Abstract
In this paper, we investigate the generalised monogamy inequalities of convexroof extended negativity (CREN) in multilevel 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 multiqubit pure states satisfies some monogamy relations. Additionally, we test the generalised monogamy inequalities for qudits by considering the partially coherent superposition of a generalised Wclass state in a vacuum and show that the generalised monogamy inequalities are satisfied in this case as well.
Introduction
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 nocloning 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 blackhole 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 CoffmanKunduWootters (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 nqubit 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 multiparty quantum entanglement and found that there was strong monogamy^{25}. Additionally, generalised monogamy relations of concurrence for Nqubit systems were also proposed by Zhu et al.^{26}.
In this paper, we study the generalised monogamy inequalities of CREN in multiqubit systems. We first recall some basic concepts of entanglement measures. Then, monogamy inequalities are given by the concurrence and negativity of the nqubit entanglement. Furthermore, we consider some states in a higherdimensional 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 Wclass 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 tradeoff 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 twoqubit system is different from the previous monogamy inequality that is typically used.
Results
This paper is organised as follows: in the first subsection, we recall some basic concepts of concurrence and negativity. We present the monogamy relations of concurrence and negativity in the second subsection. In the third subsection, the generalised monogamy inequalities of CREN are given. The fourth subsection includes some examples that verify these results.
Preliminaries: concurrence and negativity
For any bipartite pure state ψ〉_{AB} in a d ⊗ d′ (d ≤ d′) quantum system with its Schmidt decomposition,
the concurrence is defined as^{27}
where ρ_{A} = tr_{B} (ψ〉_{AB}〈ψ). For any mixed state ρ_{AB}, its concurrence is defined as
where the minimum is taken over all possible pure state decompositions {p_{i}, ψ_{i}〉_{AB}} of ρ_{AB}.
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 wellknown quantification of bipartite entanglement is negativity. For any bipartite pure state ψ〉_{AB}, the negativity is
where ρ_{A} = tr_{B}(ψ〉_{AB}〈ψ).
For any bipartite state ρ_{AB} in the Hilbert space negativity is defined as^{29}
where is a partial transposition with respect to the subsystem A, denotes the trace norm of X; i.e., . 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 twoqubit system, this negativity is equal to 1. CREN gives a perfect discrimination of positive partial transpositionbound 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 rank2^{17}, and consequently, it follows that for any twoqubit mixed state ρ_{AB} = ∑_{i}p_{i}ψ_{i}〉〈ψ_{i}:
and
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 is the concurrence of a threequbit state ρ_{ABC} 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 in an nqubit system A_{1}⊗...⊗A_{n}, where A_{i} ≅ C^{2} for i = 1, ..., n, a generalisation of the CKW inequality is
The dual inequality in terms of the COA for nqubit states has the form^{17}
when the rank of the matrix is 2, we have
Combining Eq. (10) with Eq. (11), we have
where i, j ∈ {1, ..., n}, i ≠ j.
For any nqubit pure state , we have
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 ^{33}, where is the threetangle of concurrence. is the concurrence under bipartition ABC 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}
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:
Theorem 1. For any 2 ⊗ 2 ⊗ 2 tripartite mixed state ρ_{ABC} we have
Proof. Let ρ_{ABC} = ∑_{i}p_{i}ψ_{i}〉_{ABC}〈ψ_{i} be an optimal decomposition realising ; that is,
where ρ_{BC} = Tr_{A}ψ_{i}〉_{ABC}〈ψ_{i}, ρ_{B} = Tr_{AC}ψ_{i}〉_{ABC}〈ψ_{i} and ρ_{C} = Tr_{AB}ψ_{i}〉_{ABC}〈ψ_{i}, and we have
Combining Eq. (23) with Eq. (24), we have
The third equality holds because CREN and concurrence are equal for any rank2 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 ABC cannot be greater than the sum of the entanglement BAC and the entanglement CAB. Taking an easy example, when considering a threequbit state, the following equation exists: ψ〉_{ABC} = a010〉 + b100〉 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 multiqubit quantum systems.
Corollary 1. For any multiparty mixed state in an nqubit system^{36}, the following monogamy inequality exists:
The meaning of the first inequality is clear the bipartite entanglement between and the other qubits, when taken as a group cannot be greater than the sum of the n − 1 individual bipartite entanglements between and the other remaining qubits. We now start to consider a fourqubit system. As shown in Fig. 1, the squared CRENOA with respect to the bipartition (ABCD) is not greater than the sum of the three squared CRENOAs (the three possible bipartitions are BACD, CABD and DABC).
The meaning of the second inequality is clear the sum of the bipartite entanglements between and the other remaining qubits cannot be greater than the sum of the bipartite entanglements .
Theorem 2. For any nqubit pure state , we have
where , and .
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 rank2 pure state.
For a mixed state, CRENOA is expressed as , and we have
Furthermore, when combining this with Eq. (35), we finally get
and
Combining Eq. (37) with Eq. (38), we have Eq. (32). In other words, we give an upper bound about , i.e.,
This completes the proof.
Theorem 2 shows that the entanglement between AB and the other qubits cannot be greater than the sum of 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 2 provides a polygamytype upper bound of multiqubit entanglement between the twoqubit system AB and the other (n − 2)qubit system C_{1}C_{2}...C_{n−2} in terms of the squared CRENOA. Especially under partition AB, a twoqubit system is different from the previous monogamy inequality. When , the calculation results in . Consequently, the polygamytype relation is obtained as shown in Eq. (19).
