Polygamy relation for the R\'enyi-$\alpha$ entanglement of assistance in multi-qubit systems

TL;DR

This paper establishes a new polygamy relation for multi-qubit quantum entanglement using Rényi-$\alpha$ entanglement of assistance, generalizing previous inequalities and applicable within specific parameter ranges.

Contribution

It introduces a novel polygamy inequality based on Rényi-$\alpha$ entanglement of assistance for multi-qubit systems, extending existing entanglement inequalities.

Findings

Proves a new polygamy relation for Rényi-$\alpha$ entanglement of assistance.

Shows the inequality holds for the $\mu$th power of the entanglement measure.

Reduces to known inequalities when $\alpha$ is in a specific range.

Abstract

We prove a new polygamy relation of multi-party quantum entanglement in terms of R\'{e}nyi-$\alpha$ entanglement of assistance for $\left( {\sqrt 7 - 1} \right)/2\leq\alpha \leq \left( {\sqrt 13 - 1} \right)/2$. This class of polygamy inequality reduces to the polygamy inequality based on entanglement of assistance since R\'{e}nyi-$\alpha$ entanglement is a generalization of entanglement of formation. We further show that the polygamy inequality also holds for the $\mu$th power of R\'{e}nyi-$\alpha$ entanglement of assistance.

Equations

$$C^2\left({|\psi ⟩}_{A_1|A_2{A}_3}\right)\le {\left[C^a\left({\rho }_{A_1{A}_2}\right)\right]}^2+{\left[C^a\left({\rho }_{A_1{A}_3}\right)\right]}^2,$$

$$C^2\left({|\psi ⟩}_{A_1|A_2\cdots A_n}\right)\le {\left[C^a\left({\rho }_{A_1{A}_2}\right)\right]}^2+\cdots +{\left[C^a\left({\rho }_{A_1{A}_n}\right)\right]}^2.$$

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

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

$$E_{\alpha }\left({\rho }_{AB}\right)=f_{\alpha }\left(C\left({\rho }_{AB}\right)\right),$$

$$f_{\alpha }\left(x\right)=\frac{1}{1-\alpha }\log \left[{\left(\frac{\Theta \left(x\right)}{2}\right)}^{\alpha }+{\left(\frac{\Xi \left(x\right)}{2}\right)}^{\alpha }\right],$$

$$E_{\alpha }^a\left({\rho }_{AB}\right):=\max {\sum }_i{p}_i{E}_{\alpha }\left({|{\psi }_i⟩}_{AB}\right),$$

$$E_{\alpha }^a\left({\rho }_{AB}\right)$$

$$E_{\alpha }^a\left({\rho }_{AB}\right)\ge f_{\alpha }\left(C^a\left({\rho }_{AB}\right)\right),$$

$$f_{\alpha }\left(C^a\left({\rho }_{AB}\right)\right)$$

$$f_{\alpha }(\sqrt{x^2+y^2})\le f_{\alpha }(x)+f_{\alpha }(y).$$

$$g_{\alpha }\left(x,y\right)=f_{\alpha }(\sqrt{x^2+y^2})-f_{\alpha }(x)-f_{\alpha }(y),$$

$$\nabla g_{\alpha }\left(x,y\right)=(\frac{\partial g_{\alpha }\left(x,y\right)}{\partial x},\frac{\partial g_{\alpha }\left(x,y\right)}{\partial y})=(0,0),$$

$$\frac{\partial g_{\alpha }\left(x,y\right)}{\partial x}$$

$$\frac{\partial g_{\alpha }\left(x,y\right)}{\partial y}$$

$$l_{\alpha }\left(x_0\right)=l_{\alpha }\left(y_0\right),$$

$$l_{\alpha }\left(x\right):=\frac{\left[{\left(\Theta \left(x\right)\right)}^{\alpha -1}-{\left(\Xi \left(x\right)\right)}^{\alpha -1}\right]}{\sqrt{1-x^2}\left[{\left(\Xi \left(x\right)\right)}^{\alpha }+{\left(\Theta \left(x\right)\right)}^{\alpha }\right]},$$

