Tighter monogamy relations of multiqubit entanglement in terms of R\'{e}nyi-$\alpha$ entanglement

TL;DR

This paper introduces tighter monogamy relations for multiqubit entanglement using Rényi-$ ext{alpha}$ entanglement, providing improved bounds over previous relations for certain parameter ranges.

Contribution

The authors develop a new class of tighter monogamy relations based on Rényi-$ ext{alpha}$ entanglement, extending and strengthening existing bounds for multiqubit systems.

Findings

Tighter monogamy relations are established for $ ext{alpha} ext{ } ext{in} ext{ }[rac{ ext{sqrt{7}}-1}{2}, 2)$ with $ ext{eta}>2$.

The new bounds are larger than existing monogamy bounds for specific parameter ranges.

The relations improve understanding of entanglement distribution constraints in multiqubit systems.

Abstract

We present a class of tight monogamy relations in terms of R\'{e}nyi-$\alpha$ entanglement, which are tighter than the monogamy relations of multiqubit entanglement just based on the power of the R\'{e}nyi-$\alpha$ entanglement for $\alpha\geq 2$ and the power $\eta>1$. For $2>\alpha\geq\frac{\sqrt{7}-1}{2}$ and the power $\eta>2$, we establish a class of tight monogamy relations of multiqubit entanglement with larger lower bounds than the existing monogamy relations of multiqubit entanglement.

Equations

$$E_{\alpha }(|\psi {⟩}_{AB})=\frac{1}{1-\alpha }{\log }_2(\text{tr}{\rho }_A^{\alpha })$$

$$E_{\alpha }({\rho }_{AB})=\min \sum _i{p}_i{E}_{\alpha }(|{\psi }_i{⟩}_{AB}),$$

$$C(|\varphi {⟩}_{AB})=\sqrt{2(1-\text{tr}{\rho }_A^2)},$$

$$C({\rho }_{AB})=\min \sum _j{p}_{j}C(|{\varphi }_j{⟩}_{AB}),$$

$$C^2({\rho }_{A|B_1\cdots B_{N-1}})-C^2({\rho }_{A|B_1})-\cdots -C^2({\rho }_{A|B_{N-1}})\ge 0,$$

$$E_{\alpha }({\rho }_{AB})=g_{\alpha }(C({\rho }_{AB})).$$

$$g_{\alpha }(x)=\frac{1}{1-\alpha }{\log }_2[(\frac{1-\sqrt{1-x^2}}{2}{)}^{\alpha }+(\frac{1+\sqrt{1-x^2}}{2}{)}^{\alpha }]$$

$$g_{\alpha }(\sqrt{x^2+y^2})\ge g_{\alpha }(x)+g_{\alpha }(y)$$

$$[g_{\alpha }(\sqrt{x^2+y^2}){]}^2\ge [g_{\alpha }(x){]}^2+[g_{\alpha }(y){]}^2$$

$$(1+x{)}^{\mu }\ge 1+\frac{\mu }{2}x+(2^{\mu }-\frac{\mu }{2}-1)x^{\mu }\ge 1+(2^{\mu }-1)x^{\mu }.$$

$$E_{\alpha }^{\eta }({\rho }_{A|B_1\cdots B_{N-1}})$$

$$+(2^{\eta }-1{)}^{N-3}\left\{E_{\alpha }^{\eta }({\rho }_{A|B_{N-2}})+\frac{\eta }{2}{E}_{\alpha }^{\eta -1}({\rho }_{A|B_{N-2}})E_{\alpha }({\rho }_{A|B_{N-1}})+(2^{\eta }-\frac{\eta }{2}-1)E_{\alpha }^{\eta }({\rho }_{A|B_{N-1}})\right\}$$

$$[g_{\alpha }(\sqrt{x^2+y^2}){]}^{\eta }\ge [g_{\alpha }(x)+g_{\alpha }(y){]}^{\eta }.$$

$$[g_{\alpha }(\sqrt{x^2+y^2}){]}^{\eta }\ge [g_{\alpha }(x){]}^{\eta }+\frac{\eta }{2}[g_{\alpha }(x){]}^{\eta -1}[g_{\alpha }(y)]+(2^{\eta }-\frac{\eta }{2}-1)[g_{\alpha }(y){]}^{\eta }.$$

$$E_{\alpha }^{\eta }(|\Psi {⟩}_{A|B_1\cdots B_{N-1}})$$

$$\ge {\left[g_{\alpha }\left(\sqrt{C^2({\rho }_{A|B_1})+\cdots +C^2({\rho }_{A|B_{N-1}})}\right)\right]}^{\eta }$$

