Polygamy relations of multipartite systems

TL;DR

This paper explores the polygamy relations in multipartite quantum systems, establishing inequalities involving various measures of quantum entanglement and assistance, advancing understanding of quantum correlations.

Contribution

It introduces generalized polygamy inequalities for multipartite states using powers of concurrence, entanglement of assistance, and SCRENoA, expanding theoretical frameworks.

Findings

Derived inequalities for concurrence of assistance, entanglement of assistance, and SCRENoA.

Extended polygamy relations to higher powers of quantum correlation measures.

Provided mathematical bounds for multipartite quantum entanglement.

Abstract

We investigate the polygamy relations of multipartite quantum states. General polygamy inequalities are given in the $\alpha$th $(\alpha\geq 2)$ power of concurrence of assistance, $\beta$th $(\beta \geq1)$ power of entanglement of assistance, and the squared convex-roof extended negativity of assistance (SCRENoA).

Equations

$$C^2({\rho }_{A|B_1,B_2\cdots ,B_{N-1}})\ge \sum _{i=1}^{N-1}{C}^2({\rho }_{A{B}_i}),$$

$$C^2({\rho }_{A|B_1,B_2\cdots ,B_{N-1}})\le \sum _{i=1}^{N-1}{C}_a^2({\rho }_{A{B}_i}).$$

$$C^2(|\varphi {⟩}_{AB})=2(1-\mathrm{Tr}({\rho }_A^2))=4\sum _{i<j}^{d_1}\sum _{k<l}^{d_2}|a_{ik}{a}_{jl}-a_{il}{a}_{jk}{|}^2.$$

$$C_a({\rho }_{AB})$$

$$D_1$$

$$L_A^m$$

$$L_B^n$$

$${\tau }_a^2(|\varphi {⟩}_{A_1|A_2\cdots A_n})\le \sum _{i=2}^n{\tau }_a^2({\rho }_{A_1{A}_i}),$$

$${{\tau }_a^{\alpha }}_{A|BC}\le {{\tau }_a^{\alpha }}_{AC}+(2^{\frac{\alpha }{2}}-1){{\tau }_a^{\alpha }}_{AB}$$

$${{\tau }_a^{\alpha }}_{A|BC}\le {{\tau }_a^{\alpha }}_{AB}+(2^{\frac{\alpha }{2}}-1){{\tau }_a^{\alpha }}_{AC}$$

$${{\tau }_a^{2x}}_{A|BC}$$

$$|\psi ⟩$$

$${{\tau }_a^{\alpha }}_{A|BC}-{{\tau }_a^{\alpha }}_{AB}-(2^{\frac{\alpha }{2}}-1){{\tau }_a^{\alpha }}_{AC}$$

$$\displaystyle=(2\lambda_{0})^{\alpha}\Big{[}(\lambda_{2}^{2}+\lambda_{3}^{2}+\lambda_{4}^{2})^{\frac{\alpha}{2}}-(\lambda_{2}^{2}+\lambda_{4}^{2})^{\frac{\alpha}{2}}$$

$$\displaystyle~{}~{}~{}-(2^{\frac{\alpha}{2}}-1)(\lambda_{3}^{2}+\lambda_{4}^{2})^{\frac{\alpha}{2}}\Big{]}$$

$$=2^{\alpha }{\cos }^{\alpha }{\theta }_0{\sin }^{\alpha }{\theta }_0{\sin }^{\alpha }{\theta }_1(2-{\sin }^{\alpha }{\theta }_3-2^{\frac{\alpha }{2}})$$

$$\le 0,$$

$${{\tau }_a^{\alpha }}_{A|BC}-{{\tau }_a^{\alpha }}_{AB}-(2^{\frac{\alpha }{2}}-1){{\tau }_a^{\alpha }}_{AC}$$

$$\displaystyle=(2\lambda_{0})^{\alpha}\Big{[}(\lambda_{2}^{2}+\lambda_{3}^{2}+\lambda_{4}^{2})^{\frac{\alpha}{2}}-(\lambda_{2}^{2}+\lambda_{4}^{2})^{\frac{\alpha}{2}}$$

$$\displaystyle~{}~{}~{}-(2^{\frac{\alpha}{2}}-1)(\lambda_{3}^{2}+\lambda_{4}^{2})^{\frac{\alpha}{2}}\Big{]}$$

$$\displaystyle=2^{\alpha}\cos^{\alpha}\theta_{0}\sin^{\alpha}\theta_{0}\sin^{\alpha}\theta_{1}\Big{[}1-(\cos^{2}\theta_{2}+\sin^{2}\theta_{2}\sin^{2}\theta_{3})^{\frac{\alpha}{2}}$$

