Generalised monogamy relation of convex-roof extended negativity in multi-level systems

In this paper, we investigate the generalised monogamy inequalities of convex-roof extended negativity (CREN) in multi-level systems. The generalised monogamy inequalities provide the upper and lower bounds of bipartite entanglement, which are obtained by using CREN and the CREN of assistance (CRENOA). Furthermore, we show that the CREN of multi-qubit pure states satisfies some monogamy relations. Additionally, we test the generalised monogamy inequalities for qudits by considering the partially coherent superposition of a generalised W-class state in a vacuum and show that the generalised monogamy inequalities are satisfied in this case as well.


Tian Tian, Yu Luo & Yongming Li
In this paper, we investigate the generalised monogamy inequalities of convex-roof extended negativity (CREN) in multi-level systems. The generalised monogamy inequalities provide the upper and lower bounds of bipartite entanglement, which are obtained by using CREN and the CREN of assistance (CRENOA). Furthermore, we show that the CREN of multi-qubit pure states satisfies some monogamy relations. Additionally, we test the generalised monogamy inequalities for qudits by considering the partially coherent superposition of a generalised W-class state in a vacuum and show that the generalised monogamy inequalities are satisfied in this case as well.
Quantum entanglement is one of the most important physical resources in quantum information processing [1][2][3][4] . As distinguished from classical correlations, quantum entanglement cannot be freely shared among many objects. We call this important phenomenon of quantum entanglement monogamy 5,6 . The property of monogamy may be as fundamental as the no-cloning theorem 7 , which gives rise to structures of entanglement in multipartite settings 8,9 . Some monogamy inequalities have been studied to apply entanglement to more useful quantum information processing. The property of monogamy property has been considered in many areas of physics: it can be used to extract an estimate of the quantity of information about a secret key captured by an eavesdropper in quantum cryptography 10,11 , as well as the frustration effects observed in condensed matter physics 12,13 and even black-hole physics 14,15 .
The monogamy relation of entanglement is a way to characterise different types of entanglement distribution. The first monogamy relation was named the Coffman-Kundu-Wootters (CKW) inequality 8 . The monogamy property can be interpreted as the following statement: the amount of entanglement between A and B plus the amount of entanglement between A and C cannot be greater than the amount of entanglement between A and the BC pair. Osborne and Verstraete later proved that the CKW inequality also holds in an n-qubit system 9 . Other types of monogamy relations for entanglement were also proposed. Studies have found that the monogamy inequality holds in terms of some entanglement measures, negativity 16 , squared CREN 17 , entanglement of formation [18][19][20] , Rényi entropy 21 and Tsallis entropy 22,23 . The monogamy property of other physical resources, such as discord and steering 24 , has also been discussed. There can be several inequivalent types of entanglement among the subsystems in multipartite quantum systems, and the amount of different types of entanglement might not be directly comparable to one another. Regula et al. studied multi-party quantum entanglement and found that there was strong monogamy 25 . Additionally, generalised monogamy relations of concurrence for N-qubit systems were also proposed by Zhu et al. 26 .
In this paper, we study the generalised monogamy inequalities of CREN in multi-qubit systems. We first recall some basic concepts of entanglement measures. Then, monogamy inequalities are given by the concurrence and negativity of the n-qubit entanglement. Furthermore, we consider some states in a higher-dimensional quantum system and find that the generalised monogamy inequalities also hold for these states. We specifically test the generalised monogamy inequalities for qudits by considering the partially coherent superposition of a generalised W-class state in a vacuum, and we show that the generalised monogamy inequalities are satisfied in this case as well. These relations also give rise to a type of trade-off in inequalities that is related to the upper and lower bounds of CRENOA. It shows the bipartite entanglement between AB and the other qubits: especially under partition AB, a two-qubit system is different from the previous monogamy inequality that is typically used.
Preliminaries: concurrence and negativity. For any bipartite pure state |ψ〉 AB in a d ⊗ d′ (d ≤ d′ ) quantum system with its Schmidt decomposition, is defined as 27 For any mixed state ρ AB , its concurrence is defined as where the minimum is taken over all possible pure state decompositions Similarly, the concurrence of assistance (COA) of ρ AB is defined as 28 where the maximum is taken over all possible pure state decompositions {p i , |ψ i 〉 AB } of ρ AB . Another well-known quantification of bipartite entanglement is negativity. For any bipartite pure state |ψ〉 AB , the negativity ψ where ρ AB T A is a partial transposition with respect to the subsystem A, X denotes the trace norm of X; i.e., ≡ † X Tr XX . Negativity is a computable measure of entanglement, which is a convex function of ρ AB . It disappears if, and only if, ρ AB is separable for the 2 ⊗ 2 and 2 ⊗ 3 systems 30 . For the purposes of this discussion, we use the following definition of negativity: For any maximally entangled state in a two-qubit system, this negativity is equal to 1. CREN gives a perfect discrimination of positive partial transposition-bound entangled states and separable states in any bipartite quantum system 31,32 . For any mixed state ρ AB , CREN is defined as where the minimum is taken over all possible pure state decompositions {p i , |ψ i 〉 AB } of ρ AB . For any mixed state ρ AB , CRENOA is defined as 17 where the maximum is taken over all possible pure state decompositions {p i , |ψ i 〉 AB } of ρ AB . CREN is equivalent to concurrence for any pure state with Schmidt rank-2 17 , and consequently, it follows that for any two-qubit mixed state ρ AB = ∑ i p i |ψ i 〉 〈 ψ i |: where the minimum and the maximum are taken over all pure state decompositions {p i , |ψ i 〉 AB } of ρ AB .
Monogamy relations of concurrence and negativity. The CKW inequality 8 was first defined as where  ρ ( ) A BC is the concurrence of a three-qubit state ρ A|BC for any bipartite cut of subsystems between A and BC. Similarly, the dual inequality in terms of COA is as follows 33 : For any pure state ψ ...

