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Guo Meng-Li, Li Bo, Wang Zhi-Xi, Fei Shao-Ming. Tighter constraints of multiqubit entanglement in terms of Rényi-α entropy. Chinese Physics B, 2020, 29(7): 070304  
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Tighter constraints of multiqubit entanglement in terms of Rényi-α entropy
Guo Meng-Li1, Li Bo2, †, Wang Zhi-Xi3, Fei Shao-Ming3, 4
Department of Mathematics, East China University of Technology, Nanchang 330013, China
School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
Max-Planck-Institute for Mathematics in the Sciences, 04103, Leipzig, Germany

 

† Corresponding author. E-mail: libobeijing2008@163.com

Project supported by the National Natural Science Foundation of China (Grant Nos. 11765016 and 11675113), the Natural Science Foundation of Beijing, China (Grant No. KZ201810028042), and Beijing Natural Science Foundation, China (Grant No. Z190005).

Abstract

Quantum entanglement plays essential roles in quantum information processing. The monogamy and polygamy relations characterize the entanglement distributions in the multipartite systems. We present a class of monogamy inequalities related to the μ-th power of the entanglement measure based on Rényi-α entropy, as well as polygamy relations in terms of the μ-th power of Rényi-α entanglement of assistance. These monogamy and polygamy relations are shown to be tighter than the existing ones.

PACS: ;03.67.Mn;;03.65.Ud;;03.65.Yz;
Keyword:;monogamy relations;;polygamy relations;;Rényi-α entropy;;Hamming weight;
1. Introduction

Quantum entanglement is one of the most quintessential features of quantum mechanics, which distinguishes the quantum world from the classical one and plays essential roles in quantum information processing,[1–5] revealing the basic understanding of the nature of quantum correlations. One distinct property of quantum entanglement is that a quantum system entangled with another system limits its sharing with other systems, known as the monogamy of entanglement.[6,7] The monogamy of entanglement can be used as a resource to distribute a secret key which is secure against unauthorized parties.[8,9] It also plays a significant role in many fields of physics such as foundations of quantum mechanics,[34,10] condensed matter physics,[11] statistical mechanics,[34] and even black-hole physics.[12,13]

The monogamy inequality was first introduced by Coffman–Kundu–Wootters (CKW) by using tangle as a bipartite entanglement measure in three-qubit systems,[14] and then generalized to multiqubit systems based on various entanglement measures.[15] The assisted entanglement is a dual concept to bipartite entanglement measure, which shows polygamy relations in multiparty quantum systems. For a three-qubit state ρABC, a polygamy inequality was introduced as[16] τa(ρA|BC)≤τa (ρA|B) + ρa(ρA|C), where τa(ρA|BC) = max ∑ipi τ(|ψi 〉A|B) is the tangle of assistance,[17,16] with the maximum taking over all possible pure state decompositions of ρAB = ∑i pi | ψi 〉AB 〈 ψi|. This tangle-based polygamy inequality was extended to multiqubit systems and also high-dimensional quantum systems in terms of various entropy entanglement measures.[18,19] General polygamy inequalities of entanglement are also established in arbitrary dimensional multipartite quantum systems.[20–25]

In this paper, we investigate the monogamy and polygamy constraints based on the μ-th power of entanglement measures in terms of the Rényi-α entropy for multiqubit systems. By using the Hamming weight of binary vectors, we present a class of monogamy inequalities for multiqubit entanglement based on the μ-th power of Rényi-α entanglement (RαE)[26] for μ ≥ 1. For 0 ≤ μ ≤ 1, we introduce a class of tight polygamy inequalities based on the μ-th power of the Rényi-α entanglement of assistance (RαEoA). Then, we show that both the monogamy inequalities with μ ≥1 and the polygamy inequalities with 0≤ α ≤ 1 can be further improved to be tighter under certain conditions. These monogamy and polygamy relations are shown to be tighter than the existing ones. Moreover, our monogamy inequality is shown to be more effective for the counterexamples of the CKW monogamy inequality in higher-dimensional systems.