$$\frac{d{l}_{\alpha }\left(x\right)}{dx}$$

$$m_{\alpha }\left(x\right):=1-\frac{1}{1-a}\log \left[{\left(\frac{\Theta \left(x\right)}{2}\right)}^a+{\left(\frac{\Xi \left(x\right)}{2}\right)}^a\right]$$

$$-\frac{1}{1-a}\log \left[{\left(\frac{\Theta \left(\sqrt{1-x^2}\right)}{2}\right)}^a+{\left(\frac{\Xi \left(\sqrt{1-x^2}\right)}{2}\right)}^a\right].$$

$$E_{\alpha }^a\left({\rho }_{A_1|A_2\cdots A_n}\right)\le E_{\alpha }^a\left({\rho }_{A_1{A}_2}\right)+\cdots +E_{\alpha }^a\left({\rho }_{A_1{A}_n}\right),$$

$$E_{\alpha }\left({|\psi ⟩}_{A_1|A_2\cdots A_n}\right)=f_{\alpha }\left(C\left({\rho }_{A_1|A_2\cdots A_n}\right)\right)$$

$$E_{\alpha }\left({|\psi ⟩}_{A_1|A_2\cdots A_n}\right)=f_{\alpha }\left(C\left({\rho }_{A_1|A_2\cdots A_n}\right)\right)\le f_{\alpha }\left(1\right)$$

$$E_{\alpha }^a\left({\rho }_{A_1|A_2\cdots A_n}\right)={\sum }_i{p}_j{E}_{\alpha }\left({|{\psi }_j⟩}_{A_1|A_2\cdots A_n}\right)$$

$$E_{\alpha }\left({|{\psi }_j⟩}_{A_1|A_2\cdots A_n}\right)\le E_{\alpha }^a\left({\rho }_{A_1{A}_2}^j\right)+\cdots +E_{\alpha }^a\left({\rho }_{A_1{A}_n}^j\right),$$

$${\left[E_{\alpha }^a\left({\rho }_{A_1|A_2\cdots A_n}\right)\right]}^{\mu }\le {\left[E_{\alpha }^a\left({\rho }_{A_1{A}_2}\right)\right]}^{\mu }+\cdots +{\left[E_{\alpha }^a\left({\rho }_{A_1{A}_n}\right)\right]}^{\mu }$$

Full text

One fundamental property of quantum entanglement is in its limited shareability in multi-party quantum systemshor09rmp . For example, if the two subsystems are more entangled with each other, then they will share a less amount of entanglement with the other subsystems with specific entanglement measures. This restricted shareability of entanglement is named as the monogamy of entanglement (MoE). The concept of monogamy is an essential feature allowing for security in quantum key distributionPawlowski10pra . It also plays an important role in many field of physics such as foundations of quantum mechanicsBennett14 ; Toner09 ; Seevinck10qip , condensed matter physicsMa11np ; Saez13prb , statistical mechanicsBennett14 , and even black-hole physicsSusskind13 ; Lloyd14 . Monogamy inequality was first built for three-qubit systems using tangle as the bipartite entanglement measureckw00pra , and generalized into multi-qubit systems in terms of various entanglement measuresosb06prl ; off08pra ; eos09pra ; raj10pra ; Cornelio13 ; Regula14 ; byw09pra ; Chi08 ; Yu08 ; Osterloh15 ; Eltschka15 ; ade06njp ; hir07prl ; ade07prl ; koa04pra ; fan07pra ; kim09pra ; he15pra ; choi15pra ; yluo15ap ; kim10jpa ; cor10pra ; loh06prl ; oli14pra ; bxw14prl ; zhu14pra ; bxw14pra ; gao15sr ; Song16pra ; Yuan16sr ; Luo16pra ; Tian16sr .

