1. Introduction
Entanglement is one of the most important features of quantum mechanics, which distinguishes quantum mechanics from classical theory. A key property of entanglement is known as monogamy relations [1, 2], that is, entanglement cannot be freely shared unconditionally among the multipartite quantum systems. Monogamy relation provides a way to characterize multipartite entanglement sharing and distribution. The first mathematical characterization of monogamy relation was expressed as a form of inequality for three-qubit state in terms of squared concurrence [1]. Furthermore, Osborne and Verstraete generalized this monogamy inequality to arbitrary multiqubit systems [3]. Later, the monogamy inequality was also generalized to other entanglement measures [4–10]. In fact, monogamy of entanglement is fundamentally important in the context of quantum cryptography since it restricts on the amount of information that an eavesdropper could potentially obtain about the secret key extraction. Moreover, monogamy of entanglement also has many important applications in quantum information theory [11], condensed-matter physics [12] and even black-hole physics [13].
As a generalization of entanglement of formation, the Rényi-α entanglement [14] is a well-defined entanglement measure and it has been widely used in the study of quantum information theory [15–21]. Recently it has been shown that if $\alpha \geqslant \tfrac{\sqrt{7}-1}{2}$, the squared Rényi-α entanglement satisfies the monogamy relation in N-qubit systems [9]. It has also been shown that when α ≥ 2, the Rényi-α entanglement obeys the monogamy relation in multiqubit systems [22]. In general, tightening the monogamy relations can provide a precise characterization of the entanglement in multipartite systems. In particular, the monogamy relations are saturation for W-class states and this implies that this type of multipartite entanglement can be completely characterized [23, 24]. Furthermore, a class of tight monogamy relations was derived by raising the power of the entanglement measures [25–29]. In this paper, we focus on tightening the monogamy relations in terms of Rényi-α entanglement by raising the power of the Rényi-α entanglement for multiqubit systems. It is shown that these new monogamy relations are tighter than the results in [9, 28, 29].
2. The Rényi-α entanglement
The Rényi-α entanglement of a bipartite pure state $| \psi {\rangle }_{{AB}}$, is defined as [14]
$ \begin{eqnarray}{E}_{\alpha }(| \psi {\rangle }_{{AB}})=\displaystyle \frac{1}{1-\alpha }{\mathrm{log}}_{2}(\mathrm{tr}{\rho }_{A}^{\alpha }),\end{eqnarray}$
for any α > 0 and $\alpha \ne 1,{\rho }_{A}={\mathrm{tr}}_{B}(| \psi {\rangle }_{{AB}}\langle \psi | )$. If α tends to 1, the Rényi-α entanglement converges to the von Neumann entropy. For a bipartite mixed state ρAB, the Rényi-α entanglement is defined via the convex-roof extension $ \begin{eqnarray}{E}_{\alpha }({\rho }_{{AB}})=\min \sum _{i}{p}_{i}{E}_{\alpha }(| {\psi }_{i}{\rangle }_{{AB}}),\end{eqnarray}$