$$\displaystyle~{}~{}~{}-(2^{\frac{\alpha}{2}}-1)\sin^{\alpha}\theta_{2}\Big{]}$$

$$\displaystyle\leq 2^{\alpha}\cos^{\alpha}\theta_{0}\sin^{\alpha}\theta_{0}\sin^{\alpha}\theta_{1}\Big{[}1-\cos^{\alpha}\theta_{2}-(2^{\frac{\alpha}{2}}-1)\sin^{\alpha}\theta_{2}\Big{]}$$

$$\displaystyle=2^{\alpha}\cos^{\alpha}\theta_{0}\sin^{\alpha}\theta_{0}\sin^{\alpha}\theta_{1}\Big{[}1-\cos^{\alpha}\theta_{2}\left(1+(2^{\frac{\alpha}{2}}-1)t_{1}^{\frac{\alpha}{2}}\right)\Big{]}$$

$$\displaystyle\leq 2^{\alpha}\cos^{\alpha}\theta_{0}\sin^{\alpha}\theta_{0}\sin^{\alpha}\theta_{1}\Big{[}1-\cos^{\alpha}\theta_{2}(1+t_{1})^{\frac{\alpha}{2}}\Big{]}$$

$$\displaystyle=2^{\alpha}\cos^{\alpha}\theta_{0}\sin^{\alpha}\theta_{0}\sin^{\alpha}\theta_{1}\Big{[}1-\cos^{\alpha}\theta_{2}\left(1+\frac{\sin^{2}\theta_{2}}{\cos^{2}\theta_{2}}\right)^{\frac{\alpha}{2}}\Big{]}$$

$$=0,$$

$${{\tau }_a^{\alpha }}_{A|B_0{B}_1\cdots B_{N-1}}\le$$

$${{\tau }_a^{\alpha }}_{A{B}_0}+(2^{\frac{\alpha }{2}}-1){{\tau }_a^{\alpha }}_{A{B}_1}+\cdots +(2^{\frac{\alpha }{2}}-1{)}^m{{\tau }_a^{\alpha }}_{A{B}_m}$$

$$+(2^{\frac{\alpha }{2}}-1{)}^{m+2}({{\tau }_a^{\alpha }}_{A{B}_{m+1}}+\cdots +{{\tau }_a^{\alpha }}_{A{B}_{N-2}})$$

$$+(2^{\frac{\alpha }{2}}-1{)}^{m+1}{{\tau }_a^{\alpha }}_{A{B}_{N-1}},$$

$${{\tau }_a^{\alpha }}_{A|B_0{B}_1\cdots B_{N-1}}$$

$$\le {{\tau }_a^{\alpha }}_{A{B}_0}+(2^{\frac{\alpha }{2}}-1){\left(\sum _{i=1}^{N-1}{{\tau }_a^2}_{A{B}_i}\right)}^{\frac{\alpha }{2}}$$

$$\le {{\tau }_a^{\alpha }}_{A{B}_0}+(2^{\frac{\alpha }{2}}-1){{\tau }_a^{\alpha }}_{A{B}_1}+(2^{\frac{\alpha }{2}}-1{)}^2{\left(\sum _{i=2}^{N-1}{{\tau }_a^2}_{A{B}_i}\right)}^{\frac{\alpha }{2}}$$

$$\le \cdots$$

$$\le {{\tau }_a^{\alpha }}_{A{B}_0}+(2^{\frac{\alpha }{2}}-1){{\tau }_a^{\alpha }}_{A{B}_1}+\cdots +(2^{\frac{\alpha }{2}}-1{)}^m{{\tau }_a^{\alpha }}_{A{B}_m}$$

$$+(2^{\frac{\alpha }{2}}-1{)}^{m+1}{\left(\sum _{i=m+1}^{N-1}{{\tau }_a^2}_{A{B}_i}\right)}^{\frac{\alpha }{2}}.$$

$${\left(\sum _{i=m+1}^{N-1}{{\tau }_a^2}_{A{B}_i}\right)}^{\frac{\alpha }{2}}$$

$$\le (2^{\frac{\alpha }{2}}-1){{\tau }_a^{\alpha }}_{A{B}_{m+1}}+{\left(\sum _{i=m+2}^{N-1}{{\tau }_a^2}_{A{B}_i}\right)}^{\frac{\alpha }{2}}$$

$$\le (2^{\frac{\alpha }{2}}-1)({{\tau }_a^{\alpha }}_{A{B}_{m+1}}+\cdots +{{\tau }_a^{\alpha }}_{A{B}_{N-2}})$$

$$+{{\tau }_a^{\alpha }}_{A{B}_{N-1}}.$$

$${{\tau }_a^{\alpha }}_{A|B_0{B}_1\cdots B_{N-1}}\le \sum _{j=0}^{N-1}(2^{\frac{\alpha }{2}}-1{)}^j{{\tau }_a^{\alpha }}_{A{B}_j},$$