On the other hand, the assisted entanglement, which is a dual concept to bipartite entanglement measures, is known to have a dually monogamous or polygamous property in multiparty quantum systems. The polygamous property can be regarded as another kind of entanglement constraints in multi-qubit systems, and Gour et alGour05pra established the first dual monogamy inequality or polygamy inequality for multi-qubit systems using concurrence of assistance (CoA). For a three-qubit pure state ${\left|\psi\right\rangle_{A_{1}A_{2}A_{3}}}$, a polygamy inequality was introduced as:

[TABLE]

where CoA for a bipartite state ${\rho_{AB}}$ is defined as: $C^{a}\left({\rho_{AB}}\right)=\max\sum\nolimits_{i}{p_{i}C\left({\left|{\psi_{i}}\right\rangle_{AB}}\right)}$, with the maximum is taken over all possible pure state decompositions of ${\rho_{AB}}=\sum\nolimits_{i}{p_{i}\left|{\psi_{i}}\right\rangle_{AB}\left\langle{\psi_{i}}\right|}$ and ${C\left({\left|{\psi_{i}}\right\rangle_{AB}}\right)}$ denotes the concurrencewoo98prl of ${\left|{\psi_{i}}\right\rangle_{AB}}$. Furthermore, it is shown that for any pure state $\left|\psi\right\rangle_{A_{1}A_{2}\cdots A_{n}}$ in a $n$-qubit systemGour07jmp , we have

[TABLE]

Later, polygamy inequalities was generalized in terms of Tsallis entanglement of assistance(TEoA)Kim10pra1 or unified entanglement of assistanceKim12pra , and polygamy inequalities in higher-dimensional systems were also shown using the entanglement of assistance(EoA)Buscemi09pra or TEoAKim16pra . In this paper, we establish a new polygamy relation of multi-party quantum entanglement in terms of Rényi-$\alpha$ entropy (ER$\alpha$E)kim10jpa . As an important generalization of entanglement of formation(EoF), ER$\alpha$E is a well-defined entanglement measure which has a continuous spectrum parametrized by the non-negative real parameter $\alpha$. It reduces to the standard EoF when $\alpha$ tends to $1$. Thus our polygamy inequalities including previous polygamy relation of EoF as a special caseBuscemi09pra . Furthermore, we generalize the polygamy inequalities in terms of the $\mu$th power of Rényi-$\alpha$ entanglement of assistance.

For a bipartite pure state $\left|\psi\right\rangle_{AB}$, the ER$\alpha$E is defined as

[TABLE]

where $S_{\alpha}(\rho_{A})$ is the Rényi-$\alpha$ entropy. The Rényi-$\alpha$ entropy has found important applications in characterizing quantum phases with differing computational power Cui12nc , ground state properties in many-body systems Franchini14prx , and topologically ordered states Flammia09prl ; Halasz13prl . The ER$\alpha$E of a bipartite mixed state $\rho_{AB}$ can be defined using the convex roof technique

[TABLE]

It is known that Rényi-$\alpha$ entropy converges to the von Neumann entropy when $\alpha$ tends to $1$. So the entanglement Rényi-$\alpha$ entropy reduces to the EoF when $\alpha$ tends to $1$. For any two-qubit state ${\rho_{AB}}$ with $\alpha\geq\left({\sqrt{7}-1}\right)/2$, there exist an analytic formula of ER$\alpha$Ekim10jpa ; yxwang16pra

[TABLE]

where

[TABLE]

with $\Theta\left(x\right)=1+\sqrt{1-x^{2}},\Xi\left(x\right)=1-\sqrt{1-x^{2}}$.

As a dual concept to ER$\alpha$E, we define the Rényi-$\alpha$ entanglement of assistance(REoA) as

[TABLE]

where the maximum is taken over all possible pure state decompositions of ${\rho_{AB}}=\sum\nolimits_{i}{p_{i}\left|{\psi_{i}}\right\rangle_{AB}\left\langle{\psi_{i}}\right|}$.

