1. Introduction
Quantum entanglement is a pivotal aspect within quantum mechanics, yielding a comprehensive comprehension of quantum correlations' fundamental nature and being instrumental in the field of quantum information processing. As a fundamental resource for quantum tasks, quantum entanglement has been harnessed in a variety of quantum communication protocols, including superdense coding [1], quantum cryptography [2], quantum teleportation [3] and remote-state preparation [4].
A fundamental characteristic of quantum entanglement is illustrated by the monogamy relation [5]. This principle states that a quantum subsystem, when entangled with one of the other subsystems, restricts its level of entanglement with the rest, a notion known as the monogamy of entanglement (MoE) [5]. Consider a three-qubit system where Alice and Bob share a Bell state $| {\rm{\Psi }}\rangle =\frac{1}{\sqrt{2}}(| 00\rangle +| 11\rangle )$ across systems A and B, they are precluded from sharing any entangled states with Charlie's system C in this scenario. Such inherent restricted entanglement distribution can be exploited in the realm of quantum cryptography, enabling the estimation of the amount of information that an eavesdropper might intercept about the secret key that is intended to be securely shared [6]. MoE has been extensively utilized in diverse disciplines such as condensed-matter physics [7] and even black-hole physics [8]. It highlights the concept that entanglement is not freely distributed, leading to profound implications for the manipulation and distribution of quantum information in multipartite quantum systems [9, 10]. Mathematically, the MoE is expressed by utilizing an entanglement measure denoted as ${ \mathcal E }$, applied to a tripartite system ϱABC, as follows
$\begin{eqnarray}{ \mathcal E }({\varrho }_{A| BC})\geqslant { \mathcal E }({\varrho }_{AB})+{ \mathcal E }({\varrho }_{AC}),\end{eqnarray}$
which is commonly recognized as the CKW inequality [5], where ${\varrho }_{AB}={{\rm{Tr}}}_{C}({\varrho }_{ABC})$, ${\varrho }_{AC}={{\rm{Tr}}}_{B}({\varrho }_{ABC})$, and ${ \mathcal E }$ serves as a quantum entanglement measure. It has been further shown that the squared concurrence satisfies the monogamy relations for multi-qubit states [5]. Subsequently, the monogamy relations have been extensively studied for other entanglement measures such as entanglement of formation [11], entanglement negativity [12], the Tsallis-q entanglement, and Rényi-α entanglement (RαE) [13, 14].Dual to the entanglement measures, the concept of entanglement of assistances leads to polygamy relations. Gour et al introduced in [15] the initial polygamy inequality for arbitrary tripartite systems, which is based on the squared concurrence of assistance. The polygamy relation for the tripartite state ϱABC is defined by
$\begin{eqnarray}{{ \mathcal E }}_{a}({\varrho }_{A| BC})\leqslant {{ \mathcal E }}_{a}({\varrho }_{A| B})+{{ \mathcal E }}_{a}({\varrho }_{A| C}),\end{eqnarray}$
where ${{ \mathcal E }}_{a}$ represents the entanglement measure of assistance that corresponds to ${ \mathcal E }$. Subsequently, Buscemi et al in [16] introduced a polygamy relation for any dimensional pure states. Later, Kim extended this relationship to the case of arbitrary dimensional mixed states [17].As a kind of generalized entanglement of formation [18], the RαE has the advantages to characterize quantum phases with differing computational power [19], as well as the topologically ordered states [20, 21]. It is naturally of significance to study the monogamy and polygamy relations associated with RαE and its dual RαE of assistance (RαEoA) in multipartite systems. Recently, it was demonstrated that if α ≥ 2, the RαE satisfies the monogamy relation within multi-qubit systems [22]. In general, enhancing monogamy relations allows for a more accurate depiction of multipartite entanglement. Subsequently, more tighter monogamous relations have been presented for RαE [23–27]. In [28], Song et al presented a novel polygamy relation of multipartite systems using RαEoA for $\frac{\sqrt{7}-1}{2}\,\leqslant \,\alpha \lt \frac{\sqrt{13}-1}{2}$. Then, in [23] the authors further improved the polygamy inequalities.
The following sections detail the organization of this article. In section 2 , we will review the essential foundational concepts required. In section 3 , we focus on tightening the monogamy relations satisfied by the η-th (η ≥ 2) power of RαE for α ≥ 2, and by the γ-th (γ ≥ 4) power of RαE for $\frac{\sqrt{7}-1}{2}\,\leqslant \,\alpha \lt 2$, respectively. The results indicate that the newly derived monogamy relations exhibit tighter bounds than the ones given in [22–27]. In section 4 , we give the polygamy relations satisfied by the t-th power of RαEoA with $\frac{\sqrt{7}-1}{2}\,\leqslant \,\alpha \lt \frac{\sqrt{13}-1}{2}$ for 0 ≤ t ≤ 1, which are tighter than those given in [23, 28]. In section 5 , we present a bound of RαEoA by examining the relations between the residual entanglement of RαE and the three tangle. In addition, we provide detailed examples to illustrate the benefits of our results. The conclusion is given in section 6 .
2. Preliminaries
Let HA and HB represent finite dimensional complex Hilbert spaces corresponding to the quantum subsystems A and B, respectively. For a bipartite pure state ∣ψ⟩AB ∈ HA ⨂ HB, the concurrence ${\mathscr{C}}(| \psi {\rangle }_{AB})$ is expressed as [29, 30]
$\begin{eqnarray}{\mathscr{C}}(| \psi {\rangle }_{AB})=\sqrt{2[1-{\rm{Tr}}({\varrho }_{A}^{2})]},\end{eqnarray}$
where ${\varrho }_{A}={{\rm{Tr}}}_{B}(| \psi {\rangle }_{AB}\langle \psi | )$. The concurrence of a bipartite mixed state ϱAB ∈ HA ⨂ HB is derived via the convex roof extension, $\begin{eqnarray}{\mathscr{C}}({\varrho }_{AB})=\mathop{\min }\limits_{\{{p}_{i},| {\psi }_{i}\rangle \}}\displaystyle \sum _{i}{p}_{i}{\mathscr{C}}(| {\psi }_{i}\rangle ),\end{eqnarray}$
where the minimum is taken over all possible pure decompositions of ϱAB = ∑ipi∣ψi⟩AB⟨ψi∣ with pi ≥ 0 and ∑ipi = 1.For an N-qubit state $| \psi {\rangle }_{A{B}_{1}\cdots {B}_{N-1}}\in {H}_{A}\otimes {H}_{{B}_{1}}\otimes \cdots \otimes {H}_{{B}_{N-1}}$, the concurrence ${\mathscr{C}}(| \psi {\rangle }_{A| {B}_{1}\cdots {B}_{N-1}})$ for the state $| \psi {\rangle }_{A{B}_{1}\cdots {B}_{N-1}}$ under the bipartite partition A and B1B2 ⋯ BN−1 obeys the following monogamy inequality [6],
$\begin{eqnarray}\begin{array}{r}{{\mathscr{C}}}^{2}(| \psi {\rangle }_{A| {B}_{1}{B}_{2}\cdots {B}_{N-1}})\geqslant \displaystyle \sum _{i=1}^{N-1}{{\mathscr{C}}}^{2}({\varrho }_{A| {B}_{i}}),\end{array}\end{eqnarray}$
where ${\mathscr{C}}({\varrho }_{A| {B}_{i}})$ is the concurrence of the reduced state ${\varrho }_{A{B}_{i}}={{\rm{tr}}}_{{B}_{1}\cdots {B}_{i-1}{B}_{i+1}\cdots {B}_{N-1}}(\varrho )$.For a tripartite pure state ∣ψ⟩ABC, the concurrence of assistance is expressed as [31, 32],