$$\ge [g_{\alpha }(C({\rho }_{A|B_1})){]}^{\eta }+\frac{\eta }{2}[g_{\alpha }(C({\rho }_{A|B_1})){]}^{\eta -1}{g}_{\alpha }\left(\sqrt{C^2({\rho }_{A|B_2})+\cdots +C^2({\rho }_{A|B_{N-1}})}\right)$$

$$+(2^{\eta }-\frac{\eta }{2}-1){\left[g_{\alpha }\left(\sqrt{C^2({\rho }_{A|B_2})+\cdots +C^2({\rho }_{A|B_{N-1}})}\right)\right]}^{\eta }$$

$$\ge \cdots$$

$$\ge [g_{\alpha }(C({\rho }_{A|B_1})){]}^{\eta }+(2^{\eta }-1)[g_{\alpha }(C({\rho }_{A|B_2})){]}^{\eta }+\cdots +(2^{\eta }-1{)}^{N-4}[g_{\alpha }(C({\rho }_{A|B_{N-3}})){]}^{\eta }$$

$$+(2^{\eta }-1{)}^{N-3}\{[g_{\alpha }(C({\rho }_{A|B_{N-2}})){]}^{\eta }+\frac{\eta }{2}[g_{\alpha }(C({\rho }_{A|B_{N-2}})){]}^{\eta -1}{g}_{\alpha }(C({\rho }_{A|B_{N-1}}))$$

$$+(2^{\eta }-\frac{\eta }{2}-1)[g_{\alpha }(C({\rho }_{A|B_{N-1}})){]}^{\eta }\}$$

$$=E_{\alpha }^{\eta }({\rho }_{A|B_1})+(2^{\eta }-1)E_{\alpha }^{\eta }({\rho }_{A|B_2})+\cdots +(2^{\eta }-1{)}^{N-4}{E}_{\alpha }^{\eta }({\rho }_{A|B_{N-3}})$$

$$+(2^{\eta }-1{)}^{N-3}\{E_{\alpha }^{\eta }({\rho }_{A|B_{N-2}})+\frac{\eta }{2}{E}_{\alpha }^{\eta -1}({\rho }_{A|B_{N-2}})E_{\alpha }({\rho }_{A|B_{N-1}})$$

$$+(2^{\eta }-\frac{\eta }{2}-1)E_{\alpha }^{\eta }({\rho }_{A|B_{N-1}})\},$$

$$E_{\alpha }({\rho }_{A|B_1\cdots B_{N-1}})$$

$$=\sum _k{p}_k{g}_{\alpha }(C(|{\phi }_k{⟩}_{A|B_1\cdots B_{N-1}}))$$

$$\ge g_{\alpha }\left(\sum _k{p}_{k}C(|{\phi }_k{⟩}_{A|B_1\cdots B_{N-1}})\right)$$

$$\ge g_{\alpha }\left(\sum _l{p}_{l}C(|{\chi }_l{⟩}_{A|B_1\cdots B_{N-1}})\right)$$

$$=g_{\alpha }(C({\rho }_{A|B_1\cdots B_{N-1}})),$$

$$E_{\alpha }^{\eta }({\rho }_{A|B_1\cdots B_{N-1}})$$

$$\ge {\left[g_{\alpha }\left(\sqrt{C^2({\rho }_{A|B_1})+\cdots +C^2({\rho }_{A|B_{N-1}})}\right)\right]}^{\eta }$$

$$\ge [g_{\alpha }(C({\rho }_{A|B_1})){]}^{\eta }+(2^{\eta }-1)[g_{\alpha }(C({\rho }_{A|B_2})){]}^{\eta }+\cdots +(2^{\eta }-1{)}^{N-4}[g_{\alpha }(C({\rho }_{A|B_{N-3}})){]}^{\eta }$$

$$+(2^{\eta }-1{)}^{N-3}\{[g_{\alpha }(C({\rho }_{A|B_{N-2}})){]}^{\eta }+\frac{\eta }{2}[g_{\alpha }(C({\rho }_{A|B_{N-2}})){]}^{\eta -1}{g}_{\alpha }(C({\rho }_{A|B_{N-1}}))$$

$$+(2^{\eta }-\frac{\eta }{2}-1)[g_{\alpha }(C({\rho }_{A|B_{N-1}})){]}^{\eta }\}$$