For $0<\alpha<1$, we can derive a upper bound of REoA. From the definition of entanglement of REoA, we have

[TABLE]

where ${\rho_{iA}}$ is the reduced density matrix of ${\left|{\psi_{i}}\right\rangle_{AB}}$, and the inequality holds due to the concave property of $S_{\alpha}\left(\rho\right)$ for $0<\alpha<1$Bassat78 ; Mosonyi11 ; Debarba17 . Similarly, we can derive $E_{\alpha}^{a}\left({\rho_{AB}}\right)\leq S_{\alpha}\left({\rho_{B}}\right)$. Thus we have $E_{\alpha}^{a}\left({\rho_{AB}}\right)\leq\min\{S_{\alpha}\left({\rho_{A}}\right),S_{\alpha}\left({\rho_{B}}\right)\}$

Before showing the main result of this paper, we first give two lemmas as follows.

Lemma 1. For any two-qubit state $\rho_{AB}$ and $\alpha\geq\left({\sqrt{7}-1}\right)/2$, we have

[TABLE]

where $E_{\alpha}^{a}\left({\rho_{AB}}\right)$ and ${C^{a}\left({\rho_{AB}}\right)}$ are the REoA and CoA of ${\rho_{AB}}$, respectively.

Proof. Suppose that the optimal decomposition for ${C^{a}\left({\rho_{AB}}\right)}$ is $\left\{{p_{i,}\left|{\psi_{i}}\right\rangle_{AB}}\right\}$, we have

[TABLE]

where in the first inequality we have used the convex property of $f_{\alpha}(x)$ as a function of $x$ for $\alpha\geq\left({\sqrt{7}-1}\right)/2$, and the second inequality is due to the definition of EoA. $\hfill\square$

Lemma 2. For any $\left({\sqrt{7}-1}\right)/2\leq\alpha\leq\left({\sqrt{13}-1}\right)/2$ and the function $f_{\alpha}\left(x\right)$ defined on the domain $\mathcal{D}=\left\{{\left({x,y}\right)|0\leq x,y\leq 1,0\leq x^{2}+y^{2}\leq 1}\right\}$, we have

[TABLE]

Proof. We define a two-vairable function

[TABLE]

on the domain $\mathcal{D}$. Then it is sufficient to show that $g_{\alpha}\left({x,y}\right)$ is a non-negative function on $\mathcal{D}$. Since $g_{\alpha}\left({x,y}\right)$ is analytic in the interior of $\mathcal{D}$, and continuous on $\mathcal{D}$, its maximum or minimum value arises only on the critical points or on the boundary of $\mathcal{D}$. The critical points of $g_{\alpha}\left({x,y}\right)$ satisfy the condition

[TABLE]

where

[TABLE]

and

[TABLE]

Suppose that there exists $\left({x_{0},y_{0}}\right)$ in the interior of $\mathcal{D}$ such that $\nabla g_{\alpha}\left({x_{0},y_{0}}\right)=(0,0)$. From Eq.(14) and Eq.(15), we have

[TABLE]

where $l_{\alpha}\left({x}\right)$ is defined as

[TABLE]

for $0<x<1$. We divide the proof into two cases. We first show that $l_{\alpha}\left({x}\right)$ is a strictly monotone-decreasing function for $0<x<1,1<\alpha<\left({\sqrt{13}-1}\right)/2$, then it is sufficient to consider the first-order derivative of $l_{\alpha}\left({x}\right)$. After a direct calculation, we have

[TABLE]