where the minimum is taken over all possible pure-state decompositions of ${\rho }_{{AB}}={\sum }_{i}{p}_{i}| {\psi }_{i}{\rangle }_{{AB}}\langle {\psi }_{i}| $.Let us recall the definition of concurrence. For a bipartite pure state $| \phi {\rangle }_{{AB}}$, the concurrence is [30]
$ \begin{eqnarray}C(| \phi {\rangle }_{{AB}})=\sqrt{2(1-\mathrm{tr}{\rho }_{A}^{2})},\end{eqnarray}$
where ${\rho }_{A}={\mathrm{tr}}_{B}(| \phi {\rangle }_{{AB}}\langle \phi | )$. For a mixed state ρAB, the concurrence is defined via the convex-roof extension $ \begin{eqnarray}C({\rho }_{{AB}})=\min \sum _{j}{p}_{j}C(| {\phi }_{j}{\rangle }_{{AB}}),\end{eqnarray}$
where the minimum is taken over all possible pure-state decompositions of ${\rho }_{{AB}}={\sum }_{j}{p}_{j}| {\phi }_{j}{\rangle }_{{AB}}\langle {\phi }_{j}| $.For an arbitrary N-qubit state ${\rho }_{{{AB}}_{1}\cdots {B}_{N-1}}\in {{ \mathcal H }}_{A}\,\otimes {{ \mathcal H }}_{{B}_{1}}\,\otimes \cdots \otimes \,{{ \mathcal H }}_{{B}_{N-1}},{\rho }_{A| {B}_{1}\cdots {B}_{N-1}}$ denote the state ${\rho }_{{{AB}}_{1}\cdots {B}_{N-1}}$ viewed as a bipartite state with partitions A and ${B}_{1}{B}_{2}\cdots {B}_{N-1}$. The concurrence $C({\rho }_{A| {B}_{1}\cdots {B}_{N-1}})$ satisfies [3]
$ \begin{eqnarray}{C}^{2}({\rho }_{A| {B}_{1}\cdots {B}_{N-1}})-{C}^{2}({\rho }_{{{AB}}_{1}})-\cdots -{C}^{2}({\rho }_{{{AB}}_{N-1}})\geqslant 0,\end{eqnarray}$
where ${\rho }_{{{AB}}_{i}}={\mathrm{tr}}_{{B}_{1}\cdots {B}_{i-1}{B}_{i+1}\cdots {B}_{N-1}}({\rho }_{A| {B}_{1}\cdots {B}_{N-1}})$, ${{ \mathcal H }}_{A},{{ \mathcal H }}_{{B}_{1}},\cdots ,{{ \mathcal H }}_{{B}_{N-1}}$ are Hilbert spaces of the systems $A,{B}_{1},\cdots ,\,{B}_{N-1}$, respectively.It has been proved that [20, 22], when $\alpha \geqslant \tfrac{\sqrt{7}-1}{2}$, for a two-qubit state, the Rényi-α entanglement has an analytical formula
$ \begin{eqnarray}{E}_{\alpha }({\rho }_{{AB}})={g}_{\alpha }(C({\rho }_{{AB}})),\end{eqnarray}$
here the function gα(x) is a monotonically increasing and convex function expressed as $ \begin{eqnarray}\begin{array}{rcl}{g}_{\alpha }(x) & = & \displaystyle \frac{1}{1-\alpha }{\mathrm{log}}_{2}\left[{\left(\displaystyle \frac{1-\sqrt{1-{x}^{2}}}{2}\right)}^{\alpha }\right.\\ & & \left.+{\left(\displaystyle \frac{1+\sqrt{1-{x}^{2}}}{2}\right)}^{\alpha }\right],\end{array}\end{eqnarray}$
in 0 ≤ x ≤ 1.The function gα(x) in equation (7 ) for α ≥ 2, has one important property such that [22]
$ \begin{eqnarray}{g}_{\alpha }\left(\sqrt{{x}^{2}+{y}^{2}}\right)\geqslant {g}_{\alpha }(x)+{g}_{\alpha }(y),\end{eqnarray}$
for $0\leqslant x,y,{x}^{2}+{y}^{2}\leqslant 1$.When $\alpha \geqslant \tfrac{\sqrt{7}-1}{2}$, it is easy to see in [9] that the function gα(x) satisfies the following inequality
$ \begin{eqnarray}{\left[{g}_{\alpha }\left(\sqrt{{x}^{2}+{y}^{2}}\right)\right]}^{2}\geqslant {\left[{g}_{\alpha }(x)\right]}^{2}+{\left[{g}_{\alpha }(y)\right]}^{2},\end{eqnarray}$
for $0\leqslant x,y,{x}^{2}+{y}^{2}\leqslant 1$.3. Tighter monogamy relations for Rényi-α entanglement
In the following, we establish a class of tight monogamy relations of Rényi-α entanglement related to the power η. We first provide the following lemma.