$${E_a^{\beta }}_{A|B_0{B}_1\cdots B_{N-1}}\le$$

Full text

I INTRODUCTION

Quantum entanglement is an important kind of quantum correlation, plays essential roles in quantum information processing MAN ; RPMK ; FMA ; KSS ; HPB ; HPBO ; JIV ; CYSG . One of the fundamental differences between classical and quantum correlations lies on the sharability among the subsystems. Different from the classical correlation, quantum correlation cannot be freely shared. Monogamy relation is important in the sense that it gives rise to the distribution of correlation in the multipartite quantum system and has a unique feature of keeping security in quantum key distribution MP .

For the systems of three qubits, a kind of monogamy of bipartite quantum entanglement in concurrence coffman can be described by Coffman-Kundu-Wootters CKW inequality wootters , $\mathcal{E}_{A|BC}\geq\mathcal{E}_{AB}+\mathcal{E}_{AC}$, where $\mathcal{E}_{A|BC}$ denotes the entanglement between systems $A$ and $BC$. Whereas monogamy of entanglement shows the restricted sharability of multipartite entanglement, the distribution of entanglement, or entanglement of assistance gg , in multipartite quantum systems was shown to have a dually monogamous (polygamous) property. Note that the monogamy of entanglement inequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system, and the same quantity sets a lower bound for the distribution of bipartite entanglement in a multipartite system, i.e., ${E_{a}}_{A|BC}\leq{E_{a}}_{AB}+{E_{a}}_{AC}$ for a tripartite quantum state $\rho_{ABC}$, where ${E_{a}}_{A|BC}$ is the assisted entanglement gg between $A$ and $BC$. The polygamy inequality was first obtained in terms of the tangle of assistance gg among three-qubit systems, and it was generalized to the multiqubit system with the help of additional entanglement measures 062328 ; 295303 ; jsb . In fgj ; 062302 ; 062338 , people derived a general polygamy inequality of multipartite entanglement beyond qubit based on the entanglement of assistance.

Recently, monogamy and polygamy relations of multi-qubit entanglement have been studied in terms of non-negative power of entanglement measures and assisted entanglement measures. In JZX ; jll ; ZXN , the authors have shown that the $x$th power of the entanglement of formation (($x\geq\sqrt{2}$)) and the concurrence ($x\geq 2$) satisfy multiqubit monogamy inequalities. Monogamy relations for quantum steering have also been demonstrated in hqy ; mko ; jk1 ; jk2 ; jk3 . Later, polygamy inequalities were also proposed in terms of $\alpha$th ($0\leq\alpha\leq 1$) power of square of convex-roof extended negativity (SCREN) and the entanglement of assistance j012334 ; 042332 . In gy2 , the authors introduced a definition of polygamy relations without inequalities. However, it is still not clear for the polygamy relation of the concurrence of assistance $\tau^{\alpha}_{a}$ $(\alpha\geq 2)$ and the $\beta$th $(\beta\geq 1)$ power of entanglement of assistance $E^{\beta}_{a}$ and the SCREN of assistance (SCRENoA) $({N_{sc}^{a}})^{\beta}$. In this paper, we study the general polygamy inequalities of $\tau^{\alpha}_{a}$, $E^{\beta}_{a}$ and $({N_{sc}^{a}})^{\beta}$ with $\alpha\geq 2$ and $\beta\geq 1$, respectively.

We first recall monogamy and polygamy inequalities related to concurrence and concurrence of assistance. Let $\mathds{H}_{X}$ denote a discrete finite-dimensional complex vector space associated with a quantum subsystem $X$. For a bipartite pure state $|\psi\rangle_{AB}\in\mathds{H}_{A}\otimes\mathds{H}_{B}$, the concurrence is given by AU ; PR ; SA , $C(|\psi\rangle_{AB})=\sqrt{{2\left[1-\mathrm{Tr}(\rho_{A}^{2})\right]}}$, where $\rho_{A}$ is the reduced density matrix obtained by tracing over the subsystem $B$, $\rho_{A}=\mathrm{Tr}_{B}(|\psi\rangle_{AB}\langle\psi|)$. The concurrence for a bipartite mixed state $\rho_{AB}$ is defined by the convex roof extension, $C(\rho_{AB})=\min_{\{p_{i},|\psi_{i}\rangle\}}\sum_{i}p_{i}C(|\psi_{i}\rangle)$, where the minimum is taken over all possible pure state decompositions of $\rho_{AB}=\sum\limits_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$, with $p_{i}\geq 0$, $\sum\limits_{i}p_{i}=1$ and $|\psi_{i}\rangle\in\mathds{H}_{A}\otimes\mathds{H}_{B}$.