In order to show the negativity of the first-order derivative of $l_{\alpha}\left({x}\right)$, let us consider the value of the two-variable function $h_{\alpha}\left({x}\right):={dl_{\alpha}\left(x\right)/dx}$ on the domain $\mathcal{D}_{1}=\left\{{\left({\alpha,x}\right)|1\leq\alpha\leq\left({\sqrt{13}-1}\right)/2,0\leq x\leq 1}\right\}$. The maximum or minimum values of $h_{\alpha}\left({x}\right)$ can arise only at the critical points or on the boundary of $\mathcal{D}_{1}$. The critical points of $h_{\alpha}\left({x}\right)$ satisfy the condition $\nabla h_{\alpha}\left(x\right)=(\partial h_{\alpha}\left(x\right)/\partial\alpha,\partial h_{\alpha}\left(x\right)/\partial x)=(0,0)$. It is shown in Fig.1(a) and (b) that there are no common solutions on the interior of domain $\mathcal{D}_{1}$ which indicate that $h_{\alpha}\left({x}\right)$ has no critical points on the interior of $\mathcal{D}_{1}$. Then we consider the function value of $h_{\alpha}\left({x}\right)$ on the boundary of $\mathcal{D}_{1}$. If $\alpha=1$, we have $h_{\alpha}\left(x\right)|_{\alpha=1}=0$. If $\alpha=\left({\sqrt{13}-1}\right)/2$, we plot $h_{\alpha}\left(x\right)|_{\alpha=\left({\sqrt{13}-1}\right)/2}$ as a function of $x$ in Fig.2, which illustrates that $h_{\alpha}\left(x\right)|_{\alpha=\left({\sqrt{13}-1}\right)/2}$ is a monotone-increasing function for $0\leq x\leq 1$ and obtain its maximum value [math] on $x=1$. When $x\to 1$, we have $h_{\alpha}\left(x\right)|_{x\to 1}=2\left({q^{3}-4q+3}\right)/3$ which is always negative for $1<\alpha<\left({\sqrt{13}-1}\right)/2$ as shown in Fig.3. Thus we have shown that $h_{\alpha}\left(x\right)$ is always negative on the interior of domain $\mathcal{D}_{1}$ which indicate that $l_{\alpha}\left(x\right)$ is a strictly monotone-decreasing function for $0<x<1,1<\alpha<\left({\sqrt{13}-1}\right)/2$. Similarly, we can show that $l_{\alpha}\left(x\right)$ is a strictly monotone-increasing function for $\left({\sqrt{7}-1}\right)<\alpha<1$. In this case, it is enough to prove the non-negative of the function $h_{\alpha}\left({x}\right):={dl_{\alpha}\left(x\right)/dx}$ on the domain $\mathcal{D}_{2}=\left\{{\left({\alpha,x}\right)|\left({\sqrt{7}-1}\right)/2\leq\alpha\leq 1,0\leq x\leq 1}\right\}$. Because $h_{\alpha}\left({x}\right)$ has no critical points on the interior of $\mathcal{D}_{2}$ as shown in Fig.1, we consider the function value of $h_{\alpha}\left({x}\right)$ on the boundary of $\mathcal{D}_{2}$. If $x\to 1$, we can verify that the function $h_{\alpha}\left(x\right)|_{x\to 1}$ is always positive for $\left({\sqrt{7}-1}\right)/2<\alpha<1$. If $\alpha=\left({\sqrt{7}-1}\right)/2$, it is shown in Fig.4 that $h_{\alpha}\left(x\right)|_{\alpha=\left({\sqrt{7}-1}\right)/2}$ is always positive for $0<x<1$. Therefore, $h_{\alpha}\left({x}\right)$ is always positive for $\left({\sqrt{7}-1}\right)/2<\alpha<1$ which indicates that $l_{\alpha}\left({x}\right)$ is a strictly monotone-increasing function in this case. Combining Eq.(16) we can derive $x_{0}=y_{0}$. However, from Eq.(13)-(15) and $\nabla g_{\alpha}\left({x_{0},x_{0}}\right)=(0,0)$ we have $l_{\alpha}\left({\sqrt{2}x_{0}}\right)=l_{\alpha}\left({x_{0}}\right)$ which contradicts to the strict monotonicity of $l_{\alpha}\left(x\right)$ for $0<x<1$. Therefore, we conclude that $g_{\alpha}\left({x,y}\right)$ has no critical points in the interior of $\mathcal{D}$.

Next, we consider the function value of $g_{\alpha}\left({x,y}\right)$ on the boundary of $\mathcal{D}$. If $x=0$ or $y=0$, it is direct to check that $g_{\alpha}\left({x,y}\right)=0$. When $x^{2}+y^{2}=1$, $g_{\alpha}\left({x,y}\right)$ becomes a two-variable function

[TABLE]