2. Preliminaries

We first recall the conceptions of Rényi-α entropy, Rényi-α entanglement, and multiqubit monogamy and polygamy inequalities. For any α > 0, α ≠ 1, the Rényi-α entropy of a quantum state ρ is defined as[27]

Sα(ρ) reduces to the von Neumann entropy when α approaches to 1.

The Rényi-α entanglement (RαE) Eα (|ψ 〉AB ) of a bipartite pure state |ψ 〉AB is defined as

where ρA = TrB |ψ 〉AB 〈 ψ| is the reduced state of system A. For a mixed state ρAB, the Rényi-α entanglement is given by

where the minimum is taken over all possible pure state decompositions of ρAB = ∑ipi |ψi 〉AB 〈 ψi|.

As a dual concept to RαE, the Rényi-α entanglement of assistance (RαEoA) is introduced as

where the maximum is taken over all possible pure state decompositions of ρAB.[28]

For any multiqubit state ρAB0 ⋯ BN – 1, a monogamous inequality has been presented in Ref. [28] for α ≥ 2,

where Eα (ρA|B0 ⋯ BN – 1) is the RαE of ρAB0 ⋯ BN – 1 with respect to the bipartition between A and B0 ⋯ BN – 1, and Eα(ρA| Bi) is the RαE of the reduced density matrix ρA Bi, i = 0,…,N – 1.

In addition, a class of polygamy inequalities has been obtained for multiqubit systems,

for 0 ≤ α ≤ 2, α ≠ 1, where Eρ (ρA B20 ⋯ BN – 1) is the RαEoA of ρA B0 ⋯ BN – 1 with respect to the bipartition between A and B0 ⋯ BN – 1, and Eaρ(ρA| Bi) is the RαEoA of the reduced density matrix ρA Bi, i = 0,…,N – 1.

In Ref. [29], Kim established a class of tight monogamy inequalities of multiqubit entanglement in terms of Hamming weight. For any nonnegative integer j with binary expansion , where log2j ≤ n and ji ∈ 0, 1 for i = 0, …, n – 1, one can always define a unique binary vector associated with j, j = (j0, j1, …, jn – 1). The Hamming weight ωH(j) of the binary vector j is defined to be the number of 1’s in its coordinates.[35] Moreover, the Hamming weight ωH(j) is bounded above by log2j,

Kim proposed the tight constraints of multiqubit entanglement based on Hamming weights[29]

for μ ≥ 1, and

for 0≤μ ≤ 1. Inequalities (5) and (6) are then further written as

for μ ≥ 1, and

for 0 ≤ μ ≤ 1.

In the following we show that these inequalities above can be further improved to be much tighter under certain conditions, which provide tighter constraints on the multiqubit entanglement distribution.

3. Tighter constraints of multiqubit entanglement in terms of RαE

We first present a class of tighter monogamy and polygamy inequalities of multiqubit entanglement in terms of the μ-th power of RαE. We need the following results.[32] Suppose k is a real number, 0 > k ≤1. Then for any 0≤ x ≤ k, we have

for μ ≥ 1, and

for 0≤ μ ≤ 1. Based on inequality (7), we have the following theorem for RαE.

We first show that the inequality (10) holds for the case of N = 2n. For n = 1, let ρAB0 and ρAB1} be the two-qubit reduced density matrices of a three-qubit pure state ρAB0B1. We obtain

Combining inequalities (7) and (11), we have

From formulas (12) and (13), we obtain

Therefore, the inequality (10) holds for n = 1.

We assume that the inequality (10) holds for N = 2n –1 with n≥ 2, and prove the case of N = 2n. For an (N + 1)-qubit pure state ρAB0B1 BN – 1, we have Eα(ρA|Bj + 2n − 1) ≤ k2n – 1Eα(ρA|Bj) from inequality (11). Therefore,

Thus, we have

According to the induction hypothesis, we obtain

By relabeling the subsystems, the induction hypothesis leads to

Thus, we have

Now consider a (2n + 1)-qubit state

which is the tensor product of ρAB0B1… BN – 1 and an arbitraryq (2n - N)-qubit state σBN … B2n – 1. We have

where ΓA|Bj is the two-qubit reduced density matrix of ΓAB0B1… B2n – 1, j = 0,1,…,2n – 1. Therefore,

where ΓA|B0B1…B2n – 1 is separated to the bipartition AB0… BN – 1 and BN… B2n – 1, Eα(ΓA|B0 B1 ⋯ B2n – 1) = Eα(ρA|B0 B1 ⋯ BN – 1), Eα(ΓA|Bj) = 0 for j = N, …, 2n–1, and ΓABj = ρABj for each j = 0, …, N – 1.