For a tripartite state $|\psi\rangle_{ABC}$, the concurrence of assistance is defined by TFS ; YCS , $C_{a}(|\psi\rangle_{ABC})\equiv C_{a}(\rho_{AB})=\max\limits_{\{p_{i},|\psi_{i}\rangle\}}\sum_{i}p_{i}C(|\psi_{i}\rangle),$ where the maximum is taken over all possible pure state decompositions of $\rho_{AB}=\mathrm{Tr}_{C}(|\psi\rangle_{ABC}\langle\psi|)=\sum\limits_{i}p_{i}|\psi_{i}\rangle_{AB}\langle\psi_{i}|.$ For pure states $\rho_{AB}=|\psi\rangle_{AB}\langle\psi|$, one has $C(|\psi\rangle_{AB})=C_{a}(\rho_{AB})$.

For an $N$-qubit state $\rho_{AB_{1}\cdots B_{N-1}}\in\mathds{H}_{A}\otimes\mathds{H}_{B_{1}}\otimes\cdots\otimes\mathds{H}_{B_{N-1}}$, the concurrence $C(\rho_{A|B_{1}\cdots B_{N-1}})$ of the state $\rho_{AB_{1}\cdots B_{N-1}}$, viewed as a bipartite state under the partition $A$ and $B_{1},B_{2},\cdots,B_{N-1}$, satisfies the Coffman-Kundu-Wootters inequality TJ ; YKM ,

[TABLE]

where $\rho_{AB_{i}}=\mathrm{Tr}_{B_{1}\cdots B_{i-1}B_{i+1}\cdots B_{N-1}}(\rho_{AB_{1}\cdots B_{N-1}})$. Further improved monogamy relations are presented in JZX ; ZXN . The dual inequality in terms of the concurrence of assistance for $N$-qubit states has the form GSB ,

[TABLE]

Now, let us consider a bipartite pure state of arbitrary dimension $d_{1}\times d_{2}$, $|\phi\rangle_{AB}=\sum_{i=1}^{d_{1}}\sum_{k=1}^{d_{2}}a_{ik}|ik\rangle_{AB}$ in $C^{d_{1}}\otimes C^{d_{2}}$. The squared concurrence of $|\phi\rangle_{AB}$ can be expressed as jpa6777

[TABLE]

For a mixed state $\rho_{AB}=\sum_{i}p_{i}|\phi_{i}\rangle_{AB}\langle\phi_{i}|$, its concurrence of assistance satisfies 022302

[TABLE]

where

[TABLE]

with $P_{A}^{m}\!=\!|i\rangle_{A}\langle i|+|j\rangle_{A}\langle j|$ and $P_{B}^{n}\!=\!|k\rangle_{B}\langle k|+|l\rangle_{B}\langle l|$ being the projectors to the subspaces spanned by $\{|i\rangle_{A},|j\rangle_{A}\}$ and $\{|k\rangle_{B},|l\rangle_{B}\}$, respectively. A general polygamy inequality for any multipartite pure state $|\phi\rangle_{A_{1}\cdots A_{n}}\in C^{d_{1}}\otimes\cdots\otimes C^{d_{n}}$ was established as 022302 ,

[TABLE]

where $\rho_{A_{1}A_{k}}$ is the reduced density matrix $|\phi\rangle_{A_{1}|A_{2}\cdots A_{n}}$ with respect to subsystem $A_{1}A_{k}$, $k=2,\cdots,n$.

II POLYGAMY RELATION for CONCURRENCE of assistance

[Lemma 1]. For any real numbers $x$ and $t$, $t\geq 1$, $x\geq 1$, we have $(1+t)^{x}\leq 1+(2^{x}-1)t^{x}$.

[Proof]. Let $f(x,y)=(1+y)^{x}-y^{x}$ with $x\geq 1,~{}0<y\leq 1$, $\frac{\partial f}{\partial y}=x[(1+y)^{x-1}-y^{x-1}]\geq 0$. Therefore, $f(x,y)$ is an increasing function of $y$, i.e., $f(x,y)\leq f(x,1)=2^{x}-1$. Set $y=\frac{1}{t},~{}t\geq 1$. We obtain $(1+t)^{x}\leq 1+(2^{x}-1)t^{x}$. Notice when $t=1$, the inequality is true. $\Box$

The following theorem provides a class of polygamy inequalities satisfied by the $\alpha$-power of $\tau_{a}$. For convenience, we denote ${\tau_{a}}(\rho_{AB_{i}})={\tau_{a}}_{AB_{i}}$ the concurrence of assistance $\rho_{AB_{i}}$ and ${\tau_{a}}(\rho_{A|B_{0}B_{1}\cdots B_{N-1}})={\tau_{a}}_{A|B_{0}B_{1}\cdots B_{N-1}}$.