A An
The dual inequality in terms of the COA for n-qubit states has the form 17 when the rank of the matrix is 2, we have For any n-qubit pure state ψ ...

A An
The dual inequality 17 in terms of CRENOA is as follows: Monogamy inequalities of CREN. For a 2 ⊗ 2 ⊗ m quantum pure state |ψ〉 ABC , it has been shown that is the concurrence under bipartition A|BC for pure state |ψ〉 ABC . Namely, Similarly, considering that CREN is equivalent to concurrence by Eq. (17), we have The concurrence is related to the linear entropy of a state 34 2 Given a bipartite state ρ, T(ρ) has the property 35 , From the definition of pure state concurrence in Eq. (2) together with Eq. (22), we have Now, we provide the following theorems: Combining Eq. (23) with Eq. (24), we have The third equality holds because CREN and concurrence are equal for any rank-2 pure state. Therefore, we obtain Combining Eq. (26) with Eq. (29), we finally get Thus, the proof is completed. Theorem 1 shows a simple relationship of CRENOA in a tripartite quantum system. The monogamy inequality shows that the entanglement A|BC cannot be greater than the sum of the entanglement B|AC and the entanglement C|AB. Taking an easy example, when considering a three-qubit state, the following equation exists: |ψ〉 ABC = a|010〉 + b|100〉 where |a| 2 + |b| 2 = 1. Using a simple calculation, the following equation can be obtained: where the state |ψ〉 ABC saturates the monogamy inequality in Eq. (25). Moreover, the iteration of Eq. (25) leads us to the generalized monogamy inequality in multi-qubit quantum systems. Corollary 1. For any multi-party mixed state ρ ... A A A n 1 2 in an n-qubit system 36 , the following monogamy inequality exists: The meaning of the first inequality is clear the bipartite entanglement between ρ A 1 and the other qubits, when taken as a group cannot be greater than the sum of the n − 1 individual bipartite entanglements between A i and the other remaining qubits. We now start to consider a four-qubit system. As shown in Fig. 1, the squared CRENOA with respect to the bipartition (A|BCD) is not greater than the sum of the three squared CRENOAs (the three possible bipartitions are B|ACD, C|ABD and D|ABC).
The meaning of the second inequality is clear the sum of the bipartite entanglements between ρ ≠ i (

1)
A i and the other remaining qubits cannot be greater than the sum of the bipartite entanglements ρ Proof. From the result of Theorem 1, we find that the generalised monogamy inequality can be easily obtained by using the superposition of states. We now consider . When the rank of the matrix is 2, we have Combining Eq. (23) with Eq. (24), we get the relationship The third equality follows from the fact that CREN and concurrence are equal for any rank-2 pure state.  , t he c a lc u l at i on re su lt s in . Consequently, the polygamy-type relation is obtained as shown in Eq. (19). Finally, consider the following four-qubit state: |ψ〉 ABCD = a|0100〉 + b|0010〉 + c|0001〉 where |a| 2 + |b| 2 + |c| 2 = 1. We can easily get the following equations:  . Proof. We have the following property for linear entropy 35 : By using the equivalent relation between concurrence and CREN (see Eq. (17)), we have There is a relationship between CREN and CRENOA (see Eq. (21)): Putting the above two equalities into Eq. (44), we get

