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(ρ_A|BC) ≥ ℰ(ρ_AB) + ℰ(ρ_AC), where ρ_AB = tr_C(ρ_ABC) and ρ_AC = tr_B(ρ_ABC). 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 E_a^[25] satisfies polygamy inequality E_a(ρ_A|BC) ≤ E_a(ρ_AB) + E_a(ρ_AC) for tripartite state ρ_ABC, 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 and x ≥ 2, respectively.^[20,21] Recently, various monogamy and polygamy inequalities are obtained under the non-negative power in the multipartite system.^{[17,18,20,22]}

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 ∈ ℋ_AB with its reduced density matrix ρ_A = Tr_B|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. (5) is continuous for the parameters q and s, it assures that UEoA reduces to Renyi-q entanglement of assistance (REoA)^[32] and Tsallis-q entanglement of assistance (TEoA)^[28] when s tends to 0 or 1, respectively. For any nonnegative s, as q tends to 1, UEoA reduces to the entanglement of assistance (EoA),^[34]

By using UE in Eq. (3) to quantify the bipartite quantum entanglement, the monogamy inequality is established in multiqubit systems; for any N-qubit state ρ_A1A2⋯AN and its two-qubit reduced density matrices ρ_A1Ai with i = 2,…,N, we have

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 2, without loss of generality, we have assumed that E_q,s(ρ_AB) ≥ E_q,s(ρ_AC), since the subsystems A and B are equivalent. Moreover, in the proof of Lemma 2, we have assumed E_q,s(ρ_AB) > 0. If E_q,s(ρ_AB) = 0 and E_q,s(ρ_AB) ≥ E_q,s(ρ_AC), then E_q,s(ρ_AB) = E_q,s(ρ_AC) = 0. The lower bound is trivially zero. For multipartite qubit systems, we have the following theorem.

Generally, conditions for inequalities (11) and (13) are not always satisfied. In following, we give a universal monogamy inequality.

Fig. 1.

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	Fig. 1. The y represents the lower bounds of UE of |ψ〉ABC, which are functions of α. The red solid line shows the values of UE of |ψ〉ABC in Example 1, the dashed line illustrates the lower bound in our calculation, the green dot-dashed line represents the lower bound given in Ref. [18].

In a special case, inequality (15) reduces to Eq. (7) of Ref. [27] for α = 1. For α > 1, (2^α – 1)^m > α^m, and all 1 ≤ m ≤ N – 3, formula (15) in Theorem 2 leads to a tighter monogamy relation with larger lower bounds than Eqs. (9) and ({37}) in Ref. [18]. Theorem 2 gives another monogamy relation based on the entanglement measure UE. Comparing inequality (11) in Theorem 1 with inequality (15) in Theorem 2, one may notice that inequality (11) is better than inequality (15). However, we note that for those classes of states that do not satisfy the conditions in Theorem 1, Theorem 2 is better.

Fig. 2.

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	Fig. 2. The y represents the lower bound of UE of |W〉AB1B2B3, which is a function of α. The red solid line denotes the UE of |W〉AB1B2B3 in Example 2, and the blue dashed line represents the lower bound from inequality (15) of Theorem 2.

In Ref. [20], the authors presented for with the conditions C_ABi ≥ C_{A|Bi + 1⋯BN – 1}, i = 1, 2, …, N – 2 for N-qubit states; in Ref. [21], the authors obtained for . Because EoF coincides with concurrence for any 2 ⊗ d-dimensional pure state, we get for with the same conditions as q tends to 1 in Theorem 1. As for , obviously, our result is better than that in Refs. [20,21].

Different from the property of inequality (11) of Theorem 1, we now provide a class of polygamy relations in multiqubit systems in terms of UEoA, which are tighter than the existing ones. For preparation, we first give two lemmas.

With preceding lemmas, we can then get the following Theorem for multiqubit systems.

Fig. 3.

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	Fig. 3. The y denotes the upper bound of the UEoA of |ψ〉ABC, which is a function of β. The red solid line represents the UEoA of |ψ〉ABC in Ref. (14), blue dashed line represents the upper bound of our result, green dot-dashed line represents the upper bound given in Ref. [18].

Note that the conditions of (20) in Theorem 3 and (21) in Ref. [18] are not generally satisfied. To get a widespread result, we have the following observation.

Fig. 4.

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	Fig. 4. The y denotes the upper bound of UEoA of |W〉AB1B2B3. Solid (red) line is exactly the UEoA of |W〉AB1B2B3, dashed (blue) line represents the upper bound of in inequality (22) for 0 ≤ β ≤ 1.

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]

Fig. 5.

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	Fig. 5. The y denotes the lower bound of UE of |ψ〉ABC, which is a function of α. The red solid line represents the UE of |ψ〉ABC in inequality (14), blue dashed line represents the lower bound of inequality (28).

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.

Fig. 6.

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	Fig. 6. The y denotes the upper bound of UEoA of |ψ〉ABC, which is a function of β. The red solid line represents the UEoA of |ψ〉ABC in inequality (14), blue dashed line represents the upper bound of inequality (30).

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.