[Theorem 1]. For any tripartite pure state $\rho_{ABC}\in H_{A}\otimes H_{B}\otimes H_{C}$:

$(1)$ if ${\tau_{a}}_{AB}\geq{\tau_{a}}_{AC}$, the concurrence of assistance satisfies

[TABLE]

for $\alpha\geq 2$.

$(2)$ if ${\tau_{a}}_{AB}\leq{\tau_{a}}_{AC}$, the concurrence of assistance satisfies

[TABLE]

for $\alpha\geq 2$.

[Proof]. For arbitrary tripartite pure state $\rho_{ABC}$, one has 022302 , ${\tau_{a}^{2}}_{A|BC}\leq{\tau_{a}^{2}}_{AC}+{\tau_{a}^{2}}_{AB}$. If ${\tau_{a}}_{AB}~{}({\tau_{a}}_{AC})=0$, the inequality (9) or (10) are true obviously. Therefore, assuming ${\tau_{a}}_{AB}\geq{\tau_{a}}_{AC}>0$, we have

[TABLE]

where the second inequality is true due to the inequality $(1+t)^{x}\leq 1+(2^{x}-1)t^{x}$ for $x\geq 1$ and $t=\frac{{\tau_{a}^{2}}_{AB}}{{\tau_{a}^{2}}_{AC}}\geq 1$. Denote $2x=\alpha$. We obtain $\alpha\geq 2$ as $x\geq 1$. Then we have the inequality (9). If ${\tau_{a}}_{AB}\leq{\tau_{a}}_{AC}$, Similarly we get (10).

Example 1. Let us consider the three-qubit state $|\psi\rangle$ in the generalized Schmidt decomposition form,

[TABLE]

where $\lambda_{i}\geq 0,~{}i=0,1,2,3,4$ and $\sum\limits_{i=0}\limits^{4}\lambda_{i}^{2}=1.$ We have ${\tau_{a}}_{A|BC}=2\lambda_{0}\sqrt{{\lambda_{2}^{2}+\lambda_{3}^{2}+\lambda_{4}^{2}}},$ ${\tau_{a}}_{AB}=2\lambda_{0}\sqrt{{\lambda_{2}^{2}+\lambda_{4}^{2}}}$, and ${\tau_{a}}_{AC}=2\lambda_{0}\sqrt{{\lambda_{3}^{2}+\lambda_{4}^{2}}}$. Without loss of generality, we set $\lambda_{0}=\cos\theta_{0},~{}\lambda_{1}=\sin\theta_{0}\cos\theta_{1},~{}\lambda_{2}=\sin\theta_{0}\sin\theta_{1}\cos\theta_{2},~{}\lambda_{3}=\sin\theta_{0}\sin\theta_{1}\sin\theta_{2}\cos\theta_{3}$, and $\lambda_{4}=\sin\theta_{0}\sin\theta_{1}\sin\theta_{2}\sin\theta_{3},~{}\theta_{i}\in[0,\frac{\pi}{2}]$.

For $\lambda_{3}\geq\lambda_{2}$, i.e. ${\tau_{a}}_{AC}\geq{\tau_{a}}_{AB}$:

(a) if $\theta_{2}=\frac{\pi}{2}$,

[TABLE]

where $\alpha\geq 2$ and the inequality is due to $\sin\theta_{3}\geq 0$.

(b) If $\theta_{2}\neq\frac{\pi}{2}$, we denote $t_{1}=\frac{\sin^{2}\theta_{2}}{\cos^{2}\theta_{2}}\geq 1$, then we have

[TABLE]

where $\alpha\geq 2$ and the second inequality is due to Lemma 1.

Therefore, we have ${\tau_{a}^{\alpha}}_{A|BC}\leq{\tau_{a}^{\alpha}}_{AB}+(2^{\frac{\alpha}{2}}-1){\tau_{a}^{\alpha}}_{AC}$ for $\alpha\geq 2$.

When $\lambda_{3}\leq\lambda_{2}$, i.e. ${\tau_{a}}_{AC}\leq{\tau_{a}}_{AB}$, from similar analysis we can obtain ${\tau_{a}^{\alpha}}_{A|BC}\leq{\tau_{a}^{\alpha}}_{AC}+(2^{\frac{\alpha}{2}}-1){\tau_{a}^{\alpha}}_{AB}$ for $\alpha\geq 2$.