As shown in Fig.5, $\partial m_{\alpha}\left(x\right)/\partial x=0$ has only one solution $x=1/\sqrt{2}$ on the domain $\mathcal{D}_{3}=\left\{{\left({\alpha,x}\right)|\left({\sqrt{7}-1}\right)/2\leq\alpha\leq\left({\sqrt{13}-1}\right)/2,0\leq x\leq 1}\right\}$. On the other hand, we plot $\partial m_{\alpha}\left(x\right)/\partial\alpha|_{x=1/\sqrt{2}}$ in Fig.6 and we can see that the function is always positive for $\left({\sqrt{7}-1}\right)/2\leq\alpha\leq\left({\sqrt{13}-1}\right)/2$, which shows that $m_{\alpha}\left(x\right)$ has no critical points on the interior of domain $\mathcal{D}_{3}$. Then we consider the value of $m_{\alpha}(x)$ on the boundary of $\mathcal{D}_{3}$. If $x=0$ or $1$, we have $m_{\alpha}(x)=0$. When $\alpha=\left({\sqrt{7}-1}\right)/2$ or $\alpha=\left({\sqrt{13}-1}\right)/2$, it is direct to check that $m_{\alpha}\left(x\right)$ is always a non-positive function. In Fig.7 we plot $m_{\alpha}\left(x\right)$ as a function of $x$ and $\alpha$, which illustrates our result.

Combining the case for $\alpha=1$ which has been proved in Ref.Buscemi09pra , we have completed the proof of Lemma 2. $\hfill\square$

Now we can prove the main result of this paper.

Theorem. For $\left({\sqrt{7}-1}\right)/2\leq\alpha\leq\left({\sqrt{13}-1}\right)/2$, and any $n$-qubit state $\rho_{A_{1}A_{2}\cdots A_{n}}$, we have

[TABLE]

where ${E_{\alpha}^{a}\left(\rho_{A_{1}|A_{2}\cdots A_{n}}\right)}$ denotes the REoA in the partition $A_{1}|A_{2}\cdots A_{n}$, and ${E_{\alpha}^{a}\left({\rho_{A_{1}A_{i}}}\right)}$ is the REoA of the two-qubit subsystem $A_{1}A_{i}$ for $i=2,\ldots,n$.

Proof. We first prove the polygamy relation for the pure state ${\left|\psi\right\rangle_{A_{1}|A_{2}\cdots A_{n}}}$. Assuming that ${C^{2}\left({\rho_{A_{1}|A_{2}\cdots A_{n}}}\right)}\leq\left[{C^{a}\left({\rho_{A_{1}A_{2}}}\right)}\right]^{2}+\cdots+\left[{C^{a}\left({\rho_{A_{1}A_{n}}}\right)}\right]^{2}\leq 1$ in Eq.(Polygamy relation for the Rényi-$\alpha$ entanglement of assistance in multi-qubit systems), then we have

[TABLE]

where in the first inequality we have used the monotonically increasing property of $f_{\alpha}(x)$ for $\alpha\geq\left({\sqrt{7}-1}\right)/2$, the second and third inequalities are obtained by the successive application of Lemma 2, and the last inequality is due to Lemma 1.

Then we consider the case ${C^{2}\left({\rho_{A_{1}|A_{2}\cdots A_{n}}}\right)}\leq 1\leq\left[{C^{a}\left({\rho_{A_{1}A_{2}}}\right)}\right]^{2}+\cdots+\left[{C^{a}\left({\rho_{A_{1}A_{n}}}\right)}\right]^{2}$. There must exist $k\in\left\{{2,\ldots,n-1}\right\}$ such that $\left[{C^{a}\left({\rho_{A_{1}A_{2}}}\right)}\right]^{2}+\cdots+\left[{C^{a}\left({\rho_{A_{1}A_{k}}}\right)}\right]^{2}\leq 1,\left[{C^{a}\left({\rho_{A_{1}A_{2}}}\right)}\right]^{2}+\cdots+\left[{C^{a}\left({\rho_{A_{1}A_{k+1}}}\right)}\right]^{2}>1$. By defining $T:=\left[{C^{a}\left({\rho_{A_{1}A_{2}}}\right)}\right]^{2}+\cdots+\left[{C^{a}\left({\rho_{A_{1}A_{k+1}}}\right)}\right]^{2}-1>0$, we can derive

[TABLE]

where we have used the monotonically increasing property of $f_{\alpha}(x)$ in the first equality, in the second inequality we have used Lemma 2, the third inequality is obtained by the successive application of Lemma 2, and the last inequality is due to Lemma 1.