Finally, consider the following fourqubit state: ψ〉_{ABCD} = a0100〉 + b0010〉 + c0001〉 where a^{2} + b^{2} + c^{2} = 1. We can easily get the following equations: and . Therefore, the state ψ〉_{ABCD} saturates the monogamy inequality in Eq. (32).
Theorem 3. For any nqubit pure state ,
where , and .
Proof. We have the following property for linear entropy^{35}:
Combining Eq. (24) with Eq. (41), we have
and
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
Similar to the above derivation, we give a lower bound about , i.e.,
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 monogamytype lower bound of multiqubit entanglement between the twoqubit system AB and the other (n − 2)qubit system C_{1}C_{2}...C_{n−2} in terms of the squared CRENOA. When , , and so we obtain the CWKtype relation in Eq. (18).
Finally, we consider the following fourqubit state ψ〉_{ABCD} = a1000〉 + b0010〉 + c0001〉 where a^{2} + b^{2} + c^{2} = 1, from which we can easily obtain the following equations: and . Therefore, the state ψ〉_{ABCD} saturates the monogamy inequality in Eq. (40). Therefore, a generalised monogamy inequality using negativity and CRENOA in an nqubit is proposed. These relations also give rise to a type of tradeoff 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^{37,38,39}.
Examples
In this section, we use some special states to study generalised monogamy inequalities. First, we consider the (GreenbergerHorneZeilinger) GHZ state and W state in Examples 1 and 2. Second, we consider two states in the higherdimensional system in Examples 3 and 4.
Example 1. For an arbitrary pure GHZ state in an nqubit system:
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: and , . Therefore:
Example 2. For a pure state W〉 in an nqubit system:
with . It is very important to understand the saturation of the previous CKW inequality. Using a simple calculation, we have . It is straightforward to check: , . 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 higherdimensional systems. We consider the following examples:
Example 3. For a pure, totally antisymmetric state ψ_{ABC}〉 in a 3 ⊗ 3 ⊗ 3 system^{40}:
This special quantum state is not satisfied with the previous CKW inequality^{41} but it is established in generalised monogamy inequalities. We can easily obtain and further obtain the inequalities . We now explore theorems 2 and 3. First, we have and . Therefore, we obtain the following inequalities:
Example 4. The nqudit generalised Wclass state in higherdimensional quantum systems is very useful in quantum information theory^{42}. We verify whether the generalised monogamy inequalities hold in higherdimensional systems using a special example. First, we recall the definition of nqudit generalised Wclass state^{43},
where .
Let be an nqudit pure state in a superposition of an nqudit generalised Wclass state and vacuum; that is,
for some 0 ≤ p ≤ 1.
For the squared negativity of with respect to the bipartition between A_{1} and the other qudits, the reduced density matrix of onto subsystem A_{1} is obtained as
where .
When considering the state, we need to obtain the eigenvalue of the matrix by applying the definition of pure state negativity in Eq. (5). Using a simple calculation, we find that the matrix has rank2 and we have
We now consider the case in which n = 2. The remaining cases follow analogously. The twoqudit reduced density matrix of is obtained as
where . For convenient calculation, we consider two unnormalised states:
Consequently, can be represented as where and are unnormalised states of the subsystems A_{1}A_{2}. By the HJW theorem^{44}, any purestate decomposition , with size r > 2 can be obtained by an r × r unitary matrix u_{hl} such that
for each h, for the normalized state with .
We apply the definition of mixed state negativity in Eqs (8 and 63), and then we have the twotangle based on the CREN of as
where .
From the definition of pure state negativity in Eqs (9 and 63), we have
We now try to verify the generalised monogamy inequalities of CREN in an nqudit system. For convenient calculation, we assume that , , ,
We first consider the generalisation of Theorem 1.
This special quantum state is satisfied with the generalised monogamy inequality in Eq. (25) i.e.,
For the generalisation of Theorem 2, the left of Eq. (32) is
Using Eqs (8 and 62) we can simplify the calculation to
and
After some calculations, we have
Second, taking Eq. (67) to the right side of Eq. (32), we then have
After a straightforward calculation, we obtain
Therefore, this nqudit 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. (40):
Calculating the absolute value of the difference between Eqs (72 and 76), we obtain
It is easy to check 4p^{2} (a − a^{2} − ab + b^{2} − b) > 0, as
After a straightforward calculation, we have
Therefore, this nqudit 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 nqudit 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 CKWlike inequalities and generalised monogamy inequalities. Furthermore, the generalised monogamy inequalities related to CREN and CRENOA were obtained by nqubit states. These relations also give rise to a type of tradeoff 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 Wclass 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.
Additional Information
How to cite this article: Tian, T. et al. Generalised monogamy relation of convexroof extended negativity in multilevel systems. Sci. Rep. 6, 36700; doi: 10.1038/srep36700 (2016).
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Acknowledgements
It is a pleasure to thank F. G. Zhang for inspiring discussions. We thank the anonymous referees for their valuable comments. This work was supported by the National Nature Science Foundation of China (Grant No. 1127123), the Higher School Doctoral Subject Foundation of Ministry of Education of China (Grant No. 20130202110001) and the Fundamental Research Funds for the Central Universitie (Grant No. 2016CBY003).
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College of Computer Science, Shaanxi Normal University, Xi’an, 710062, China
 Tian Tian
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Contributions
T.T. and Y. Luo contributed the idea. T.T. performed the calculations and wrote the main manuscript. Y. Luo checked the calculations. Y. Li improved the manuscript. All authors contributed to the discussion and reviewed the manuscript.
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The authors declare no competing financial interests.
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Correspondence to Yongming Li.
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