Specially, when $\theta_{2}=\frac{\pi}{2},\alpha=2,\theta_{3}=0$, i.e. $|\psi\rangle=\cos\theta_{0}|000\rangle+\sin\theta_{0}\cos\theta_{1}e^{i{\varphi}}|100\rangle+\sin\theta_{0}\sin\theta_{1}|110\rangle,$ the inequality in (9) is saturated. Generalizing the conclusion in Theorem 1 to $N$ partite case, we have the following result.

[Theorem 2]. For any multipartite pure state $\rho_{AB_{0}\cdots B_{N-1}}$, if ${\tau_{a}^{2}}_{AB_{i}}\leq\sum_{k=i+1}^{N-1}{\tau_{a}^{2}}_{AB_{k}}$ for $i=0,1,\cdots,m$, and ${\tau_{a}^{2}}_{AB_{j}}\geq\sum_{k=j+1}^{N-1}{\tau_{a}^{2}}_{AB_{k}}$ for $j=m+1,\cdots,N-2$, $\forall$ $1\leq m\leq N-3$, $N\geq 4$, we have

[TABLE]

for $\alpha\geq 2$.

[Proof]. From the inequality (8) and Theorem 1, we have

[TABLE]

Similarly, as ${\tau_{a}^{2}}_{AB_{j}}\geq\sum_{k=j+1}^{N-1}{\tau_{a}^{2}}_{AB_{k}}$ for $j=m+1,\cdots,N-2$, we get

[TABLE]

Combining (II) and (II), we have Theorem 2. $\Box$

In Theorem 2, if ${\tau_{a}^{2}}_{AB_{i}}\leq\sum_{j=i+1}^{N-1}{\tau_{a}^{2}}_{AB_{j}}$ for all $i=0,1,\cdots N-2$, then we have the following conclusion:

[Theorem 3]. For any multipartite pure state $\rho_{AB_{0}\cdots B_{N-1}}$, if ${\tau_{a}^{2}}_{AB_{i}}\leq\sum_{j=i+1}^{N-1}{\tau_{a}^{2}}_{AB_{j}}$ for all $i=0,1,\cdots N-2$, we have

[TABLE]

for $\alpha\geq 2$.

Example 2. We consider again the pure state (12). Setting $\lambda_{0}=\lambda_{1}=\frac{1}{2}$, $\lambda_{2}=\lambda_{3}=\lambda_{4}=\frac{\sqrt{6}}{6}$, one has ${\tau_{a}}_{A|BC}=\frac{\sqrt{2}}{2}$, ${\tau_{a}}_{AB}={\tau_{a}}_{AC}=\frac{\sqrt{3}}{3}$. Let $y={\tau_{a}^{\alpha}}_{AB}+(2^{\frac{\alpha}{2}}-1){\tau_{a}^{\alpha}}_{AC}-{\tau_{a}^{\alpha}}_{A|BC}=2^{\frac{\alpha}{2}}\left(\frac{\sqrt{3}}{3}\right)^{\alpha}$, $\alpha\geq 2$, be the residual concurrence of assistance. From our results, one can see that $y\geq 0$ for $\alpha\geq 2$, which is the case that does not given in 042332 , see Fig. 1.

III POLYGAMY RELATIONS FOR entanglement of assistance

For polygamy inequality beyond qubits, it was shown that the von Neumann entropy can be used to establish a polygamy inequality of tripartite quantum systems fgj . For any arbitrary dimensional tripartite pure state $|\psi\rangle_{ABC}$, one has $E(|\psi\rangle_{A|BC})\leq E_{a}(\rho_{AB})+E_{a}(\rho_{AC})$, where $E(|\psi\rangle_{A|BC})=S(\rho_{A})$ is the entropy of entanglement between $A$ and $BC$ in terms of the von Neumann entropy $S(\rho)=-\mathrm{Tr}\rho\ln\rho$, and $E_{a}(\rho_{AB})=\max\sum_{i}p_{i}E(|\psi_{i}\rangle_{AB})$, with the maximization taking over all possible pure state decompositions of $\rho_{AB}=\sum_{i}p_{i}|\psi_{i}\rangle_{AB}\langle\psi_{i}|$. Later, a general polygamy inequality for any multipartite state $\rho_{A_{1}|A_{2}\cdots A_{n}}$ was established, $E_{a}(\rho_{A_{1}A_{2}\cdots A_{n}})\leq\sum_{i=2}^{n}E_{a}(\rho_{A_{1}A_{i}})$ 062302 .