Using the polygamy relation for the pure state we can prove the Theorem in the mixed state. Suppose that the optimal decomposition for $E_{\alpha}^{a}\left({\rho_{A_{1}|A_{2}\cdots A_{n}}}\right)$ is $\left\{{p_{j},\left|{\psi_{j}}\right\rangle_{A_{1}|A_{2}\cdots A_{n}}}\right\}$, then we have

[TABLE]

where ${\rho_{A_{1}A_{j}}^{j}}$ is the reduced density matrix of ${\left|{\psi_{j}}\right\rangle_{A_{1}A_{2}\cdots A_{n}}}$ onto the two-qubit subsystem $A_{1}A_{j}$ for each $i=2,\ldots,n$, in the first inequality we have used the polygamy relation for each pure state decomposition state ${\left|{\psi_{j}}\right\rangle_{A_{1}|A_{2}\cdots A_{n}}}$

[TABLE]

and the last inequality is due to the definition of REoA for each ${\rho_{A_{1}A_{i}}}$. Thus we have completed the proof of Theorem. $\hfill\square$

Furthermore, we can establish the following $\mu$th power polygamy inequalities for the Rényi-$\alpha$ entanglement of assistance.

Corollary. For $\left({\sqrt{7}-1}\right)/2\leq\alpha\leq\left({\sqrt{13}-1}\right)/2,0\leq\mu\leq 1$, and any $n$-qubit state $\rho_{A_{1}A_{2}\cdots A_{n}}$, we have

[TABLE]

This inequality holds because $\left[{E_{\alpha}^{a}\left({\rho_{A_{1}|A_{2}\cdots A_{n}}}\right)}\right]^{\mu}\leq\left[{E_{\alpha}^{a}\left({\rho_{A_{1}A_{2}}}\right)+\cdots+E_{\alpha}^{a}\left({\rho_{A_{1}A_{n}}}\right)}\right]^{\mu}\leq\left[{E_{\alpha}^{a}\left({\rho_{A_{1}A_{2}}}\right)}\right]^{\mu}+\cdots+\left[{E_{\alpha}^{a}\left({\rho_{A_{1}A_{n}}}\right)}\right]^{\mu}$, where the last inequality is due to the concave property of $x^{\mu}$ for $0\leq\mu\leq 1$.

By introducing the dual concept of REoA, we have established polygamy relations for the Rényi-$\alpha$ entanglement of assistance in multi-qubit systems. We have also generalized the polygamy inequalities into the $\mu$th power of REoA. These derived polygamy relations provide a lower bound for distribution of bipartite REoA in a multi-party system. The monogamy and polygamy relations are not only fundamental property of entanglement in multi-party systems but also provide us an efficient way of characterizing multipartite entanglement. In Ref. Song16pra , we have proved that squared Rényi-$\alpha$ entanglement with the order $\alpha\geq(\sqrt{7}-1)/2$ obeys a general monogamy relation in an arbitrary $n$-qubit mixed state. It is further shown that we can construct the multipartite entanglement indicators in terms of ER$\alpha$E which still work well even when the indicators based on the concurrence and EoF lose their efficacy. Thus our polygamy inequalities together with previous monogamy inequalities in terms of ER$\alpha$E might provide a useful tool to understand the property of multi-party quantum entanglement.

This work was supported by NSF-China under Grant Nos.11374085, 11274010, the Anhui Provincial Natural Science Foundation under Grant Nos.1708085MA12, 1708085MA10, the Key Program of the Education Department of Anhui Province under Grant Nos. KJ2017A922, KJ2016A583, the discipline top-notch talents Foundation of Anhui Provincial Universities under Grant Nos.gxbjZD2017024, gxbjZD2016078, the Anhui Provincial Candidates for academic and technical leaders Foundation under Grant No.2015H052 and the Excellent Young Talents Support Plan of Anhui Provincial Universities.