BC1⋯CN−2)−C2(|ψ⟩B|AC1⋯CN−2)| ≥ max{N−2∑i=1[˜N2(ρACi)−˜N2a(ρBCi)],N−2∑i=1[˜N2(ρBCi)−˜N2a(ρACi)]},

where the second inequality is due to Eq.(19), the third inequality is due to Eq.(15). ∎

A monogamy-type lower bound of is given by Theorem 1. According to the relation between negativity and concurrence, we will give an upper bound of .

###### Theorem 2.

For any -qubit pure state we have

 N2(|ψ⟩AB|C1⋯CN−2)≤r(r−1)2[2˜N2a(ρAB)+N−2∑i=1(˜N2a(ρACi)+˜N2a(ρBCi)], (24)

where is the Schmidt rank of the pure state , , and .

###### Proof.

From Eq. (32) in [27], we have

 N(|ψ⟩AB|C1⋯CN−2)≤√r(r−1)2C(|ψ⟩AB|C1⋯CN−2). (25)

In addition, we have the fact that

 C2(|ψ⟩AB|C1⋯CN−2) ≤ C2(|ψ⟩A|BC1⋯CN−2)+C2(|ψ⟩B|AC1⋯CN−2) (26) ≤ 2˜N2a(ρAB)+N−2∑i=1(˜N2a(ρACi)+˜N2a(ρBCi)), (27)

where the first inequality is due to Eq.(18), the second inequality is due to the right inequality of Eq.(15).

From inequalities Eq.(25) and Eq.(27), the inequality Eq.(24) can be deserved. ∎

###### Corollary 1.

If the Schmidt rank of pure state is two, then we have

 N2(|ψ⟩AB|C1⋯CN−2)≤2˜N2a(ρAB)+N−2∑i=1(˜N2a(ρACi)+˜N2a(ρBCi), (28)

where and

###### Example 1.

Consider the -qubit generalized -class states [28]:

 |W⟩A1A2⋯AN=a1|10⋯0⟩A1A2⋯AN+a2|01⋯0⟩A1A2⋯AN+⋯+aN|00⋯1⟩A1A2⋯AN, (29)

where The state , viewed as a bipartite state, has the form

 |W⟩A1A2|A3⋯AN=√|a1|2+|a2|2(a1√|a1|2+|a2|2|10⟩+a2√|a1|2+|a2|2|01⟩)⊗|0⋯0⟩+ (N∑i=3|ai|2)12|00⟩⊗(a3(N∑i=3|ai|2)12|10⋯0⟩+⋯+aN(N∑i=3|ai|2)12|00⋯1⟩).

Hence, .

For any , we have

 ρAiAj=(ai|10⟩+aj|01⟩)(a∗i⟨10|+a∗j⟨01|)+∑k≠i,j|ak|2|00⟩⟨00|.

Furthermore, from Eqs. (5), (6), (12) and (13), we have

 ˜N(ρAiAj)=˜Na(ρAiAj)=2|ai||aj|.

The lower bound of , that is, the right hand side of Eq.(20) is equal to . And the upper bound in Eq.(24) is equal to .

When either or , the lower bound of is equal to upper bound.

For , , suppose , then the lower and upper bounds of are shown in the following figure:

###### Corollary 2.

For any -qubit pure state , if the Schmidt rank of state is two, then

• we have monogamy relations

 |N2(|ψ⟩A|BC1⋯CN−2)−N2(|ψ⟩B|AC1⋯CN−2)| ≤ N2(|ψ⟩AB|C1⋯CN−2) ≤N2(|ψ⟩A|BC1⋯CN−2)+N2(|ψ⟩B|AC1⋯CN−2). (30)

Specially, if the systems and are not entangled, both the two equalities hold.