$$=E_{\alpha }^{\eta }({\rho }_{A|B_1})+(2^{\eta }-1)E_{\alpha }^{\eta }({\rho }_{A|B_2})+\cdots +(2^{\eta }-1{)}^{N-4}{E}_{\alpha }^{\eta }({\rho }_{A|B_{N-3}})$$

$$+(2^{\eta }-1{)}^{N-3}\{E_{\alpha }^{\eta }({\rho }_{A|B_{N-2}})+\frac{\eta }{2}{E}_{\alpha }^{\eta -1}({\rho }_{A|B_{N-2}})E_{\alpha }({\rho }_{A|B_{N-1}})$$

$$+(2^{\eta }-\frac{\eta }{2}-1)E_{\alpha }^{\eta }({\rho }_{A|B_{N-1}})\}.$$

$$E_{\alpha }^{\eta }({\rho }_{A|B_1\cdots B_{N-1}})\ge$$

$$+(2^{\eta }-1{)}^{m+1}[E_{\alpha }^{\eta }({\rho }_{A|B_{m+1}})+\cdots +E_{\alpha }^{\eta }({\rho }_{A|B_{N-3}})]$$

$$+(2^{\eta }-1{)}^m\{(2^{\eta }-\frac{\eta }{2}-1)[E_{\alpha }({\rho }_{A|B_{N-2}}){]}^{\eta }+\frac{\eta }{2}{E}_{\alpha }({\rho }_{A|B_{N-2}})E_{\alpha }^{\eta -1}({\rho }_{A|B_{N-1}})$$

$$+[E_{\alpha }({\rho }_{A|B_{N-1}}){]}^{\eta }\}$$

$$E_{\alpha }^{\eta }({\rho }_{A|B_1\cdots B_{N-1}})$$

Full text

I INTRODUCTION

A key property of entanglement is known as monogamy relations [1, 2], that is, entanglement cannot be freely shared unconditionally among the multipartite quantum systems. The first mathematical characterization of monogamy relation was known as the monogamy inequality for three-qubit quantum state in terms of squared concurrence, and it is called CKW-inequality [1]. Furthermore, Osborne and Verstraete generalized this monogamy inequality to arbitrary multiqubit systems [3]. Later, the same monogamy inequality was also generalized to other entanglement measures such as entanglement negativity [4] and entanglement of formation [5]. Monogamy relations are used to characterize the distribution of entanglement in multipartite systems. Moreover, the monogamy property has many important applications in quantum information theory [6], condensed-matter physics [7] and even black-hole physics [8].

As a generalization of entanglement of formation, the Rényi-$\alpha$ entanglement is a well-defined entanglement measure, and has been widely used in the study of quantum information theory [9, 10]. It has been shown that if $\alpha\geq 2$, the Rényi-$\alpha$ entanglement satisfies the monogamy relations in $N$-qubit systems [11]. When $\alpha\geq\frac{\sqrt{7}-1}{2}$, the squared Rényi-$\alpha$ entanglement satisfies the monogamy relations in $N$-qubit systems [12]. Recently, a class of tight monogamy relations were derived in multiqubit systems [13-16]. In this paper, we establish a class of tight monogamy relations of multiqubit entanglement in terms of Rényi-$\alpha$ entanglement related to the power of the Rényi-$\alpha$ entanglement, which are tighter than the results in [12, 15, 16].

II THE RÉNYI-$\alpha$ ENTANGLEMENT

The Rényi-$\alpha$ entanglement of a bipartite pure state $|\psi\rangle_{AB}$, is defined as [17]

[TABLE]

for any $\alpha>0$ and $\alpha\neq 1$, $\rho_{A}=\text{tr}_{B}(|\psi\rangle_{AB}\langle\psi|)$. If $\alpha$ tends to 1, the Rényi-$\alpha$ entanglement converges to the von Neumann entropy. For a bipartite mixed state $\rho_{AB}$, the Rényi-$\alpha$ entanglement is defined via the convex-roof extension

[TABLE]

where the minimum taken over all possible pure-state decompositions of $\rho_{AB}=\sum_{i}p_{i}|\psi_{i}\rangle_{AB}\langle\psi_{i}|$.

Let us recall the definition of concurrence. For a bipartite pure state $|\phi\rangle_{AB}$, the concurrence is [18]

[TABLE]

where $\rho_{A}=\text{tr}_{B}(|\phi\rangle_{AB}\langle\phi|)$. For a mixed state $\rho_{AB}$, the concurrence is defined via the convex-roof extension

[TABLE]

where the minimum taken over all possible pure-state decompositions of $\rho_{AB}=\sum_{j}p_{j}|\phi_{j}\rangle_{AB}\langle\phi_{j}|$.