$\begin{eqnarray*}\begin{array}{r}{{\mathscr{C}}}_{a}(| \psi {\rangle }_{ABC})\equiv {{\mathscr{C}}}_{a}({\varrho }_{AB})=\mathop{\max }\limits_{\{{p}_{i},| {\psi }_{i}\rangle \}}\displaystyle \sum _{i}{p}_{i}{\mathscr{C}}(| {\psi }_{i}\rangle ),\end{array}\end{eqnarray*}$
where the maximum is taken over all possible pure state decompositions of ${\varrho }_{AB}={{\rm{Tr}}}_{C}(| \psi {\rangle }_{ABC}{\langle \psi | )={\sum }_{i}{p}_{i}| {\psi }_{i}\rangle }_{AB}\langle {\psi }_{i}| $ with pi ≥ 0 and ∑ipi = 1. For a pure state ϱAB, it holds that ${\mathscr{C}}(| \psi {\rangle }_{AB})={{\mathscr{C}}}_{a}({\varrho }_{AB})$.For a two-qubit mixed state ϱ, the concurrence of ϱ is expressed as [33]
$\begin{eqnarray*}\begin{array}{r}{\mathscr{C}}(\varrho )=\max \{{\vartheta }_{1}-{\vartheta }_{2}-{\vartheta }_{3}-{\vartheta }_{4},0\},\end{array}\end{eqnarray*}$
where ϑ1 ≥ ϑ2 ≥ ϑ3 ≥ ϑ4 represents the eigenvalues of the matrix $\sqrt{\sqrt{\varrho }\widetilde{\varrho }\sqrt{\varrho }}$. The expression $\widetilde{\varrho }$ is defined as (ϱy ⨂ ϱy)ϱ*(ϱy ⨂ ϱy), with ϱ* denoting the complex conjugation of ϱ, and ϱy denoting the standard Pauli matrix.RαE is a widely recognized entanglement measure that has found numerous applications in various fields such as condensed matter physics [34]. For a bipartite pure state ∣φ⟩AB, the RαE is defined as [22]
$\begin{eqnarray}{R}_{\alpha }(| \varphi {\rangle }_{AB})=\frac{1}{1-\alpha }{{\rm{log}}}_{2}({\rm{Tr}}{\varrho }_{A}^{\alpha }),\end{eqnarray}$
where α > 0, α ≠ 1 and ${\varrho }_{A}={{\rm{Tr}}}_{B}(| \varphi {\rangle }_{AB}\langle \varphi | )$. When α tends to 1, the RαE converges to the von Neumann entropy. The RαE of a bipartite mixed state ϱAB is defined by $\begin{eqnarray}{R}_{\alpha }({\varrho }_{AB})={\rm{\min }}\displaystyle \sum _{i}{p}_{i}{R}_{\alpha }(| {\varphi }_{i}{\rangle }_{AB}),\end{eqnarray}$
where the minimum is taken over all possible pure state decompositions $\{{p}_{i},{\varphi }_{AB}^{i}\}$ of ϱAB.It has been proved in [22, 35] that when $\alpha \,\geqslant \,\frac{\sqrt{7}-1}{2}$, the RαE of any two-qubit state ϱAB has an analytical formula,
$\begin{eqnarray}{R}_{\alpha }({\varrho }_{AB})={f}_{\alpha }({\mathscr{C}}({\varrho }_{AB})),\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{f}_{\alpha }(\xi ) & = & \frac{1}{1-\alpha }{{\rm{log}}}_{2}\left[{\left(\frac{1-\sqrt{1-{\xi }^{2}}}{2}\right)}^{\alpha }\right.\\ & & \left.+{\left(\frac{1+\sqrt{1-{\xi }^{2}}}{2}\right)}^{\alpha }\right]\end{array}\end{eqnarray}$
is a monotonically increasing convex function, 0 ≤ ξ ≤ 1. When α ≥ 2, one has [22], $\begin{eqnarray}\begin{array}{r}{f}_{\alpha }(\sqrt{{\xi }_{1}^{2}+{\xi }_{2}^{2}})\geqslant {f}_{\alpha }({\xi }_{1})+{f}_{\alpha }({\xi }_{2}),\end{array}\end{eqnarray}$
for $0\,\leqslant \,{\xi }_{1},{\xi }_{2},{\xi }_{1}^{2}+{\xi }_{2}^{2}\,\leqslant \,1$. When $\frac{\sqrt{7}-1}{2}\,\leqslant \,\alpha \lt 2$, it has been shown that [26], $\begin{eqnarray}{f}_{\alpha }^{2}(\sqrt{{\xi }_{1}^{2}+{\xi }_{2}^{2}})\geqslant {f}_{\alpha }^{2}({\xi }_{1})+{f}_{\alpha }^{2}({\xi }_{2}),\end{eqnarray}$
for $0\,\leqslant \,{\xi }_{1},{\xi }_{2},{\xi }_{1}^{2}+{\xi }_{2}^{2}\,\leqslant \,1$.For any N-qubit state ${\varrho }_{A{B}_{1}\cdots {B}_{N-1}}\in {H}_{A}\otimes {H}_{{B}_{1}}\otimes \cdots \otimes {H}_{{B}_{N-1}}$, RαE satisfies the following monogamy relation [22],
$\begin{eqnarray}{R}_{\alpha }^{\mu }\left({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}}\right)\geqslant {R}_{\alpha }^{\mu }\left({\varrho }_{A| {B}_{1}}\right)+\cdots +{R}_{\alpha }^{\mu }\left({\varrho }_{A| {B}_{N-1}}\right),\end{eqnarray}$
for α ≥ 2 and μ ≥ 1, and the following monogamy relation [26], $\begin{eqnarray}{R}_{\alpha }^{\omega }\left({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}}\right)\geqslant {R}_{\alpha }^{\omega }\left({\varrho }_{A| {B}_{1}}\right)+\cdots +{R}_{\alpha }^{\omega }\left({\varrho }_{A| {B}_{N-1}}\right),\end{eqnarray}$
for $\alpha \,\geqslant \,\frac{\sqrt{7}-1}{2}$ and ω ≥ 2, where ${R}_{\alpha }\left({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}}\right)$ is the RαE of ${\varrho }_{A{B}_{1}\cdots {B}_{N-1}}$ related to the bipartition A and the rest, and ${R}_{\alpha }\left({\varrho }_{A| {B}_{i-1}}\right)$ is the RαE of the reduced density matrix ${\varrho }_{A{B}_{i-1}}$ of the two-qubit system ABi−1, i = 2, …, n.Dual to RαE, the RαEoA is expressed as
$\begin{eqnarray}{R}_{\alpha }^{a}({\varrho }_{AB})={\rm{\max }}\displaystyle \sum _{i}{p}_{i}{R}_{\alpha }(| {\varphi }_{i}{\rangle }_{AB}),\end{eqnarray}$
where the maximum is taken over all possible pure state decompositions $\{{p}_{i},{\varphi }_{AB}^{i}\}$ of ϱAB. For $\frac{\sqrt{7}-1}{2}\,\leqslant \,\alpha \,\leqslant \,\frac{\sqrt{13}-1}{2}$, 0 ≤ t ≤ 1 and any N-qubit state ${\varrho }_{A{B}_{1}\cdots {B}_{N-1}}$, one has [28] $\begin{eqnarray}{({R}_{\alpha }^{a})}^{t}\left({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}}\right)\leqslant {({R}_{\alpha }^{a})}^{t}\left({\varrho }_{A| {B}_{1}}\right)+\cdots +{({R}_{\alpha }^{a})}^{t}\left({\varrho }_{A| {B}_{N-1}}\right),\end{eqnarray}$
where ${R}_{\alpha }^{a}\left({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}}\right)$ denotes the RαEoA in the partition A∣B1 ⋯ BN−1, and ${R}_{\alpha }^{a}\left({\varrho }_{A| {B}_{i-1}}\right)$ is the RαEoA of the reduced density matrix ${\varrho }_{A{B}_{i-1}}$ of the two-qubit system ABi−1,i = 2, …, n.3. Tighter monogamy relations for RαE
Monogamy relations serve as a pivotal tool in advancing our comprehension of entanglement distribution within multipartite systems. Tighter monogamy relations enable a more detailed delineation of the distribution of quantum correlations within the subsystems. This could result in tighter security constraints within the realm of quantum cryptography [36]. In the following, we establish new classes of tighter monogamy relations of RαE. We first introduce some related lemmas.
[37] For λ≥2 and ς≥0, one has
$\begin{eqnarray}\begin{array}{l}{(1+\varsigma )}^{\lambda -1}\geqslant 1+(\lambda -1)\varsigma .\end{array}\end{eqnarray}$
For 0 ≤ θ ≤ 1 and λ≥2, we see that
$\begin{eqnarray}\begin{array}{l}{(1+\theta )}^{\lambda }\geqslant 1+\lambda \theta +({2}^{\lambda }-\lambda -1){\theta }^{\lambda }.\end{array}\end{eqnarray}$
If θ = 0, the inequality holds. If θ ≠ 0, the core idea is to reformulate the desired inequality. Observing that equation (
$\begin{eqnarray*}\begin{array}{l}\frac{\partial f}{\partial \theta }=\frac{\lambda {\theta }^{\lambda -1}\{[{(1+\theta )}^{\lambda -1}-1]\theta -[{(1+\theta )}^{\lambda }-\lambda \theta -1]\}}{{\theta }^{2\lambda }}\\ =\,\frac{[\lambda {(1+\theta )}^{\lambda -1}-\lambda ]{\theta }^{\lambda }-\left[{(1+\theta )}^{\lambda }-\lambda \theta -1\right]\lambda {\theta }^{\lambda -1}}{{\theta }^{2\lambda }}\\ =\,\frac{\lambda {\theta }^{\lambda -1}[1+(\lambda -1)\theta -{(1+\theta )}^{\lambda -1}]}{{\theta }^{2\lambda }}.\end{array}\end{eqnarray*}$
By lemma We present below the tighter monogamy inequality in terms of the RαE for α ≥ 2.
