† Corresponding author. E-mail:
Project supported by the National Basic Research Program of China (Grant No. 2015CB856703), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB23030100), the National Natural Science Foundation of China (Grant Nos. 11847209, 11375200, and 11635009), and the China Postdoctoral Science Foundation.
Monogamy relation is one of the essential properties of quantum entanglement, which characterizes the distribution of entanglement in a multipartite system. By virtual of the unified-(q,s) entropy, we obtain some novel monogamy and polygamy inequalities in general class of entanglement measures. For the multiqubit system, a class of tighter monogamy relations are established in term of the α-th power of unified-(q,s) entanglement for α ≥ 1. We also obtain a class of tighter polygamy relations in the β-th (0 ≤ β ≤ 1) power of unified-(q,s) entanglement of assistance. Applying these results to specific quantum correlations, e.g., entanglement of formation, Renyi-q entanglement of assistance, and Tsallis-q entanglement of assistance, we obtain the corresponding monogamy and polygamy relations. Typical examples are presented for illustration. Furthermore, the complementary monogamy and polygamy relations are investigated for the α-th (0 ≤ α ≤ 1) and β-th (β ≥ 1) powers of unified entropy, respectively, and the corresponding monogamy and polygamy inequalities are obtained.
Quantum entanglement[1–8] plays an important role in quantum information processing, which is considered as one of the most significant resources of quantum communication. It has been proved to be completely safe for any interior or exterior eavesdropper. In the past decades, the bipartite entanglement system has been investigated extensively, and abundant precious results are obtained. The usual monogamy inequality[9,10] may write as ℰ(ρA|BC) ≥ ℰ(ρAB) + ℰ(ρAC), where ρAB = trC(ρABC) and ρAC = trB(ρABC). The ℰ here stands for one kind of entanglement measure, which is essentially different from the classical correlation. The monogamy relation indicates that the entanglement between subsystems constraints each other. In Ref. [11], the authors found that the monogamy relation may play an important role in quantum key distribution security.
To understand more about the relations among quantum entanglements of multipartite systems is crucial for their application in quantum processing. The application of monogamy relation performed in quantum information theory may be helpful in this aim. The monogamy inequality, which is also called CKW inequality,[9] was first proposed by Coffman–Kundu–Wootters based on the concurrence of the qubit system.[12] Later on, many other forms of monogamy relations were established, including those for multiqubit and high-dimensional systems.[13–22] Recently, in Refs. [23,24], the authors gave an alternative definition of the monogamy relation with no inequality employed.
Qualitatively, for a composite quantum system, the monogamy inequality implies a trade-off relation of entanglement among subsystems, i.e., the more two subsystems are entangled the less this pair will interfere with the rest of others within the system. Different from the CKW inequality fulfilled by the concurrence, the assisted entanglement Ea[25] satisfies polygamy inequality Ea(ρA|BC) ≤ Ea(ρAB) + Ea(ρAC) for tripartite state ρABC, which is the dual property of the monogamy relation in the multipartite system. For a three-qubit system, reference [25] described a class of polygamy of bipartite entanglement in term of tangle of assistance, which was later generalized to the multipartite system based on different kinds of assisted entanglements.[26–28] Generally, the entanglement of assistance does not satisfy the usual monogamy relations. In Refs. [16,20], the authors found special classes of quantum states satisfying the monogamy relation for both concurrence of assistance and negativity of assistance. It was found that for the multiqubit system, the monogamy relations for the x-th power of entanglement of formation and concurrence are satisfied for
In this paper, using the unified entropy,[29,30] we give new monogamy and polygamy inequalities, which are tighter than the existing ones in some classes of quantum states. Applying these results to the quantum correlations such as entanglement of formation, Renyi-q entanglement of assistance, and Tsallis-q entanglement of assistance, we then obtain the corresponding monogamy and polygamy relations. By using the two-qubit entanglement of individual subsystems, a class of tighter monogamy relations are established for multiqubit entanglement, in term of the α-th power of unified-(q,s) entanglement for α ≥ 1. We present also a class of tighter polygamy relations for the multiqubit entanglement, in term of the β-th (0 ≤ β ≤ 1) power of unified-(q,s) entanglement of assistance. The yet unknown complementary monogamy and polygamy relations are investigated for the α-th (0 ≤ α ≤ 1) and β-th (β ≥ 1) powers of unified entropy, respectively. Several illustrating examples are given.