For an arbitrary $N$-qubit state $\rho_{A|B_{1}\cdots B_{N-1}}\in\mathcal{H}_{A}\otimes\mathcal{H}_{B_{1}}\otimes\cdots\otimes\ \mathcal{H}_{B_{N-1}}$, the concurrence $C(\rho_{A|B_{1}\cdots B_{N-1}})$ of the state $\rho_{A|B_{1}\cdots B_{N-1}}$ in the partition $A$ and $B_{1}\cdots\ B_{N-1}$, satisfies [3]

[TABLE]

where $\rho_{A|B_{i}}=\text{tr}_{B_{1}\cdots B_{i-1}B_{i+1}\cdots B_{N-1}}(\rho_{A|B_{1}\cdots B_{N-1}})$, $\mathcal{H}_{A},\mathcal{H}_{B_{1}},\cdots,\mathcal{H}_{B_{N-1}}$ are Hilbert spaces of the systems $A,B_{1},\cdots,B_{N-1}$, respectively.

It has been proved that [11, 19], when $\alpha\geq\frac{\sqrt{7}-1}{2}$, for a two-qubit state, the Rényi-$\alpha$ entanglement has an analytical formula

[TABLE]

Here the function $g_{\alpha}(x)$ is a monotonically increasing and convex function expressed as

[TABLE]

in $0\leq x\leq 1$.

The function $g_{\alpha}(x)$ in Eq. (7) for $\alpha\geq 2$, has one important property such that [11]

[TABLE]

for $0\leq x,y,x^{2}+y^{2}\leq 1$.

When $\alpha\geq\frac{\sqrt{7}-1}{2}$, it is easy to see in [12] that the function $g_{\alpha}(x)$ satisfies the following relations

[TABLE]

for $0\leq x,y,x^{2}+y^{2}\leq 1$.

III TIGHTER MONOGAMY RELATIONS FOR RÉNYI-$\alpha$ ENTANGLEMENT

In the following, we establish a class of tight Rényi-$\alpha$ entanglement monogamy relations related to the power $\eta$. We first provide the following lemma.

Lemma 1. For $x\in[0,1]$ and $\mu\geq 1$, then

[TABLE]

Proof. If $x=0$, the inequality is trivial. Otherwise, let $f(\mu,x)=\frac{(1+x)^{\mu}-\frac{\mu}{2}x-1}{x^{\mu}}$. Then, $\frac{\partial f}{\partial x}=\frac{\mu x^{\mu-1}[1+\frac{(\mu-1)}{2}x-(1+x)^{\mu-1}]}{x^{2\mu}}$. When $\mu\geq 1$ and $0\leq x\leq 1$, it is obviously that $1+\frac{(\mu-1)}{2}x\leq(1+x)^{\mu-1}$. Thus, $\frac{\partial f}{\partial x}\leq 0$, $f(\mu,x)$ is a decreasing function of $x$, i.e. $f(\mu,x)\geq f(\mu,1)=2^{\mu}-\frac{\mu}{2}-1$. Thus we have $(1+x)^{\mu}\geq 1+\frac{\mu}{2}x+(2^{\mu}-\frac{\mu}{2}-1)x^{\mu}$.

Since $x\geq x^{\mu}$, for $x\in[0,1]$ and $\mu\geq 1$, one gets $1+\frac{\mu}{2}x+(2^{\mu}-\frac{\mu}{2}-1)x^{\mu}=1+\frac{\mu}{2}(x-x^{\mu})+(2^{\mu}-1)x^{\mu}\geq 1+(2^{\mu}-1)x^{\mu}$. Altogether, we can get $(1+x)^{\mu}\geq 1+\frac{\mu}{2}x+(2^{\mu}-\frac{\mu}{2}-1)x^{\mu}\geq 1+(2^{\mu}-1)x^{\mu}$.

Now we provide our main results of this paper.

Lemma 2. For an $N$-qubit state $\rho_{A|B_{1}\cdots B_{N-1}}\in\mathcal{H}_{A}\otimes\mathcal{H}_{B_{1}}\otimes\cdots\otimes\mathcal{H}_{B_{N-1}}$, if $C(\rho_{A|B_{i}})\geq C(\rho_{A|B_{i+1}\cdots B_{N-1}})$ for $i=1,2,\cdots,N-2$, then

[TABLE]

for $\alpha\geq 2$ and the power $\eta\geq 1$.