For any tripartite quantum state ϱABC, we have for α ≥ 2 and η ≥ 2,
(1) If Rα(ϱA∣B) ≥ Rα(ϱA∣C), then
$\begin{eqnarray}\begin{array}{rcl}{R}_{\alpha }^{\eta }({\varrho }_{A| BC}) & \geqslant & {R}_{\alpha }^{\eta }({\varrho }_{A| B})+\eta {R}_{\alpha }^{\eta -1}({\varrho }_{A| B}){R}_{\alpha }({\varrho }_{A| C})\\ & & +({2}^{\eta }-\eta -1){R}_{\alpha }^{\eta }({\varrho }_{A| C}).\end{array}\end{eqnarray}$
(2) If Rα(ϱA∣B) ≤ Rα(ϱA∣C), then $\begin{eqnarray}\begin{array}{rcl}{R}_{\alpha }^{\eta }({\varrho }_{A| BC}) & \geqslant & {R}_{\alpha }^{\eta }({\varrho }_{A| C})+\eta {R}_{\alpha }^{\eta -1}({\varrho }_{A| C}){R}_{\alpha }({\varrho }_{A| B})\\ & & +({2}^{\eta }-\eta -1){R}_{\alpha }^{\eta }({\varrho }_{A| B}).\end{array}\end{eqnarray}$
If Rα(ϱA∣B) ≥ Rα(ϱA∣C), we have
$\begin{eqnarray}\begin{array}{rcl}{R}_{\alpha }^{\eta }({\varrho }_{A| BC}) & \geqslant & {[{R}_{\alpha }({\varrho }_{A| B})+{R}_{\alpha }({\varrho }_{A| C})]}^{\eta }\\ & = & {R}_{\alpha }^{\eta }({\varrho }_{A| B}){\left(1+\frac{{R}_{\alpha }({\varrho }_{A| C})}{{R}_{\alpha }({\varrho }_{A| B})}\right)}^{\eta }\\ & \geqslant & {R}_{\alpha }^{\eta }({\varrho }_{A| B})+\eta {R}_{\alpha }^{\eta -1}({\varrho }_{A| B}){R}_{\alpha }({\varrho }_{A| C})\\ & & +({2}^{\eta }-\eta -1){R}_{\alpha }^{\eta }({\varrho }_{A| C}),\end{array}\end{eqnarray}$
where the first inequality is derived from the inequality (From lemma
$\begin{eqnarray*}\begin{array}{rcl}{(1+\theta )}^{\lambda } & \geqslant & 1+\lambda \theta +({2}^{\lambda }-\lambda -1){\theta }^{\lambda }\\ & \geqslant & 1+\frac{{\lambda }^{2}}{\lambda +1}\theta +({2}^{\lambda }-\frac{{\lambda }^{2}}{\lambda +1}-1){\theta }^{\lambda }\\ & & (\,\rm{The inequality in [27]}\,)\\ & \geqslant & 1+\frac{\lambda }{2}\theta +({2}^{\lambda }-\frac{\lambda }{2}-1){\theta }^{\lambda }\\ & & \quad (\,\rm{The inequality in [26]}\,)\\ & \geqslant & 1+({2}^{\lambda }-1){\theta }^{\lambda }\quad (\,\rm{The inequality in [25]}\,).\end{array}\end{eqnarray*}$
Since $\lambda \gt \frac{{\lambda }^{2}}{\lambda +1}\,\geqslant \,\frac{\lambda }{2}$, θ − θλ > 0 for 0 ≤ θ ≤ 1 and λ ≥ 2, obviously our inequality given in theorem Generalizing theorem 1 to N-qubit quantum systems, we obtain the following theorem. The conditions in theorem 2 describe the distribution order of entanglement among the subsystems: the first z subsystems (i.e. B1 to Bz, ∀ 1 ≤ z ≤ N − 3, N ≥ 4) have strong entanglement with subsystem A, sufficient to dominate the combined entanglement of all subsequent subsystems; starting from Bz+1, the entanglement is weaker and cannot dominate the subsequent parts. This ordering allows us to iteratively simplify the derivation of the monogamy inequality, ensuring a tighter bound.
For any N-qubit quantum state ${\varrho }_{A{B}_{1}\cdots {B}_{N-1}}\in {H}_{A}\otimes {H}_{{B}_{1}}\otimes \cdots \otimes {H}_{{B}_{N-1}}$, if ${R}_{\alpha }({\varrho }_{A| {B}_{i}})\,\geqslant \,\displaystyle {\sum }_{l=i+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})$ for i = 1, 2, ⋯ , z, and ${R}_{\alpha }({\varrho }_{A| {B}_{j}})\,\leqslant \,\displaystyle {\sum }_{l=j+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})$ for j = z + 1, ⋯ , N − 2, ∀ 1 ≤ z ≤ N − 3, N ≥ 4, then
$\begin{array}{l}R_{\alpha}^{\eta}\left(\varrho_{A \mid B_{1} \cdots B_{N-1}}\right) \\\geqslant \sum_{i=1}^{z}\left\{( 2 ^ { \eta } - \eta - 1 ) ^ { i - 1 } \left[R_{\alpha}^{\eta}\left(\varrho_{A \mid B_{i}}\right)\right.\right. \\\left.\left.+\eta R_{\alpha}^{\eta-1}\left(\varrho_{A \mid B_{i}}\right)\left(\sum_{l=i+1}^{N-1} R_{\alpha}\left(\varrho_{A \mid B_{l}}\right)\right)\right]\right\} \\+\left(2^{\eta}-\eta-1\right)^{z}\left(2^{\eta}-1\right)\left[R_{\alpha}^{\eta}\left(\varrho_{A \mid B_{z+1}}\right)\right. \\\left.+\cdots+R_{\alpha}^{\eta}\left(\varrho_{A \mid B_{N-3}}\right)\right] \\+\left(2^{\eta}-\eta-1\right)^{z}\left\{\left(2^{\eta}-\eta-1\right) R_{\alpha}^{\eta}\left(\varrho_{A \mid B_{N-2}}\right)\right. \\\left.+\eta R_{\alpha}\left(\varrho_{A \mid B_{N-2}}\right) R_{\alpha}^{\eta-1}\left(\varrho_{A \mid B_{N-1}}\right)+R_{\alpha}^{\eta}\left(\varrho_{A \mid B_{N-1}}\right)\right\}\end{array}$
where η ≥ 2 and α ≥ 2.When ${R}_{\alpha }({\varrho }_{A| {B}_{i}})\,\geqslant \,\displaystyle {\sum }_{l=i+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})$ for i = 1, 2, ⋯ , z, set ${\theta }_{i}=\displaystyle {\sum }_{l=i+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})/{R}_{\alpha }({\varrho }_{A| {B}_{i}})$. Substituting θi into the inequality (
$\begin{eqnarray}\begin{array}{rcl}{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{i}\cdots {B}_{N-1}}) & \geqslant & {\left(\displaystyle \sum _{l=i}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)}^{\eta }\\ & \geqslant & {R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{i}})+\eta {R}_{\alpha }^{\eta -1}({\varrho }_{A| {B}_{i}})\\ & & \times \displaystyle \sum _{l=i+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})+({2}^{\eta }-\eta -1)\\ & & \times \left(\displaystyle \sum _{l=i+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right).\end{array}\end{eqnarray}$
When i = 1, we obtain $\begin{eqnarray}\begin{array}{l}{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}})\\ \geqslant \,{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{1}})+\eta {R}_{\alpha }^{\eta -1}({\varrho }_{A| {B}_{1}})\left(\displaystyle \sum _{l=2}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)\\ +\,({2}^{\eta }-\eta -1){\left(\displaystyle \sum _{l=2}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)}^{\eta }\\ \geqslant \,{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{1}})+\eta {R}_{\alpha }^{\eta -1}({\varrho }_{A| {B}_{1}})\left(\displaystyle \sum _{l=2}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)\\ +\,({2}^{\eta }-\eta -1){R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{2}})\\ +\,({2}^{\eta }-\eta -1)\eta {R}_{\alpha }^{\eta -1}({\varrho }_{A| {B}_{2}})\left(\displaystyle \sum _{l=3}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)\\ +\,{({2}^{\eta }-\eta -1)}^{2}{\left(\displaystyle \sum _{l=3}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)}^{\eta }\\ \geqslant \,\cdots \\ \geqslant \,\displaystyle \sum _{i=1}^{z}\left\{{({2}^{\eta }-\eta -1)}^{i-1}\left[{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{i}})\right.\right.\\ \left.\left.+\,\eta {R}_{\alpha }^{\eta -1}({\varrho }_{A| {B}_{i}})\left(\displaystyle \sum _{l=i+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)\right]\right\}\\ +\,{({2}^{\eta }-\eta -1)}^{z}{\left(\displaystyle \sum _{l=z+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)}^{\eta },\end{array}\end{eqnarray}$
by utilizing the inequality (When ${R}_{\alpha }({\varrho }_{A| {B}_{j}})\,\leqslant \,\displaystyle {\sum }_{l=j+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})$ for j = z + 1, ⋯ , N − 2, we use a strengthened version of lemma
$\begin{eqnarray}\begin{array}{rcl}{\left(\displaystyle \sum _{l=j}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)}^{\eta } & \geqslant & {\left(\displaystyle \sum _{l=j+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)}^{\eta }\\ & & +({2}^{\eta }-1){R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{j}}).\end{array}\end{eqnarray}$
When j = z + 1, we see that $\begin{eqnarray}\begin{array}{l}{\left(\displaystyle \sum _{l=z+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)}^{\eta }\\ \geqslant \,{\left(\displaystyle \sum _{l=z+2}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)}^{\eta }+({2}^{\eta }-1){R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{z}+1})\\ \geqslant \,{[{R}_{\alpha }({\varrho }_{A| {B}_{N-2}})+{R}_{\alpha }({\varrho }_{A| {B}_{N-1}})]}^{\eta }\\ +\,({2}^{\eta }-1)[{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{z+1}})+\cdots +{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{N-3}})]\\ \geqslant \,{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{N-1}})+\eta {R}_{\alpha }({\varrho }_{A| {B}_{N-2}}){R}_{\alpha }^{\eta -1}({\varrho }_{A| {B}_{N-1}})\\ +\,({2}^{\eta }-\eta -1){R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{N-2}})\\ +\,({2}^{\eta }-1)[{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{z+1}})+\cdots +{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{N-3}})],\end{array}\end{eqnarray}$
where the first and second inequalities hold because of the inequality (In particular, we have the following monogamy relation.