Since for μ ≥1, for any multiqubit state ρAB0 B1 ⋯ BN – 1 we have the following relation:

Therefore, our inequality (9) in Theorem 1 is always tighter than the inequality (5).

In fact, the tighter monogamy inequality (9) holds not only for multiqubit systems, but also for some multipartite higher-dimensional quantum systems, which can be proved in a similar way as in Ref. [29]. Here, we show that inequality (9) is also more efficient than inequality (5) for such higher-dimensional quantum systems. Let us consider the counterexample of the CKW inequality in tripartite quantum systems[30]

One has Eα(|ψ〉A|BC) = Sα(ρ). Taking α = 3, we have Eα(|ψ〉A|BC) = log 3 and the RαE of the two-qubit reduced density matrices are

In the case k = 1, for μ≥1, we have

Therefore, one gets

where μ≥1, see Fig. 1. In other words, our new monogamy inequality is indeed tighter than the previous one given in Ref. [29].

Fig. 1. Rényi-α entanglement with respect to μ: the solid line is for y1 and the dashed line for y2 from the result in Ref. [29].

Under certain conditions, the inequality (9) can even be improved further to become a much tighter inequality.

For any multiqubit state ρAB0… BN – 1, it is easy to show that

Thus,

where the second inequality is due to the induction hypothesis.

In fact, according to Eq. (4), for any μ ≥ 1, one has

For the case of μ < 0, we can also derive a tighter upper bound of .

4. Tighter constraints of multiqubit entanglement in terms of RαEoA

We consider now the RαEoA defined in Eq. (1), and provide a class of polygamy inequalities satisfied by the multiqubit entanglement in terms of RαEoA.

Then we assume that the inequality (24) holds for N = 2n–1 with n≥ 2. Consider the case of N = 2n. For an (N + 1)-qubit pure state ρAB0B1… BN – 1 with its two-qubit reduced density matrices ρABj, j = 0,1,…,N− - 1, we have due to the ordering of subsystems in the inequality (25). Then, we obtain

Hence,

According to the induction hypothesis, we obtain

By relabeling the subsystems, the induction hypothesis leads to

Therefore,

Consider the (2n+1)-qubit state (17). We have

Since for 0 ≤ μ ≤ 1, it is easy to see that inequality (23) is tighter than inequality (6).

As an example, let us consider the three-qubit W-state[33]

We have and

In the case of k = 1 and 0≤ μ ≤1, we have

Therefore, we obtain

where 0≤ μ≤1, see Fig. 2.

Similar to the improvement from the inequality (9) to the inequality (16), we can also improve the polygamy inequality in Theorem 4. The proof is similar to that of Theorem 2.

Since ωH(j)≤ j, for 0≤μ≤1 we obtain

Therefore, for any multiqubit state ρAB0 … BN – 1 satisfying the condition (28), the inequality (27) of Theorem 5 is tighter than the inequality (23) of Theorem 4.

Fig. 2. Rényi-α entanglement y with respect to μ: the solid line is for y3 and the dashed line for y4 from the result in Ref. [29].
5. Conclusion

Quantum entanglement is the essential resource in quantum information. The monogamy and polygamy relations characterize the entanglement distributions in the multipartite systems. Tighter monogamy and polygamy inequalities give finer characterization of the entanglement distribution. In this article, by using the Hamming weights of binary vectors we have proposed a class of monogamy inequalities related to the μ-th power of the entanglement measure based on Rényi-α entropy, polygamy relations in terms of the μ-th powered of RαEoA for 0≤ μ ≤ 1. These new monogamy and polygamy relations are shown to be tighter than the existing ones. Moreover, it has been shown that our monogamy inequality is effective for the counterexamples of the CKW monogamy inequality in higher-dimensional systems. Our results may motivate further investigations on the entanglement distribution in multipartite systems.

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