Proof. From the inequality (8), for $\eta\geq 1$, we have

[TABLE]

Without loss of generality, we assume $x\geq y$, then we obtain

[TABLE]

Here the inequality is due to Lemma 1.

Let us first consider an $N$-qubit pure state $|\Psi\rangle_{A|B_{1}\cdots B_{N-1}}$. The entanglement $E_{\alpha}(|\Psi\rangle_{A|B_{1}\cdots B_{N-1}})$ and $C(|\Psi\rangle_{A|B_{1}\cdots B_{N-1}})$ are related by the function $g_{\alpha}(x)$ in Eq. (7) since the subsystem $B_{1}\cdots B_{N-1}$ can be regarded as a logic qubit. Thus, we can obtain

[TABLE]

where we have utilized the monogamy inequality (5) and the monotonically increasing property of the function $g_{\alpha}(x)$ to obtain the the first inequality, the second inequality is due to Eq. (13) by letting $x=C(\rho_{A|B_{1}})$ and $y=\sqrt{C^{2}(\rho_{A|B_{2}})+\cdots+C^{2}(\rho_{A|B_{N-1}})}$, the other inequalities are from iterative use of Eq. (13) and Eq. (8). In fact, we also use the conditions $1+\frac{\mu}{2}x+(2^{\mu}-\frac{\mu}{2}-1)x^{\mu}\geq 1+(2^{\mu}-1)x^{\mu}$ and $C(\rho_{A|B_{i}})\geq C(\rho_{A|B_{i+1}\cdots B_{N-1}})\geq\sqrt{C^{2}(\rho_{A|B_{i+1}})+\cdots+C^{2}(\rho_{A|B_{N-1}})}$, $i=1,2,\cdots,N-2$. Since for any two-qubit state $\rho_{AB}$, when $\alpha\geq\frac{\sqrt{7}-1}{2}$, $E_{\alpha}(\rho_{AB})=g_{\alpha}(C(\rho_{AB}))$, we obtain the last equality.

Next, let us consider an $N$-qubit mixed state $\rho_{A|B_{1}\cdots B_{N-1}}$. Suppose that the optimal decomposition for $E_{\alpha}(\rho_{A|B_{1}\cdots B_{N-1}})$ is $\rho_{A|B_{1}\cdots B_{N-1}}=\sum_{k}p_{k}|\varphi_{k}\rangle_{A|B_{1}\cdots B_{N-1}}\langle\varphi_{k}|\in\mathcal{H}_{A}\otimes\mathcal{H}_{B_{1}}\otimes\cdots\otimes\mathcal{H}_{B_{N-1}}$. Thus, we have

[TABLE]

where the first inequality follows from the convex property of the function $g_{\alpha}(x)$, the second equality is satisfied due to $\{p_{l},|\chi_{l}\rangle_{A|B_{1}\cdots B_{N-1}}\}$ being the optimal decomposition for $C(\rho_{A|B_{1}\cdots B_{N-1}})$.

Now we can derive

[TABLE]

Here in the second inequality we have used the monogamy inequality (5) and the monotonically increasing property of the function $g_{\alpha}(x)$. By iterative use of inequality (13), we have the third inequality. As a matter of fact, the conditions $1+\frac{\mu}{2}x+(2^{\mu}-\frac{\mu}{2}-1)x^{\mu}\geq 1+(2^{\mu}-1)x^{\mu}$ and $C(\rho_{A|B_{i}})\geq C(\rho_{A|B_{i+1}\cdots B_{N-1}})\geq\sqrt{C^{2}(\rho_{A|B_{i+1}})+\cdots+C^{2}(\rho_{A|B_{N-1}})}$, $i=1,2,\cdots,N-2$, have been used. Since for any two-qubit state $\rho_{AB}$, when $\alpha\geq\frac{\sqrt{7}-1}{2}$, $E_{\alpha}(\rho_{AB})=g_{\alpha}(C(\rho_{AB}))$, one gets the last equality, and the proof is completed.

Based on the Lemma 2, if some $C(\rho_{A|B_{i}})\geq C(\rho_{A|B_{i+1}\cdots B_{N-1}})$ and some $C(\rho_{A|B_{j}})\leq C(\rho_{A|B_{j+1}\cdots B_{N-1}})$, we have the following conclusion.