For any N-qubit quantum state ${\varrho }_{A{B}_{1}\cdots {B}_{N-1}}$ $\in {H}_{A}\otimes {H}_{{B}_{1}}\otimes $ $\cdots \otimes {H}_{{B}_{N-1}}$, if ${R}_{\alpha }({\varrho }_{A| {B}_{i}})\,\geqslant \,\displaystyle {\sum }_{l=i+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})$ for i = 1, 2, ⋯ , N − 2, then
$\begin{eqnarray}\begin{array}{rcl}{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}}) & \geqslant & \displaystyle \sum _{i=1}^{N-2}\left\{{({2}^{\eta }-\eta -1)}^{i-1}\right.\\ & & \times \left[{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{i}})+\eta {R}_{\alpha }^{\eta -1}({\varrho }_{A| {B}_{i}})\right.\\ & & \left.\left.\times \left(\displaystyle \sum _{l=i+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{l}})\right)\right]\right\}\\ & & +{({2}^{\eta }-\eta -1)}^{N-2}{R}_{\alpha }^{\eta }({\varrho }_{A| {B}_{N}-1}),\end{array}\end{eqnarray}$
where η ≥ 2 and α ≥ 2.To illustrate the tightness of the monogamy inequality, we consider the following example.
Under local unitary operations, any three-qubit pure state can be written as [38, 39],
$\begin{eqnarray}\begin{array}{rcl}| \psi {\rangle }_{A| BC} & = & {\lambda }_{0}| 000\rangle +{\lambda }_{1}{{\rm{e}}}^{{\rm{i}}\varphi }| 100\rangle +{\lambda }_{2}| 101\rangle \\ & & +{\lambda }_{3}| 110\rangle +{\lambda }_{4}| 111\rangle ,\end{array}\end{eqnarray}$
where 0 ≤ φ ≤ π, λi ≥ 0, i = 0, 1, 2, 3, 4, and ${\sum }_{i=0}^{4}{\lambda }_{i}^{2}=1$. Set ${\lambda }_{0}=\frac{\sqrt{5}}{3}$, λ1 = 0, λ4 = 0, ${\lambda }_{2}=\frac{\sqrt{3}}{3}$ and ${\lambda }_{3}=\frac{1}{3}$. When α = 2, we get R2(ψA∣BC) = 0.98230, R2(ϱA∣B) = 0.66742 and R2(ϱA∣C) = 0.19010. Thus, $\begin{eqnarray}\begin{array}{r}{R}_{2}^{\eta }({\varrho }_{A| B})+({2}^{\eta }-\eta -1){R}_{2}^{\eta }({\varrho }_{A| C})\\ \quad +\,\eta {R}_{2}^{\eta -1}({\varrho }_{A| B}){R}_{2}({\varrho }_{A| C})\\ \quad =\,{(0.66742)}^{\eta }+\eta {(0.66742)}^{\eta -1}(0.19010)\\ \quad +\,({2}^{\eta }-\eta -1){(0.19010)}^{\eta },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{R}_{2}^{\eta }({\varrho }_{A| B})+{R}_{2}^{\eta }({\varrho }_{A| C})={(0.66742)}^{\eta }+{(0.19010)}^{\eta },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{R}_{2}^{\eta }({\varrho }_{A| B})+({2}^{\eta }-1){R}_{2}^{\eta }({\varrho }_{A| C})\\ \quad =\,{(0.66742)}^{\eta }+({2}^{\eta }-1){(0.19010)}^{\eta },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{R}_{2}^{\eta }({\varrho }_{A| B})+\frac{\eta }{2}{R}_{2}^{\eta -1}({\varrho }_{A| B}){R}_{2}({\varrho }_{A| C})\\ \quad +({2}^{\eta }-\frac{\eta }{2}-1){R}_{2}^{\eta }({\varrho }_{A| C})\\ \quad =\,{(0.66742)}^{\eta }+\frac{\eta }{2}{(0.66742)}^{\eta -1}(0.19010)\\ \quad +\,({2}^{\eta }-\frac{\eta }{2}-1){(0.19010)}^{\eta },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{R}_{2}^{\eta }({\varrho }_{A| B})+\frac{{\eta }^{2}}{\eta +1}{R}_{2}^{\eta -1}({\varrho }_{A| B}){R}_{2}({\varrho }_{A| C})\\ \quad +\,({2}^{\eta }-\frac{{\eta }^{2}}{\eta +1}-1){R}_{2}^{\eta }({\varrho }_{A| C})\\ \quad =\,{(0.66742)}^{\eta }+\frac{{\eta }^{2}}{\eta +1}{(0.66742)}^{\eta -1}(0.19010)\\ \quad +\,({2}^{\eta }-\frac{{\eta }^{2}}{\eta +1}-1){(0.19010)}^{\eta }.\end{array}\end{eqnarray}$
Equation (
Figure 1. The y axis is the lower bound of Rα(ψA∣BC). Solid black line denotes R2(ψA∣BC) for the state given in equation ( |
For $\frac{\sqrt{7}-1}{2}\,\leqslant \,\alpha \,\leqslant \,2$, we have the following conclusion.
For any tripartite quantum state ϱABC, we have when $\frac{\sqrt{7}-1}{2}\,\leqslant \,\alpha \lt 2$.
(1) If Rα(ϱA∣B) ≥ Rα(ϱA∣C), then
$\begin{eqnarray}\begin{array}{l}{R}_{\alpha }^{\gamma }({\varrho }_{A| BC})\geqslant {R}_{\alpha }^{\gamma }({\varrho }_{A| B})+\lambda {R}_{\alpha }^{\gamma -2}({\varrho }_{A| B}){R}_{\alpha }^{2}({\varrho }_{A| C})\\ +\,({2}^{\lambda }-\lambda -1){R}_{\alpha }^{\gamma }({\varrho }_{A| C}),\end{array}\end{eqnarray}$
for γ ≥ 4. (2) If Rα(ϱA∣B) ≤ Rα(ϱA∣C), then
$\begin{eqnarray}\begin{array}{l}{R}_{\alpha }^{\gamma }({\varrho }_{A| BC})\geqslant {R}_{\alpha }^{\gamma }({\varrho }_{A| C})+\lambda {R}_{\alpha }^{\gamma -2}({\varrho }_{A| C}){R}_{\alpha }^{2}({\varrho }_{A| B})\\ \,+\,({2}^{\lambda }-\lambda -1){R}_{\alpha }^{\gamma }({\varrho }_{A| B}),\end{array}\end{eqnarray}$
for γ ≥ 4, where $\lambda =\frac{\gamma }{2}(\geqslant 2)$.If Rα(ϱA∣B) ≥ Rα(ϱA∣C), we have
$\begin{eqnarray*}\begin{array}{l}{R}_{\alpha }^{\gamma }({\varrho }_{A| BC})\geqslant {[{R}_{\alpha }^{2}({\varrho }_{A| B})+{R}_{\alpha }^{2}({\varrho }_{A| C})]}^{\lambda }\\ =\,{R}_{\alpha }^{\gamma }({\varrho }_{A| B}){\left(1+\frac{{R}_{\alpha }^{2}({\varrho }_{A| C})}{{R}_{\alpha }^{2}({\varrho }_{A| B})}\right)}^{\lambda }\\ \geqslant \,{R}_{\alpha }^{\gamma }({\varrho }_{A| B})+\lambda {R}_{\alpha }^{\gamma -2}({\varrho }_{A| B}){R}_{\alpha }^{2}({\varrho }_{A| C})\\ +\,({2}^{\lambda }-\lambda -1){R}_{\alpha }^{\gamma }({\varrho }_{A| C}),\end{array}\end{eqnarray*}$
where the first inequality is obtained from the inequality (From theorem 4 , we get the following theorem for multi-qubit quantum systems. The conditions in theorem 5 reflect the distribution characteristics of entanglement assistance in multipartite systems. The first z (∀ 1 ≤ z ≤ N − 3, N ≥ 4) subsystems have sufficiently large squared entanglement values with A, enough to cover the sum of the subsequent subsystems, indicating that these subsystems play a dominant role in the polygamy relations. In contrast, the subsequent subsystems have smaller squared entanglement values, suggesting a more uniform distribution of entanglement resources. This condition allows us to decompose the polygamy inequality into a dominant part and a residual part, thereby obtaining a more precise upper bound.