For $x\in [0,1]$ and $t\geqslant 1$, then
$ \begin{eqnarray}{\left(1+x\right)}^{t}\geqslant 1+\displaystyle \frac{t}{2}x+\left({2}^{t}-\displaystyle \frac{t}{2}-1\right){x}^{t}\geqslant 1+({2}^{t}-1){x}^{t}.\end{eqnarray}$
Note that the inequality (
Since $x\geqslant {x}^{t}$, for $x\in [0,1]$ and $t\geqslant 1$, one gets $1\,+\tfrac{t}{2}x\,+\,\left({2}^{t}\,-\,\tfrac{t}{2}\,-\,1\right){x}^{t}\,=\,1\,+\,\tfrac{t}{2}(x\,-\,{x}^{t})\,+\,({2}^{t}\,-\,1){x}^{t}\,\geqslant 1+({2}^{t}-1){x}^{t}.$ Altogether, we can get ${\left(1+x\right)}^{t}\geqslant 1+\tfrac{t}{2}x\,+\left({2}^{t}-\tfrac{t}{2}-1\right){x}^{t}\geqslant 1+({2}^{t}-1){x}^{t}$.
Now we provide our main results of this paper.
For an $N$-qubit state ${\rho }_{{{AB}}_{1}\cdots {B}_{N-1}}\in {{ \mathcal H }}_{A}\,\otimes {{ \mathcal H }}_{{B}_{1}}\,\otimes \cdots \otimes \,{{ \mathcal H }}_{{B}_{N-1}}$, if $C({\rho }_{{{AB}}_{i}})\geqslant C({\rho }_{A| {B}_{i+1}\cdots {B}_{N-1}})$ for $i=1,2,\cdots ,N-2,N\geqslant 3$, then
$ \begin{eqnarray}\begin{array}{l}{E}_{\alpha }^{\eta }({\rho }_{A| {B}_{1}\cdots {B}_{N-1}})\geqslant {E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{1}})+\left({2}^{\eta }-1\right){E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{2}})\\ \quad +\,\cdots +\,{\left({2}^{\eta }-1\right)}^{N-4}{E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{N-3}})\\ \quad +\,{\left({2}^{\eta }-1\right)}^{N-3}\left.\Space{0ex}{3.0ex}{0ex}\{{E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{N-2}})\\ \quad +\,\displaystyle \frac{\eta }{2}{E}_{\alpha }^{\eta -1}({\rho }_{{{AB}}_{N-2}}){E}_{\alpha }({\rho }_{{{AB}}_{N-1}})\\ \quad +\,\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{N-1}})\right\},\end{array}\end{eqnarray}$
for $\alpha \geqslant 2$ and the power $\eta \geqslant 1;$ and $ \begin{eqnarray}\begin{array}{l}{E}_{\alpha }^{\gamma }({\rho }_{A| {B}_{1}\cdots {B}_{N-1}})\geqslant {E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{1}})+({2}^{t}-1){E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{2}})+\cdots \\ +\,{\left({2}^{t}-1\right)}^{N-4}{E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{N-3}})+{\left({2}^{t}-1\right)}^{N-3}\left\{\Space{0ex}{3.0ex}{0ex}{E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{N-2}})\right.\\ +\,\displaystyle \frac{t}{2}{E}_{\alpha }^{\gamma -2}({\rho }_{{{AB}}_{N-2}}){E}_{\alpha }^{2}({\rho }_{{{AB}}_{N-1}})\\ \left.+\,\left({2}^{t}-\displaystyle \frac{t}{2}-1\right){E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{N-1}})\right\},\end{array}\end{eqnarray}$
for $2\gt \alpha \geqslant \tfrac{\sqrt{7}-1}{2}$ and the power $\gamma \geqslant 2$, where $t=\tfrac{\gamma }{2}$. For α ≥ 2, by the inequality (
$ \begin{eqnarray}{\left[{g}_{\alpha }\left(\sqrt{{x}^{2}+{y}^{2}}\right)\right]}^{\eta }\geqslant {\left[{g}_{\alpha }(x)+{g}_{\alpha }(y)\right]}^{\eta }.\end{eqnarray}$
Without loss of generality, we assume x ≥ y, the inequality (10 ) of lemma 1 ensures