Lemma 3. For an $N$-qubit state $\rho_{A|B_{1}\cdots B_{N-1}}\in\mathcal{H}_{A}\otimes\mathcal{H}_{B_{1}}\otimes\cdots\otimes\mathcal{H}_{B_{N-1}}$, if $C(\rho_{A|B_{i}})\geq C(\rho_{A|B_{i+1}\cdots B_{N-1}})$ for $i=1,2,\cdots m$, and $C(\rho_{A|B_{j}})\leq C(\rho_{A|B_{j+1}\cdots B_{N-1}})$ for $j=m+1,\cdots,N-2,\forall\ 1\leq m\leq N-3,N\geq 4$, then

[TABLE]

for $\alpha\geq 2$ and the power $\eta\geq 1$.

Proof. From Lemma 2, we have

[TABLE]

With a similar procedure as $C(\rho_{A|B_{j}})\leq C(\rho_{A|B_{j+1}\cdots B_{N-1}})$ for $j=m+1,\cdots,N-2$, we have

[TABLE]

Combining inequalities (18) and (19), we have Lemma 3.

Now, we present a tight monogamy relation for $2>\alpha\geq\frac{\sqrt{7}-1}{2}$.

Lemma 4. For an $N$-qubit state $\rho_{A|B_{1}\cdots B_{N-1}}\in\mathcal{H}_{A}\otimes\mathcal{H}_{B_{1}}\otimes\cdots\otimes\mathcal{H}_{B_{N-1}}$, if $C(\rho_{A|B_{i}})\geq C(\rho_{A|B_{i+1}\cdots B_{N-1}})$ for $i=1,2,\cdots,N-2$, then

[TABLE]

for $2>\alpha\geq\frac{\sqrt{7}-1}{2}$ and the power $\eta\geq 2$, where $t=\frac{\eta}{2}$.

Proof. From the inequality (9), for $\eta\geq 2$, we have

[TABLE]

Without loss of generality, we assume $x\geq y$, according to Lemma 1, we have

[TABLE]

By using inequalities (13) and (14) and the similar consideration in the proof of Lemma 2, we obtain Lemma 4. If some $C(\rho_{A|B_{i}})\geq C(\rho_{A|B_{i+1}\cdots B_{N-1}})$ and some $C(\rho_{A|B_{j}})\leq C(\rho_{A|B_{j+1}\cdots B_{N-1}})$, then we have the following lemma.

Lemma 5. For an $N$-qubit state $\rho_{A|B_{1}\cdots B_{N-1}}\in\mathcal{H}_{A}\otimes\mathcal{H}_{B_{1}}\otimes\cdots\otimes\mathcal{H}_{B_{N-1}}$, if $C(\rho_{A|B_{i}})\geq C(\rho_{A|B_{i+1}\cdots B_{N-1}})$ for $i=1,2,\cdots,m$, and $C(\rho_{A|B_{j}})\leq C(\rho_{A|B_{j+1}\cdots B_{N-1}})$ for $j=m+1,\cdots,N-2,\forall\ 1\leq m\leq N-3,N\geq 4$, then

[TABLE]

for $2>\alpha\geq\frac{\sqrt{7}-1}{2}$ and the power $\eta\geq 2$, where $t=\frac{\eta}{2}$.

We note that for $\eta>2$, $t=\frac{\eta}{2}>1$, $1+\frac{t}{2}x+(2^{t}-\frac{t}{2}-1)x^{t}>1+(2^{t}-1)x^{t}$ for all $x\in(0,1)$, Lemma 4 and Lemma 5 give a tighter monogamy relation with larger lower bounds than the result in [12, 15, 16].

IV CONCLUSION

As a fundamental problem in quantum entanglement theory, multipartite entanglement has attracted increasing interest over the last 20 years. We have investigated the monogamy relations related to the power of the Rényi-$\alpha$ entanglement for $N$-qubit state. When $\alpha\geq 2$ and the power $\eta>1$, we derive a tighter monogamy relation than the monogamy relations just based on the power of the Rényi-$\alpha$ entanglement. For $2>\alpha\geq\frac{\sqrt{7}-1}{2}$ and the power $\eta>2$, we give a tighter monogamy relation with larger lower bounds than the result in [12, 15, 16]. Our result can provide a useful methodology to study further the monogamy properties of the multiparty quantum entanglement.

Acknowledgements.

This work was supported by the National Natural Science Foundation of China under Grant No: 11475054, the Hebei Natural Science Foundation of China under Grant No: A2018205125.