For any N-qubit quantum state ${\varrho }_{A{B}_{1}\cdots {B}_{N-1}}\in {H}_{A}\otimes {H}_{{B}_{1}}\otimes \cdots \otimes {H}_{{B}_{N-1}}$, if ${R}_{\alpha }^{2}({\varrho }_{A| {B}_{i}})\,\geqslant \displaystyle {\sum }_{l=i+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})$ for i = 1, 2, ⋯ , z, and ${R}_{\alpha }^{2}({\varrho }_{A| {B}_{j}})\,\leqslant \,\displaystyle {\sum }_{l=j+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})$ for j = z + 1, ⋯ , N − 2, ∀ 1 ≤ z ≤ N − 3 and N ≥ 4, we see that
$\begin{eqnarray}\begin{array}{l}{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}})\geqslant \displaystyle \sum _{i=1}^{z}\left\{{({2}^{\lambda }-\lambda -1)}^{i-1}\right.\\ \times \,\left[{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{i}})+\lambda {R}_{\alpha }^{\gamma -2}({\varrho }_{A| {B}_{i}})\right.\\ \times \,\left.\left.\left(\displaystyle \sum _{l=i+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)\right]\right\}\\ +\,{({2}^{\lambda }-\lambda -1)}^{z}({2}^{\lambda }-1)[{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{z+1}})\\ +\,\cdots +{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{N-3}})]\\ +\,{({2}^{\lambda }-\lambda -1)}^{z}\left\{({2}^{\lambda }-\lambda -1){R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{N-2}})\right.\\ +\,\lambda {R}_{\alpha }^{2}({\varrho }_{A| {B}_{N-2}}){R}_{\alpha }^{\gamma -2}({\varrho }_{A| {B}_{N-1}})\\ \left.+\,{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{N-1}})\right\},\end{array}\end{eqnarray}$
where γ ≥ 4, $\frac{\sqrt{7}-1}{2}\,\leqslant \,\alpha \lt 2$ and $\lambda =\frac{\gamma }{2}$(≥2).When ${R}_{\alpha }^{2}({\varrho }_{A| {B}_{i}})\,\geqslant \,\displaystyle {\sum }_{l=i+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})$ for i = 1, 2, ⋯ , z, set ${\theta }_{i}^{{\prime} }=\displaystyle {\sum }_{l=i+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})/{R}_{\alpha }^{2}({\varrho }_{A| {B}_{i}})$. Substituting ${\theta }_{i}^{{\prime} }$ into the inequality (
$\begin{eqnarray}\begin{array}{l}{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{i}\cdots {B}_{N-1}})\geqslant {\left(\displaystyle \sum _{l=i}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)}^{\lambda }\\ \geqslant {R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{i}})+\lambda {R}_{\alpha }^{\gamma -2}({\varrho }_{A| {B}_{i}})\\ \times \displaystyle \sum _{l=i+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\\ +({2}^{\lambda }-\lambda -1){\left(\displaystyle \sum _{l=i+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)}^{\lambda }.\end{array}\end{eqnarray}$
When i = 1, we obtain $\begin{eqnarray}\begin{array}{l}{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}})\\ \geqslant \,{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{1}})+\lambda {R}_{\alpha }^{\gamma -2}({\varrho }_{A| {B}_{1}})\left(\displaystyle \sum _{l=2}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)\\ +\,({2}^{\lambda }-\lambda -1){\left(\displaystyle \sum _{l=2}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)}^{\lambda }\\ \geqslant \,{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{1}})+\lambda {R}_{\alpha }^{\gamma -2}({\varrho }_{A| {B}_{1}})\left(\displaystyle \sum _{l=2}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)\\ +\,({2}^{\lambda }-\lambda -1){R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{2}})\\ +\,({2}^{\lambda }-\lambda -1)\lambda {R}_{\alpha }^{\gamma -2}({\varrho }_{A| {B}_{2}})\left(\displaystyle \sum _{l=3}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)\\ +\,{({2}^{\lambda }-\lambda -1)}^{2}{\left(\displaystyle \sum _{l=3}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)}^{\lambda }\\ \geqslant \,\cdots \\ \geqslant \,\displaystyle \sum _{i=1}^{z}\left\{{({2}^{\lambda }-\lambda -1)}^{i-1}\left[{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{i}})\right.\right.\\ +\,\left.\left.\lambda {R}_{\alpha }^{\gamma -2}({\varrho }_{A| {B}_{i}})\\ \times \,\left(\displaystyle \sum _{l=i+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)\right]\right\}\\ +\,{({2}^{\lambda }-\lambda -1)}^{z}{\left(\displaystyle \sum _{l=z+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)}^{\lambda },\end{array}\end{eqnarray}$
by using the inequality (When ${R}_{\alpha }^{2}({\varrho }_{A| {B}_{j}})\,\leqslant \,\displaystyle {\sum }_{l=j+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})$ for j = z + 1, ⋯ , N − 2, we use a strengthened version of lemma
$\begin{eqnarray}\begin{array}{l}{\left(\displaystyle \sum _{l=j}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)}^{\lambda }\geqslant {\left(\displaystyle \sum _{l=j+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)}^{\lambda }\\ \,+\,({2}^{\lambda }-1){R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{j}}).\end{array}\end{eqnarray}$
When j = z + 1, we get $\begin{eqnarray}\begin{array}{l}{\left(\displaystyle \sum _{l=z+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)}^{\lambda }\geqslant {\left(\displaystyle \sum _{l=z+2}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)}^{\lambda }\\ +\,({2}^{\lambda }-1){R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{z}+1})\\ \geqslant \,{[{R}_{\alpha }^{2}({\varrho }_{A| {B}_{N-2}})+{R}_{\alpha }^{2}({\varrho }_{A| {B}_{N-1}})]}^{\lambda }+({2}^{\lambda }-1)\\ \times \,[{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{z+1}})+\cdots +{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{N-3}})]\\ \geqslant \,{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{N-1}})+\lambda {R}_{\alpha }^{2}({\varrho }_{A| {B}_{N-2}}){R}_{\alpha }^{\gamma -2}({\varrho }_{A| {B}_{N-1}})\\ +\,({2}^{\lambda }-\lambda -1){R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{N-2}})\\ +\,({2}^{\lambda }-1)[{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{z+1}})+\cdots +{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{N-3}})],\end{array}\end{eqnarray}$
where the first and second inequalities hold because of the inequality (In particular, the following monogamy relations hold.