Recently, another class of multipartite polygamy inequalities in terms of the $\beta$th power of entanglement of assistance (EOA) has been introduced 042332 . For any multipartite state $\rho_{AB_{0}B_{1}\cdots B_{N-1}}$ and $0\leq\beta\leq 1$, if ${E_{a}}_{AB_{i}}\geq\sum_{j=i+1}^{N-1}{E_{a}}_{AB_{j}}$ for $i=0,1,\cdots,N-2$, then ${E_{a}^{\beta}}_{A|B_{0}B_{1}\cdots B_{N-1}}\leq\sum_{j=0}^{N-1}\beta^{j}{E_{a}^{\beta}}_{AB_{j}}$, where ${E_{a}}(\rho_{AB_{i}})={E_{a}}_{AB_{i}}$ is the entanglement of assistance $\rho_{AB_{i}}$ and ${E_{a}}(\rho_{A|B_{0}B_{1}\cdots B_{N-1}})={E_{a}}_{A|B_{0}B_{1}\cdots B_{N-1}}$. But, for $\beta\geq 1$ the polygamy relations for the $\beta$th power of the entanglement of assistance is still not clear.

[Theorem 4]. For any multipartite state $\rho_{AB_{0}\cdots B_{N-1}}$, if ${E_{a}}_{AB_{i}}\leq\sum_{k=i+1}^{N-1}{E_{a}}_{AB_{k}}$ for $i=0,1,\cdots,m$, and ${E_{a}}_{AB_{j}}\geq\sum_{k=j+1}^{N-1}{E_{a}}_{AB_{k}}$ for $j=m+1,\cdots,N-2$, $\forall$ $1\leq m\leq N-3$, $N\geq 4$, we have

[TABLE]

for $\beta\geq 1$.

[Proof]. From Lemma 1, we have

[TABLE]

Similarly, as ${E_{a}}_{AB_{j}}\geq\sum_{k=j+1}^{N-1}{E_{a}}_{AB_{k}}$ for $j=m+1,\cdots,N-2$, we get

[TABLE]

Combining (III) and (III), we have Theorem 4. $\Box$

As a special case of Theorem 4, if ${E_{a}}_{AB_{i}}\leq\sum_{j=i+1}^{N-1}{E_{a}}_{AB_{j}}$ for all $i=0,1,\cdots N-2$, we have the following conclusion:

[Theorem 5]. For any multipartite state $\rho_{AB_{0}\cdots B_{N-1}}$, if ${E_{a}}_{AB_{i}}\leq\sum_{j=i+1}^{N-1}{E_{a}}_{AB_{j}}$ for all $i=0,1,\cdots N-2$, we have

[TABLE]

for $\beta\geq 1$.

Example 3. Let consider the three-qubit $W$ state $|W\rangle_{ABC}=\frac{1}{\sqrt{3}}(|100\rangle+|010\rangle+|001\rangle)$. We have $E_{a}(|W\rangle_{A|BC})=S(\rho_{A})=\log_{2}3-\frac{2}{3}$ and $E_{a}(\rho_{AB})=E_{a}(\rho_{AC})=\frac{2}{3}$. Set $y={E_{a}^{\beta}}(\rho_{AB})+(2^{\beta}-1){E_{a}^{\beta}}(\rho_{AC})-E_{a}^{\beta}(|W\rangle_{A|BC})=2^{\beta}(\frac{2}{3})^{\beta}-(\log_{2}3-\frac{2}{3})^{\beta}$ to be the residual entanglement of assistance. Fig. 2 shows our polygamy inequality for $\beta\geq 1$.

IV POLYGAMY RELATIONS FOR SCRENoA

Given a bipartite state $\rho_{AB}$ in $H_{A}\otimes H_{B}$, the negativity is defined by GRF , $N(\rho_{AB})=(||\rho_{AB}^{T_{A}}||-1)/2$, where $\rho_{AB}^{T_{A}}$ is the partially transposed $\rho_{AB}$ with respect to the subsystem $A$, $||X||$ denotes the trace norm of $X$, i.e., $||X||=\mathrm{Tr}\sqrt{XX^{\dagger}}$. For the purpose of discussion, we use the following definition of negativity, $N(\rho_{AB})=||\rho_{AB}^{T_{A}}||-1$. For any bipartite pure state $|\psi\rangle_{AB}$, the negativity $N(\rho_{AB})$ is given by $N(|\psi\rangle_{AB})=2\sum_{i<j}\sqrt{\lambda_{i}\lambda_{j}}=(\mathrm{Tr}\sqrt{\rho_{A}})^{2}-1$, where $\lambda_{i}$ are the eigenvalues for the reduced density matrix $\rho_{A}$ of $|\psi\rangle_{AB}$. For a mixed state $\rho_{AB}$, the square of convex-roof extended negativity (SCREN) is defined by

[TABLE]

where the minimum is taken over all possible pure state decompositions $\{p_{i},~{}|\psi_{i}\rangle_{AB}\}$ of $\rho_{AB}$. Similar to the duality between concurrence and concurrence of assistance, we also define a dual quantity to SCREN as

[TABLE]

which we refer to as the SCREN of assistance (SCRENoA), where the maximum is taken over all possible pure state decompositions $\{p_{i},~{}|\psi_{i}\rangle_{AB}\}$ of $\rho_{AB}$. For convenience, we denote ${N_{sc}^{a}}_{AB_{i}}=N_{sc}^{a}(\rho_{AB_{i}})$ the SCRENoA of $\rho_{AB_{i}}$ and ${N_{sc}^{a}}_{AB_{0}\cdots B_{N-1}}=N_{sc}^{a}(|\psi\rangle_{AB_{0}\cdots B_{N-1}})$.