• the three terms and have following relations:

 N2(|ψ⟩A|BC1⋯CN−2) ≤ N2(|ψ⟩B|AC1⋯CN−2)+N2(|ψ⟩AB|C1⋯CN−2), (31) N2(|ψ⟩B|AC1⋯CN−2) ≤ N2(|ψ⟩A|BC1⋯CN−2)+N2(|ψ⟩AB|C1⋯CN−2), (32) N2(|ψ⟩AB|C1⋯CN−2) ≤ N2(|ψ⟩A|BC1⋯CN−2)+N2(|ψ⟩B|AC1⋯CN−2). (33)

For the -class states (29), we have for any . Clearly, relations (31)-(33) are satisfied. And the first inequality (second inequality) of Eq.(30) is just the Eq.(20) in Theorem 1 (Eq.(24) in Theorem 2).

The above results can generalized to the negativity under partition for pure state .

###### Theorem 3.

For any -qubit pure state we have

 N2(|ψ⟩ABC1|C2⋯CN−2)≥2r(r−1)max{N−2∑i=1[˜N2(ρACi)−˜N2a(ρBCi)],N−2∑i=1[˜N2(ρBCi)−˜N2a(ρACi)]}−∑j∈J˜N2a(ρC1j), (34)
 N2(|ψ⟩ABC1|C2⋯CN−2)≥∑j∈J˜N2(ρC1j)−r(r−1)2[2˜N2a(ρAB)+N−2∑i=1(˜N2a(ρACi)+˜N2a(ρBCi)], (35)

where , is the Schmidt rank of the pure state , and is the reduced density matrix by tracing over the subsystems except for and .

###### Proof.

For any -qubit pure state we have a Schmidt decomposition Then from Eq.(1), we get

 C(|ψ⟩ABC1|C2⋯CN−2)=√2(1−Trρ2ABC1) (36)

where .

Hence we obtain

 C(|ψ⟩ABC1|C2⋯CN−2)=2√∑i

From Eq.(9),

 N(|ψ⟩ABC1|C2⋯CN−2)=2(∑i

so we get

 N2(|ψ⟩ABC1|C2⋯CN−2) ≥ C2(|ψ⟩ABC1|C2⋯CN−2) = 2(1−Tr(ρ2ABC1)) ≥ |C2(|ψ⟩AB|C1⋯CN−2)−C2(|ψ⟩C1|ABC2⋯CN−2)|.

If , then

 N2(|ψ⟩ABC1|C2⋯CN−2) ≥ C2(|ψ⟩AB|C1⋯CN−2)−C2(|ψ⟩C1|ABC2⋯CN−2) ≥ 2r(r−1)˜N2(|ψ⟩AB|C1⋯CN−2)−∑j∈J˜N2a(ρC1j),

where the second inequality is due to Eq.(25) and Eq.(15). Combine with Eq.(20), we can obtain the inequality (34).

If , then

 N2(|ψ⟩ABC1|C2⋯CN−2) ≥ C2(|ψ⟩C1|ABC2⋯CN−2)−C2(|ψ⟩AB|C1⋯CN−2) ≥ ∑j∈J˜N2(ρC1j)−˜N2(|ψ⟩AB|C1⋯CN−2),

where the second inequality is due to Eq.(15). Combine with Eq.(24), the inequality (35) holds. ∎

Similar to Theorem 2, we also have an upper bound of .

###### Theorem 4.

For any -qubit pure state we have

 N2(|ψ⟩ABC1|C2⋯CN−2) ≤ r(r−1)2[2˜N2a(ρAB)+N−2∑i=1˜N2a(ρACi) (39) +N−2∑i=1˜N2a(ρBCi)+∑j∈J˜N2a(ρC1j)],

where and are defined as in Theorem 3, and is the Schmidt rank of the pure state

###### Proof.

From Eq.(32) in [27], we have

 N2(|ψ⟩ABC1|C2⋯CN−2) ≤ r(r−1)2C2(|ψ⟩ABC1|C2⋯CN−2) ≤ r(r−1)2(C2(|ψ⟩AB|C1⋯CN−2)+C2(|ψ⟩C1|ABC2⋯CN−2)).

Combine with Eq.(24) and Eq.(15), the inequality (39) can be deserved. ∎

###### Example 2.

For the -qubit generalized W-class states (29), we have

 N2(|W⟩A1A2A3|A4⋯AN)=43∑i=1|ai|2N∑j=4|aj|2. (40)

The lower bound of