For any N-qubit quantum state ${\varrho }_{A{B}_{1}\cdots {B}_{N-1}}\in {H}_{A}\otimes {H}_{{B}_{1}}\otimes \cdots \otimes {H}_{{B}_{N-1}}$, if ${R}_{\alpha }^{2}({\varrho }_{A| {B}_{i}})\,\geqslant \,\displaystyle {\sum }_{l=i+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})$ for i = 1, 2, ⋯ , N − 2, then
$\begin{eqnarray}\begin{array}{l}{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}})\geqslant \displaystyle \sum _{i=1}^{N-2}\left\{{({2}^{\lambda }-\lambda -1)}^{i-1}\right.\\ \left[{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{i}})+\lambda {R}_{\alpha }^{\gamma -2}({\varrho }_{A| {B}_{i}})\right.\\ \times \,\left.\left.\left(\displaystyle \sum _{l=i+1}^{N-1}{R}_{\alpha }^{2}({\varrho }_{A| {B}_{l}})\right)\right]\right\}\\ +\,{({2}^{\lambda }-\lambda -1)}^{N-2}{R}_{\alpha }^{\gamma }({\varrho }_{A| {B}_{N}-1}),\end{array}\end{eqnarray}$
where γ ≥ 4, $\frac{\sqrt{7}-1}{2}\,\leqslant \,\alpha \lt 2$ and $\lambda =\frac{\gamma }{2}$(≥2).For state (
$\begin{eqnarray}\begin{array}{r}{R}_{\alpha }^{\gamma }({\varrho }_{A| B})+({2}^{\frac{\gamma }{2}}-\frac{\gamma }{2}-1){R}_{\alpha }^{\gamma }({\varrho }_{A| C})\\ +\frac{\gamma }{2}{R}_{\alpha }^{\gamma -2}({\varrho }_{A| B}){R}_{\alpha }({\varrho }_{A| C})\\ =\,{(0.83477)}^{\gamma }+\frac{\gamma }{2}{(0.41466)}^{2}{(0.83477)}^{\gamma -2}\\ +({2}^{\frac{\gamma }{2}}-\frac{\gamma }{2}-1){(0.41466)}^{\gamma },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{R}_{\alpha }^{\gamma }({\varrho }_{A| B})+{R}_{\alpha }^{\gamma }({\varrho }_{A| C})={(0.83477)}^{\gamma }+{(0.41466)}^{\gamma },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{R}_{\alpha }^{\gamma }({\varrho }_{A| B})+({2}^{\gamma }-1){R}_{\alpha }^{\gamma }({\varrho }_{A| C})\\ =\,{(0.83477)}^{\gamma }+({2}^{\gamma }-1){(0.41466)}^{\gamma },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{R}_{\alpha }^{\gamma }({\varrho }_{A| B})+\frac{\gamma }{4}{R}_{\alpha }^{\gamma -2}({\varrho }_{A| B}){R}_{\alpha }({\varrho }_{A| C})\\ +\,({2}^{\frac{\gamma }{2}}-\frac{\gamma }{4}-1){R}_{\alpha }^{\gamma }({\varrho }_{A| C})\\ =\,{(0.83477)}^{\gamma }+\frac{\gamma }{4}{(0.41466)}^{2}{(0.83477)}^{\gamma -2}\\ +\,({2}^{\frac{\gamma }{2}}-\frac{\gamma }{4}-1){(0.41466)}^{\gamma },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{R}_{\alpha }^{\gamma }({\varrho }_{A| B})+\frac{{\gamma }^{2}}{2(\gamma +2)}{R}_{\alpha }^{\gamma -2}({\varrho }_{A| B}){R}_{\alpha }({\varrho }_{A| C})\\ +\,({2}^{\frac{\gamma }{2}}-\frac{{\gamma }^{2}}{2(\gamma +2)}-1){R}_{\alpha }^{\gamma }({\varrho }_{A| C})\\ =\,{(0.83477)}^{\gamma }+\frac{{\gamma }^{2}}{2(\gamma +2)}{(0.41466)}^{2}{(0.83477)}^{\gamma -2}\\ +\,({2}^{\frac{\gamma }{2}}-\frac{{\gamma }^{2}}{2(\gamma +2)}-1){(0.41466)}^{\gamma }.\end{array}\end{eqnarray}$
Equation (
4. Tighter polygamy relations for RαEoA
In this section, we study the polygamy relations of RαEoA with $\frac{\sqrt{7}-1}{2}\,\leqslant \,\alpha \,\leqslant \,\frac{\sqrt{13}-1}{2}$ for N-qubit systems. In [23], the relation (15 ) has been further improved to be
$\begin{eqnarray}\begin{array}{r}{[{R}_{\alpha }^{a}({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}})]}^{t}\leqslant {[{R}_{\alpha }^{a}({\varrho }_{A| {B}_{1}})]}^{t}\\ +\,({2}^{t}-1){[{R}_{\alpha }^{a}({\varrho }_{A| {B}_{2}})]}^{t}+\cdots \\ \,+\,{({2}^{t}-1)}^{z-1}{[{R}_{\alpha }^{a}({\varrho }_{A| {B}_{z}})]}^{t}\\ \,+\,{({2}^{t}-1)}^{z+1}\left({[{R}_{\alpha }^{a}({\varrho }_{A| {B}_{z+1}})]}^{t}\right.\\ \,\left.+\cdots +{[{R}_{\alpha }^{a}({\varrho }_{A| {B}_{N-2}})]}^{t}\right)\\ \,+\,{({2}^{t}-1)}^{z}{[{R}_{\alpha }^{a}({\varrho }_{A| {B}_{N-1}})]}^{t},\end{array}\end{eqnarray}$
for 0 ≤ t ≤ 1, where ${R}_{\alpha }^{a}({\varrho }_{A{B}_{i}})\,\geqslant \,{R}_{\alpha }^{a}({\varrho }_{A| {B}_{i+1}\cdots {B}_{N-1}})$ for i = 1, 2, ⋯ , z, and ${R}_{\alpha }^{a}({\varrho }_{A{B}_{j}})\,\leqslant \,{R}_{\alpha }^{a}({\varrho }_{A| {B}_{j+1}\cdots {B}_{N-1}})$ for j = z + 1, ⋯ , N − 2, ∀1 ≤ z ≤ N − 3, N ≥ 4 and $\frac{\sqrt{7}-1}{2}\,\leqslant \,\alpha \,\leqslant \,\frac{\sqrt{13}-1}{2}$.Next, we illustrate that the above polygamy relations can be tightened under some conditions. We first propose the following lemma.
For any 0 ≤ θ ≤ 1 and 0 ≤ v ≤ 1, then
$\begin{eqnarray}\begin{array}{l}{(1+\theta )}^{v}\leqslant 1+{2}^{v-2}v\theta +({2}^{v}-{2}^{v-2}v-1){\theta }^{v}\\ \leqslant 1+({2}^{v}-1){\theta }^{v}.\end{array}\end{eqnarray}$
If θ = 0, the inequality holds. If θ ≠ 0, the core idea is to reformulate the desired inequality. Observing that equation (
$\begin{eqnarray*}\begin{array}{l}\frac{\partial f(\theta ,v)}{\partial \theta }=\frac{[v{(1+\theta )}^{v-1}-{2}^{v-2}v]{\theta }^{v}-\left[{(1+\theta )}^{v}-{2}^{v-2}v\theta -1\right]v{\theta }^{v-1}}{{\theta }^{2v}}\\ =\,\frac{v{\theta }^{v-1}[1+{2}^{v-2}\theta (v-1)-{(1+\theta )}^{v-1}]}{{\theta }^{2v}}.\end{array}\end{eqnarray*}$
Let g(θ, v) = (1 + θ)v−1 − 1 − 2v−2θ(v − 1). Then $\frac{\partial g(\theta ,v)}{\partial \theta }=(v-1)[{(1+\theta )}^{v-2}-{2}^{v-2}]\,\leqslant \,0$ for 0 ≤ θ ≤ 1 and 0 ≤ v ≤ 1. Hence, g(θ, v) is a decreasing function of θ for fixed v, and g(θ, v) ≤ g(0, v) = 0. Thus, $\frac{\partial f(\theta ,v)}{\partial \theta }\,\geqslant \,0$. f(θ, v) is an increasing function of θ, i.e., f(θ, v) ≤ f(1, v) = 2v − 2v−2v − 1. Then, we have (1 + θ)v ≤ 1 + 2v−2vθ + (2v − 2v−2v − 1)θv. Since 2v−2v(θ − θv) ≤ 0 for 0 ≤ θ ≤ 1 and 0 ≤ v ≤ 1, we get 2v−2vθ + (2v − 2v−2v − 1)θv ≤ 1 + (2v − 1)θv. □For any tripartite quantum state ϱABC, if ${R}_{\alpha }^{a}({\varrho }_{A| B})\,\geqslant \,{R}_{\alpha }^{a}({\varrho }_{A| C})$, we see that
$\begin{eqnarray}\begin{array}{l}{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| BC})\leqslant {({R}_{\alpha }^{a})}^{t}({\varrho }_{A| B})+h{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| C})\\ \quad +\,{2}^{t-2}t[{R}_{\alpha }^{a}({\varrho }_{A| C}){({R}_{\alpha }^{a})}^{t-1}({\varrho }_{A| B})\\ \quad -\,{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| C})],\end{array}\end{eqnarray}$
for 0 ≤ t ≤ 1, where h = 2t − 1.It has been shown that ${R}_{\alpha }^{a}({\varrho }_{A| BC})\,\leqslant \,{R}_{\alpha }^{a}({\varrho }_{A| B})\,+{R}_{\alpha }^{a}({\varrho }_{A| C})$ in (
$\begin{eqnarray}\begin{array}{l}{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| BC})\leqslant {[{R}_{\alpha }^{a}({\varrho }_{A| B})+{R}_{\alpha }^{a}({\varrho }_{A| C})]}^{t}\\ \quad =\,{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| B}){\left(1+\frac{{R}_{\alpha }^{a}({\varrho }_{A| C})}{{R}_{\alpha }^{a}({\varrho }_{A| B})}\right)}^{t}\\ \quad \leqslant \,{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| B})\left[1+{2}^{t-2}t\frac{{R}_{\alpha }^{a}({\varrho }_{A| C})}{{R}_{\alpha }^{a}({\varrho }_{A| B})}\right.\\ \quad \left.+\,({2}^{t}-{2}^{t-2}t-1)\frac{{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| C})}{{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| B})}\right]\\ \quad =\,{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| B})+h{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| C})\\ \quad +\,{2}^{t-2}t[{R}_{\alpha }^{a}({\varrho }_{A| C}){({R}_{\alpha }^{a})}^{t-1}({\varrho }_{A| B})\\ \quad -\,{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| C})],\end{array}\end{eqnarray}$
where the second inequality is due to lemma From lemma
$\begin{eqnarray*}\begin{array}{l}{(1+\theta )}^{v}\leqslant 1+{2}^{v-2}v\theta +({2}^{v}-{2}^{v-2}v-1){\theta }^{v}\\ \leqslant 1+({2}^{v}-1){\theta }^{v}\quad (\,\rm{The inequality in [25]}\,).\end{array}\end{eqnarray*}$
Since θ − θλ ≤ 0 for 0 ≤ θ ≤ 1 and 0 ≤ v ≤ 1, obviously our inequality given in theorem The following theorem gives tighter upper bounds of ${({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}})$ for 0 ≤ t ≤ 1.