In j012334 it has been shown that ${N_{sc}^{a}}_{A|B_{0}B_{1}\cdots B_{N-1}}\leq\sum_{j=0}^{N-1}{N_{sc}^{a}}_{AB_{j}}$. It is further improved that for $0\leq\beta\leq 1$, $({N_{sc}^{a}}_{A|B_{0}B_{1}\cdots B_{N-1}})^{\beta}\leq\sum_{j=0}^{N-1}\beta^{j}({N_{sc}^{a}}_{AB_{j}})^{\beta}$. But, it is still not clear whether the polygamy relation still holds for the $\beta$th $(\beta\geq 1)$ power of SCRENoA. With a similar consideration to $\tau_{AB_{0}\cdots B_{N-1}}$, we have the following result of SCRENoA for $\beta\geq 1$.

[Theorem 6]. For any multipartite state $\rho_{AB_{0}\cdots B_{N-1}}$, if ${N_{sc}^{a}}_{AB_{i}}\leq\sum_{k=i+1}^{N-1}{N_{sc}^{a}}_{AB_{k}}$ for $i=0,1,\cdots,m$, and ${N_{sc}^{a}}_{AB_{j}}\geq\sum_{k=j+1}^{N-1}{N_{sc}^{a}}_{AB_{k}}$ for $j=m+1,\cdots,N-2$, $\forall$ $1\leq m\leq N-3$, $N\geq 4$, we have

[TABLE]

for $\beta\geq 1$.

[Proof]. From Lemma 1, we have

[TABLE]

Similarly, as ${N_{sc}^{a}}_{AB_{j}}\geq\sum_{k=j+1}^{N-1}{N_{sc}^{a}}_{AB_{k}}$ for $j=m+1,\cdots,N-2$, we get

[TABLE]

Combining (IV) and (IV), we have the Theorem 6. $\Box$

Particularly, the equality in (IV) can be established for 4-qubit generlized $W$-class states $|W\rangle_{AB_{1}B_{2}B_{3}}=\frac{1}{2}(|1000\rangle+|0100\rangle+|0010\rangle+|0001\rangle)$, with $\beta=1$, which can be seen clearly in example 4 below Theorem 7. Specially, from Theorem 6 we have

[Theorem 7]. For any multipartite state $\rho_{AB_{0}\cdots B_{N-1}}$, if ${E_{a}}_{AB_{i}}\leq\sum_{j=i+1}^{N-1}{E_{a}}_{AB_{j}}$ for all $i=0,1,\cdots N-2$, we have

[TABLE]

for $\beta\geq 1$.

Example 4. Let us consider the 4-qubit generlized $W$-class states,

[TABLE]

We have ${N_{sc}^{a}}_{A|B_{1}B_{2}B_{3}}=\frac{3}{4}$, ${N_{sc}^{a}}_{AB_{i}}=\frac{1}{4}$, $i=1,2,3$. Let $y$ be the difference between the right and left hand of inequality (26). One has $y=[2^{\beta}+(2^{\beta}-1)^{2}](\frac{1}{4})^{\beta}-(\frac{3}{4})^{\beta}$; see Fig. 3.

V conclusion

Entanglement monogamy and polygamy are fundamental properties of multipartite entangled states. We have investigated in this work the polygamy relations related to the concurrence of assistance, entanglement of assistance, and SCREN generally for multipartite states. We have found a class of polygamy inequalities of multipartite entanglement in arbitrary-dimensional quantum systems in the $\alpha$th ($\alpha\geq 2$) power of concurrence of assistance, a case that has not been studied before. Moreover, the $\beta$th power of polygamy inequalities have been obtained for the entanglement of assistance and SCRENoA for $\beta\geq 1$. The approach developed in this work is applicable to the study of monogamy properties in other quantum entanglement measures and quantum correlations.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China(NSFC) under Grants 11847209; 11675113 and 11635009; Key Project of Beijing Municipal Commission of Education under No. KZ201810028042; the Ministry of Science and Technology of the Peoples’ Republic of China(2015CB856703); and the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDB23030100.