  
This lower bound is a direct consequence of CREN. Theorem 3 shows that the entanglement between AB and the other qubits cannot be less than the absolute value of the difference between both the individual entanglements between A and each of the n − 1 remaining qubits and the individual entanglements between B and each of the n − 1 remaining qubits. Theorem 3 provides a monogamy-type lower bound of multi-qubit entanglement between the two-qubit system AB and the other (n − 2)-qubit system C 1 C 2 ...C n−2 in terms of the squared CRENOA.
, and so we obtain the CWK-type relation in Eq. (18).
Finally, we consider the following four-qubit state |ψ〉 ABCD = a|1000〉 + b|0010〉 + c|0001〉 where |a| 2 + |b| 2 + |c| 2 = 1, from which we can easily obtain the following equations: . Therefore, the state |ψ〉 ABCD saturates the monogamy inequality in Eq. (40). Therefore, a generalised monogamy inequality using negativity and CRENOA in an n-qubit is proposed. These relations also give rise to a type of trade-off in inequalities that is related to the upper and lower bounds of CRENOA.
Remark. It is interesting to note that the properties of CREN are based on the subadditivity of linear entropy. However, negativity violates this subadditivity in general conditions 37-39 . Examples. In this section, we use some special states to study generalised monogamy inequalities. First, we consider the (Greenberger-Horne-Zeilinger) GHZ state and W state in Examples 1 and 2. Second, we consider two states in the higher-dimensional system in Examples 3 and 4. Example 1. For an arbitrary pure GHZ state in an n-qubit system: n n where |a| 2 + |b| 2 = 1. The generalized GHZ state is satisfied with the previous CKW inequality. We will now show that the generalised GHZ state satisfies the generalised monogamy inequalities. We have ρ 1 = ρ 2 = … = ρ n = a 2 |0〉 〈 0| + b 2 |1〉 〈 1|. It is straightforward to check: . Therefore: In the same way, we get the following inequalities: From the above results, we discover that the generalised GHZ state and W state satisfy our inequalities. We further explore the condition of the generalised inequalities in higher-dimensional systems. We consider the following examples: Example 3. For a pure, totally antisymmetric state |ψ ABC 〉 in a 3 ⊗ 3 ⊗ 3 system 40 : and further obtain the inequalities . We now explore theorems 2 and 3. First, we have . Therefore, we obtain the following inequalities: Example 4. The n-qudit generalised W-class state in higher-dimensional quantum systems is very useful in quantum information theory 42 . We verify whether the generalised monogamy inequalities hold in higher-dimensional systems using a special example. First, we recall the definition of n-qudit generalised W-class state 43 , be an n-qudit pure state in a superposition of an n-qudit generalised W-class state and vacuum; that is,  , with size r > 2 can be obtained by an r × r unitary matrix u hl such that We apply the definition of mixed state negativity in Eqs (8 and 63), and then we have the two-tangle based on We first consider the generalisation of Theorem 1. Therefore, this n-qudit pure state is satisfied with the generalised monogamy inequality in Eq. (32). In other words, the test of the Theorem 2 has been accomplished. Next, we verify Theorem 3. First, we consider the term CREN from Eq.

  
Therefore, this n-qudit pure state satisfies the generalised monogamy inequality in Eq. (40). We have now verified the generalised monogamy inequalities. In other words, the generalised monogamy inequality are satisfied with the n-qudit pure state for all three of our theorems.

Conclusions
In this paper, we have used CREN to study different types of monogamy relations. In particular, we have shown that CREN satisfies the generalised monogamy inequalities. We have investigated the CKW-like inequalities and generalised monogamy inequalities. Furthermore, the generalised monogamy inequalities related to CREN and CRENOA were obtained by n-qubit states. These relations also give rise to a type of trade-off in inequalities that is related to the upper and lower bounds of CRENOA. Finally, we have shown that the partially coherent superposition of the generalised W-class state and vacuum extensions of CREN satisfies the generalised monogamy inequalities. We believe that the generalised monogamy inequalities can be useful in quantum information theory. This paper was based on the linear entropy. To continue this work, we will study the nature of other entropy further in the future work. We hope that our work will be useful to the quantum physics.