For any N-qubit mixed state ${\varrho }_{A{B}_{1}\cdots {B}_{N-1}}\in {H}_{A}\otimes {H}_{{B}_{1}}\otimes \cdots \otimes {H}_{{B}_{N-1}}$, if ${R}_{\alpha }^{a}({\varrho }_{A| {B}_{i}})\,\geqslant \,\displaystyle {\sum }_{k=i+1}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})$ for i = 1, 2, ⋯ , N − 2, we obtain
$\begin{eqnarray}\begin{array}{rcl}{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}}) & \leqslant & \displaystyle \sum _{i=1}^{N-2}{h}^{i-1}[{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{i}})+{N}_{A{B}_{i}}]\\ & & +{h}^{N-2}{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{N-1}}),\end{array}\end{eqnarray}$
for 0 ≤ t ≤ 1, where h = 2t − 1 and ${N}_{A{B}_{i}}$ = ${2}^{t-2}t\left(\displaystyle {\sum }_{k=i+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{k}})\right.$ ${({R}_{\alpha }^{a})}^{t-1}({\varrho }_{A| {B}_{i}})$ $-\left.{[\displaystyle {\sum }_{k=i+1}^{N-1}{R}_{\alpha }({\varrho }_{A| {B}_{k}})]}^{t}\right)$.According to (
$\begin{eqnarray}\begin{array}{l}{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}})\leqslant [{R}_{\alpha }^{a}({\varrho }_{A| {B}_{1}})\\ +\,{R}_{\alpha }^{a}({\varrho }_{A| {B}_{2}})+\cdots +{R}_{\alpha }^{a}({\varrho }_{A| {B}_{N-1}}){]}^{t}\\ \leqslant \,{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{1}})+h{\left[\displaystyle \sum _{k=2}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right]}^{t}+{2}^{t-2}t\left(\displaystyle \sum _{k=2}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right.\\ \times \left.{({R}_{\alpha }^{a})}^{t-1}({\varrho }_{A| {B}_{1}})-{\left[\displaystyle \sum _{k=2}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right]}^{t}\right)\\ \leqslant \,{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{1}})+h\left[{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{2}})+h{\left[\displaystyle \sum _{k=3}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right]}^{t}\right.\\ +\,{2}^{t-2}t\left(\displaystyle \sum _{k=3}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right.\\ \times \,\left.\left.{({R}_{\alpha }^{a})}^{t-1}({\varrho }_{A| {B}_{2}})-{\left[\displaystyle \sum _{k=3}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right]}^{t}\right)\right]\\ +\,{2}^{t-2}t\left(\displaystyle \sum _{k=2}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right.\\ \times \,\left.{({R}_{\alpha }^{a})}^{t-1}({\varrho }_{A| {B}_{1}})-{\left[\displaystyle \sum _{k=2}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right]}^{t}\right)\\ \leqslant \,\cdots \\ \leqslant \,{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{1}})+h{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{2}})+\cdots +{h}^{N-2}{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{N-1}})\\ +\,{h}^{N-3}{2}^{t-2}t\left({R}_{\alpha }^{a}({\varrho }_{A| {B}_{N-1}})\right.\\ \times \,\left.{({R}_{\alpha }^{a})}^{t-1}({\varrho }_{A| {B}_{N-2}})-{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{N-1}})\right)\\ +\,\cdots \\ +\,h{2}^{t-2}t\left(\displaystyle \sum _{k=3}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right.\\ \times \,\left.{({R}_{\alpha }^{a})}^{t-1}({\varrho }_{A| {B}_{2}})-{\left[\displaystyle \sum _{k=3}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right]}^{t}\right)\\ +\,{2}^{t-2}t\left(\displaystyle \sum _{k=2}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right.\\ \times \,\left.{({R}_{\alpha }^{a})}^{t-1}({\varrho }_{A| {B}_{1}})-{\left[\displaystyle \sum _{k=2}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right]}^{t}\right),\end{array}\end{eqnarray}$
where the second inequality is due to lemma Typically, the conditions for inequalities (15 ) are not fulfilled in every case. As a consequence, we propose a general polygamy inequality.
For any N-qubit mixed state ${\varrho }_{A{B}_{1}\cdots {B}_{N-1}}\in {H}_{A}\otimes {H}_{{B}_{1}}\otimes \cdots \otimes {H}_{{B}_{N-1}}$, if ${R}_{\alpha }^{a}({\varrho }_{A| {B}_{i}})\,\geqslant \,\displaystyle {\sum }_{k=i+1}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})$ for i = 1, 2, ⋯ , z, and ${R}_{\alpha }^{a}({\varrho }_{A| {B}_{j}})\,\leqslant \,\displaystyle {\sum }_{k=j+1}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})$ for j = z + 1, ⋯ , N − 2, ∀ 1 ≤ z ≤ N − 3 and N ≥ 4, then
$\begin{eqnarray}\begin{array}{l}{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}})\\ \leqslant \,\displaystyle \sum _{i=1}^{z}{h}^{i-1}[{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{i}})+{N}_{A{B}_{i}}]\\ +\,{h}^{z}\displaystyle \sum _{j=z+1}^{N-2}[h{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{j}})+{\overline{N}}_{A{B}_{j}}]\\ +\,{h}^{z}{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{N-1}}),\end{array}\end{eqnarray}$
for 0 ≤ t ≤ 1, where h = 2t − 1, ${N}_{A{B}_{i}}$ = ${2}^{t-2}t\left(\displaystyle {\sum }_{k=i+1}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right.$ $\times {({R}_{\alpha }^{a})}^{t-1}({\varrho }_{A| {B}_{i}})$ $-\left.{[\displaystyle {\sum }_{k=i+1}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})]}^{t}\right)$ and ${\overline{N}}_{A{B}_{j}}$ = ${2}^{t-2}t\left({R}_{\alpha }^{a}({\varrho }_{A| {B}_{j}})\right.$ ${[\displaystyle {\sum }_{k=j+1}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})]}^{t-1}$ $-\left.{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{j}})\right)$.From the proof of theorem
$\begin{eqnarray}\begin{array}{l}{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{1}\cdots {B}_{N-1}})\leqslant \displaystyle \sum _{i=1}^{z}{h}^{i-1}[{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{i}})+{N}_{A{B}_{i}}]\\ \,+\,{h}^{z}{\left[\displaystyle \sum _{k=z+1}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right]}^{t}.\end{array}\end{eqnarray}$
In addition, since ${R}_{\alpha }^{a}({\varrho }_{A| {B}_{j}})\,\leqslant \,\displaystyle \sum _{k=j+1}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})$ for j = z + 1, ⋯ , N − 2, we obtain
$\begin{eqnarray}\begin{array}{l}{\left[\displaystyle \sum _{k=z+1}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right]}^{t}\\ \leqslant {\left[\displaystyle \sum _{k=z+2}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right]}^{t}+h{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{z+1}})\\ +{2}^{t-2}t\left({R}_{\alpha }^{a}({\varrho }_{A| {B}_{z+1}})\right.\\ \times \,\left.{\left[\displaystyle \sum _{k=z+2}^{N-1}{R}_{\alpha }^{a}({\varrho }_{A| {B}_{k}})\right]}^{t-1}-{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{z+1}})\right)\\ \leqslant \cdots \\ \leqslant \displaystyle \sum _{j=z+1}^{N-2}[h{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{j}})+{\overline{N}}_{A{B}_{j}}]\\ +{({R}_{\alpha }^{a})}^{t}({\varrho }_{A| {B}_{N-1}}).\end{array}\end{eqnarray}$
By the denotations of ${N}_{A{B}_{j}}$ and ${\overline{N}}_{A{B}_{j}}$, from the inequalities (
Consider the following example to demonstrate the tightness of our polygamy relations of multi-qubit entanglement.