For any quantum state ρ ∈ ℋ, its unified-(q,s) entropy is defined as[29,30]
For a bipartite pure state |ψ〉AB ∈ ℋAB with its reduced density matrix ρA = TrB|ψ〉AB〈ψ| onto subsystem A and each q, s ≥ 0, its unified-(q,s) entanglement (UE) is defined as[27]
As a dual quantity to UE, unified-(q,s) entanglement of assistance (UEoA) is defined as[34]
Similarly, as UEoA in Eq. (
By using UE in Eq. (
It is also shown that UEoA can be used to characterize the polygamy of multiqubit entanglement as[34]
Monogamy of entanglement is an intriguing feature of quantum entanglement, which characterizes the distributions of quantum entanglement among multipartite quantum systems, and becomes one of the hot issues in the study of quantum information theory in recent years. And the optimized monogamy relations give rise to finer characterizations of the entanglement distributions, which are tightly related to the security of quantum cryptographic protocols based on entanglement[11] (it limits the amount of correlations that an eavesdropper can have with the honest parties). Therefore, obtaining tighter monogamy relations of entanglement is necessary to understand the whole picture of quantum entanglement. Here, in term of the α-th power of UE, we provide a class of tighter monogamy relations for the multiqubit entanglement. First, we give two lemmas.
In Lemma
Generally, conditions for inequalities (
In a special case, inequality (
In Ref. [20], the authors presented
Different from the property of inequality (
With preceding lemmas, we can then get the following Theorem for multiqubit systems.
Note that the conditions of (
In this section, we investigate monogamy and polygamy relations of multiqubit states. General monogamy and polygamy inequalities are given in the α-th (0 ≤ α ≤ 1) power and the β-th (β ≥ 1) power of unified entropy, respectively, which are complementary to the preceding two sections for different regions of parameter chosen in α and β.[35,36]
By improving an inequality and using the β-th (β ≥ 1) power of UEoA, we obtain a new class of weighted polygamy inequalities of multipartite entanglement in the multiqubit quantum system.
Monogamy and polygamy of entanglement can restrict the possible correlations between the authorized users and the eavesdroppers, thus tightening the security bounds in quantum cryptography. The optimized monogamy and polygamy relations give rise to finer characterizations of the entanglement distributions. The complementary monogamy and polygamy relations may help to investigate the efficiency of entanglement used in quantum cryptography and characterizations of the entanglement distributions. These results may highlight future works on the study of quantum key distribution based on multipartite quantum entanglement distributions.
Entanglement monogamy and polygamy are two fundamental properties of multipartite entangled states. We provide in this work a characterization of multiqubit entanglement constraints in terms of UE. Employing the two-qubit entanglement of individual subsystems, a class of tighter monogamy inequalities are established for multiqubit entanglement based on the α-th (α ≥ 1) power of UE. We also provide a class of tighter polygamy inequalities of multiqubit entanglement in terms of the β-th (0 ≤ β ≤ 1) power of UEoA. Applying these results to specific quantum correlations, such as entanglement of formation, Renyi-q entanglement of assistance, and Tsallis-q entanglement of assistance, we obtain the corresponding tighter monogamy or polygamy relation for some classes of quantum states. Moreover, we investigate the α-th (0 ≤ α ≤ 1) and β-th (β ≥ 1) power of monogamy and polygamy relations based on the UE and UEoA. These monogamy and polygamy relations are complementary to the existing ones in different regions of parameters α and β. Last, it is worth mentioning that the approach used in this work is applicable to the study of monogamy and polygamy natures in other entanglement measures, such as entanglement of formation, Renyi-q entanglement of assistance, and Tsallis-q entanglement of assistance, or other high-dimensional quantum systems.
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