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

Generalized monogamy inequalities of convex-roof extended negativity in -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]

(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]

(2)

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

(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]

(4)
(5)

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

(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

(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,

(8)

then [16]

(9)

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

(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.

(11)

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

(12)
(13)

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

(14)

where .

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

(15)

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

(16)

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

(17)

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

(18)
(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

(20)

where and

Proof.

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

(21)

where

We thus obtain

(22)

From Eq.(9), we have

(23)

Consequently, we have

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

(24)

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

Proof.

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

(25)

In addition, we have the fact that

(26)
(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

(28)

where and

Example 1.

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

(29)

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

Hence, .

For any , we have

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

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:

(a) The dotted line is the upper bound of , the solid line is . And the abscissa represents the value range of from to .
(b) The dotted line is the lower bound of , the solid line is . And the abscissa represents the value range of from to .
Figure 1: The monogamy relation of .
Corollary 2.

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

  • we have monogamy relations

    (30)

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

  • the three terms and have following relations:

    (31)
    (32)
    (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

(34)
(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

(36)

where .

Hence we obtain

(37)

From Eq.(9),

(38)

so we get

If , then

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

If , then

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

(39)

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

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

(40)

The lower bound of