Consider a three-qubit generalised W-class state,
$\begin{eqnarray}| \psi {\rangle }_{A| BC}=\frac{1}{\sqrt{6}}| 100\rangle +\frac{1}{\sqrt{6}}| 010\rangle +\frac{2}{\sqrt{6}}| 001\rangle .\end{eqnarray}$
We get ${R}_{\frac{\sqrt{7}-1}{2}}({\psi }_{A| BC})=0.698944$, ${R}_{\frac{\sqrt{7}-1}{2}}({\varrho }_{A| B})\,=0.241379$ and ${R}_{\frac{\sqrt{7}-1}{2}}({\varrho }_{A| C})=0.607261$. Thus one gets $\begin{eqnarray}\begin{array}{r}{[{R}_{\alpha }^{a}]}^{t}({\varrho }_{A| B})+({2}^{t}-{2}^{t-2}t-1){[{R}_{\alpha }^{a}]}^{t}({\varrho }_{A| C})\\ +\,{2}^{t-2}t{[{R}_{\alpha }^{a}]}^{t-1}({\varrho }_{A| B}){R}_{\alpha }^{a}({\varrho }_{A| C})\\ =\,{(0.241379)}^{t}+({2}^{t}-{2}^{t-2}t-1){(0.607261)}^{t}\\ +\,{2}^{t-2}t{(0.241379)}^{t-1}(0.607261),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{[{R}_{\alpha }^{a}]}^{t}({\varrho }_{AB})+{[{R}_{\alpha }^{a}]}^{t}({\varrho }_{AC})\\ =\,{(0.241379)}^{t}+{(0.607261)}^{t},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{[{R}_{\alpha }^{a}]}^{t}({\varrho }_{AB})+({2}^{t}-1){[{R}_{\alpha }^{a}]}^{t}({\varrho }_{AC})\\ =\,{(0.241379)}^{t}+({2}^{t}-1){(0.607261)}^{t}.\end{array}\end{eqnarray}$
Equation (
Figure 3. The y axis is the upper bound of ${R}_{\alpha }^{a}({\psi }_{A| BC})$. Solid black line denotes ${R}_{\alpha }^{a}({\psi }_{A| BC})$ for the state ( |
5. Residual entanglement of RαE
For a 2 ⨂ 2 ⨂ d quantum pure state ∣ψ⟩ABC, the three tangle is expressed as [40],
$\begin{eqnarray}\tau (| \psi {\rangle }_{ABC})={{\mathscr{C}}}^{2}(| \psi {\rangle }_{A| BC})-{{\mathscr{C}}}^{2}({\varrho }_{A| B})-{{\mathscr{C}}}^{2}({\varrho }_{A| C}),\end{eqnarray}$
where ${\varrho }_{AB(AC)}={{\rm{Tr}}}_{C(B)}(| \psi {\rangle }_{ABC}\langle \psi | )$. Additionally, the three-tangle can also be characterized by the concurrence of assistance, an important entanglement measure that quantifies the maximum average concurrence shared by two parties with the assistance of the third party [40]: $\begin{eqnarray}{{\mathscr{C}}}_{a}^{2}({\varrho }_{A| B})={{\mathscr{C}}}^{2}({\varrho }_{A| B})+\tau (| \psi {\rangle }_{ABC}).\end{eqnarray}$
For a mixed state ϱABC, the three tangle is expressed as $\begin{eqnarray}\tau ({\varrho }_{ABC})=\min \displaystyle \sum _{i}{p}_{i}\sqrt{\tau (| {\psi }^{i}{\rangle }_{ABC})},\end{eqnarray}$
where the minimum is taken over all possible pure state decompositions {pi, ∣ψi⟩AB} of ϱABC.Similar to the three tangle determined by concurrence, for the three-qubit state ∣ψ⟩ABC ∈ HA ⨂ HB ⨂ HC, we define the residual entanglement given by the μth (μ ≥ 1) power of RαE by using the relation (12 ),
$\begin{eqnarray}{\tau }_{\mu }^{R}(| \psi {\rangle }_{ABC})={R}_{\alpha }^{\mu }(| \psi {\rangle }_{A| BC})-{R}_{\alpha }^{\mu }({\varrho }_{A| B})-{R}_{\alpha }^{\mu }({\varrho }_{A| C}),\end{eqnarray}$
where α ≥ 2. We have the following relations among the residual entanglement of RαE, RαEoA and three tangle.For a three-qubit pure state ∣ψ⟩ABC, we see that
$\begin{eqnarray}{\tau }_{\mu }^{R}(| \psi {\rangle }_{ABC})\geqslant {f}_{\alpha }^{\mu }\left(\sqrt{\tau (| \psi {\rangle }_{ABC}})\right),\end{eqnarray}$
and $\begin{eqnarray}{({R}_{\alpha }^{a})}^{\mu }({\varrho }_{A| B})\geqslant {R}_{\alpha }^{\mu }({\varrho }_{A| B})+{f}_{\alpha }^{\mu }\left(\sqrt{\tau (| \psi {\rangle }_{ABC}})\right),\end{eqnarray}$
where α ≥ 2, μ ≥ 1, fα is given in (For α ≥ 2 and μ ≥ 1 we get
$\begin{eqnarray}\begin{array}{rcl}{f}_{\alpha }^{\mu }(\sqrt{{\xi }_{1}^{2}+{\xi }_{2}^{2}}) & = & {({f}_{\alpha }(\sqrt{{\xi }_{1}^{2}+{\xi }_{2}^{2}}))}^{\mu }\\ & \geqslant & {({f}_{\alpha }({\xi }_{1})+{f}_{\alpha }({\xi }_{2}))}^{\mu }\\ & \geqslant & {f}_{\alpha }^{\mu }({\xi }_{1})+{f}_{\alpha }^{\mu }({\xi }_{2}),\end{array}\end{eqnarray}$
where the first inequality is due to the inequality (Based on the definition of ${\tau }_{\mu }^{R}(| \psi {\rangle }_{ABC})$, we have
$\begin{eqnarray*}\begin{array}{l}{\tau }_{\mu }^{R}(| \psi {\rangle }_{ABC})={R}_{\alpha }^{\mu }(| \psi {\rangle }_{A| BC})-{R}_{\alpha }^{\mu }({\varrho }_{A| B})-{R}_{\alpha }^{\mu }({\varrho }_{A| C})\\ \,=\,{f}_{\alpha }^{\mu }({\mathscr{C}}(| \psi {\rangle }_{A| BC}))-{f}_{\alpha }^{\mu }({\mathscr{C}}({\varrho }_{A| B}))-{f}_{\alpha }^{\mu }({\mathscr{C}}({\varrho }_{A| C}))\\ \,=\,{f}_{\alpha }^{\mu }\left(\sqrt{{{\mathscr{C}}}^{2}({\varrho }_{A| B})+{{\mathscr{C}}}^{2}({\varrho }_{A| C})+\tau (| \psi {\rangle }_{A| BC})}\right)\\ \quad -\,{f}_{\alpha }^{\mu }({\mathscr{C}}({\varrho }_{A| B}))-{f}_{\alpha }^{\mu }({\mathscr{C}}({\varrho }_{A| C}))\\ \,\geqslant \,{f}_{\alpha }^{\mu }\left(\sqrt{\tau (| \psi {\rangle }_{ABC}})\right),\end{array}\end{eqnarray*}$
where the third equality is obtained from the equation (According to equation (
$\begin{eqnarray*}\begin{array}{l}\,{({R}_{\alpha }^{a})}^{\mu }({\varrho }_{A| B})\,\geqslant \,{f}_{\alpha }^{\mu }({{\mathscr{C}}}_{a}({\varrho }_{A| B}))\\ \,=\,{f}_{\alpha }^{\mu }(\sqrt{{{\mathscr{C}}}^{2}({\varrho }_{A| B})+\tau (| \psi {\rangle }_{ABC})})\\ \,\geqslant \,{f}_{\alpha }^{\mu }({\mathscr{C}}({\varrho }_{A| B}))+{f}_{\alpha }^{\mu }\left(\sqrt{\tau (| \psi {\rangle }_{ABC}})\right)\\ \,=\,{R}_{\alpha }^{\mu }({\varrho }_{A| B})+{f}_{\alpha }^{\mu }\left(\sqrt{\tau (| \psi {\rangle }_{ABC}})\right),\end{array}\end{eqnarray*}$
where the first inequality is obtained from the relation ${R}_{\alpha }^{a}({\varrho }_{AB})\,\geqslant \,{f}_{\alpha }({{\mathscr{C}}}_{a}({\varrho }_{AB}))$ with $\alpha \,\geqslant \,\frac{\sqrt{7}-1}{2}$ in [28], the second inequality is derived from the inequality (The relations among RαE, RαEoA and the three tangle given in theorem 10 can be utilized to derive a lower bound of RαEoA. Let us consider the following example.
For the superpositions of the Greenberger–Horne–Zeilinger-state and the W-state,
$\begin{eqnarray}\begin{array}{r}| {\rm{\Psi }}{\rangle }_{ABC}=\sqrt{\frac{1}{2}}| {\rm{G}}{\rm{H}}{\rm{Z}}\rangle -\sqrt{\frac{1}{2}}| W\rangle ,\end{array}\end{eqnarray}$
where $| {\rm{G}}{\rm{H}}{\rm{Z}}\rangle =\frac{1}{\sqrt{2}}(| 000\rangle +| 111\rangle )$ and $| W\rangle =\frac{1}{\sqrt{3}}(| 100\rangle \,+| 010\rangle +| 001\rangle )$, according to theorem 
Figure 4. The purple surface represents the lower bound of ${R}_{\alpha }^{a}({\varrho }_{AB})$ for the state ( |
Figure 5. The green line represents the lower bound of ${R}_{\alpha }^{a}({\varrho }_{AB})$ for the state ( |
6. Conclusion
Entanglement monogamy and polygamy relations are two intrinsic characteristics of multipartite entangled states. Our results refine the characterization of multi-qubit entanglement shareability and distribution through RαE. We have provided classes of tighter monogamy relations in terms of the η-th (η ≥ 2) power of RαE for α ≥ 2, and the γ-th (γ ≥ 4) power of RαE for $\frac{\sqrt{7-1}}{2}\,\leqslant \,\alpha \lt 2$, respectively. Moreover, we have presented polygamy relations satisfied by the tth power of RαEoA with $\frac{\sqrt{7}-1}{2}\,\leqslant \,\alpha \lt \frac{\sqrt{13}-1}{2}$ for 0 ≤ t ≤ 1. We have also established that these new inequalities impose finer constraints than those of the existing ones. In addition, we have provided relationships among the residual entanglements of RαE and RαEoA, and the three tangle based on our general monogamy relations. Our approach may be also applied to the investigation on polygamy and monogamy relations in terms of other entanglement measures [12, 41, 42].