$ \begin{eqnarray}\begin{array}{l}{\left[{g}_{\alpha }\left(\sqrt{{x}^{2}+{y}^{2}}\right)\right]}^{\eta }\geqslant {\left[{g}_{\alpha }(x)\right]}^{\eta }+\displaystyle \frac{\eta }{2}{\left[{g}_{\alpha }(x)\right]}^{\eta -1}{g}_{\alpha }(y)\\ \quad +\,\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){\left[{g}_{\alpha }(y)\right]}^{\eta }.\end{array}\end{eqnarray}$
Let us first consider an N-qubit pure state $| {\rm{\Psi }}{\rangle }_{A| {B}_{1}\cdots {B}_{N-1}}$. The entanglement ${E}_{\alpha }(| {\rm{\Psi }}{\rangle }_{A| {B}_{1}\cdots {B}_{N-1}})$ and $C(| {\rm{\Psi }}{\rangle }_{A| {B}_{1}\cdots {B}_{N-1}})$ are related by the function gα(x) in equation (7 ) since the subsystem ${B}_{1}\cdots {B}_{N-1}$ can be regarded as a logic qubit. Thus, we can obtain5 ) and the monotonically increasing property of the function gα(x) to obtain the first inequality, the second inequality is due to inequality (14 ) by letting $x=C({\rho }_{{{AB}}_{1}})$ and $y\,=\sqrt{{C}^{2}({\rho }_{{{AB}}_{2}})\,+\cdots +\,{C}^{2}({\rho }_{{{AB}}_{N-1}})}$. The third inequality is obtained from the iterative use of inequality (14 ). Here we are using the fact that $C({\rho }_{{{AB}}_{i}})\geqslant C({\rho }_{A| {B}_{i+1}\cdots {B}_{N-1}})\geqslant \sqrt{{C}^{2}({\rho }_{{{AB}}_{i+1}})\,+\cdots +\,{C}^{2}({\rho }_{{{AB}}_{N-1}})}$, $i=1,2,\cdots ,N-2$ and $1+\tfrac{\eta }{2}x+\left({2}^{\eta }-\tfrac{\eta }{2}-1\right){x}^{\eta }\,\geqslant 1\,+\left({2}^{\eta }-1\right){x}^{\eta }$ for $\eta \geqslant 1$. Since for any two-qubit state ${\rho }_{{AB}}$, when $\alpha \geqslant \tfrac{\sqrt{7}-1}{2}$, ${E}_{\alpha }({\rho }_{{AB}})={g}_{\alpha }(C({\rho }_{{AB}}))$, we obtain the last equality.
$ \begin{eqnarray}\begin{array}{l}{E}_{\alpha }^{\eta }(| {\rm{\Psi }}{\rangle }_{A| {B}_{1}\cdots {B}_{N-1}})\\ \quad =\,{[{g}_{\alpha }(C(| {\rm{\Psi }}{\rangle }_{A| {B}_{1}\cdots {B}_{N-1}}))]}^{\eta }\\ \quad \geqslant \,{\left[{g}_{\alpha }\left(\sqrt{{C}^{2}({\rho }_{{{AB}}_{1}})+\cdots +{C}^{2}({\rho }_{{{AB}}_{N-1}})}\right)\right]}^{\eta }\\ \quad \geqslant \,{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{1}}))\right]}^{\eta }\\ \qquad +\,\displaystyle \frac{\eta }{2}{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{1}}))\right]}^{\eta -1}\\ \qquad \times \,{g}_{\alpha }\left(\sqrt{{C}^{2}({\rho }_{{{AB}}_{2}})\,+\cdots +\,{C}^{2}({\rho }_{{{AB}}_{N-1}})}\right)\\ \qquad +\,\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){\left[{g}_{\alpha }\left(\sqrt{{C}^{2}({\rho }_{{{AB}}_{2}})+\cdots +{C}^{2}({\rho }_{{{AB}}_{N-1}})}\right)\right]}^{\eta }\\ \quad \geqslant \,{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{1}}))\right]}^{\eta }+\left({2}^{\eta }-1\right){\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{2}}))\right]}^{\eta }+\cdots \\ \qquad +\,{\left({2}^{\eta }-1\right)}^{N-4}{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{N-3}}))\right]}^{\eta }\\ \qquad +\,{\left({2}^{\eta }-1\right)}^{N-3}\{{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{N-2}}))\right]}^{\eta }\\ \qquad +\,\displaystyle \frac{\eta }{2}{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{N-2}}))\right]}^{\eta -1}{g}_{\alpha }(C({\rho }_{{{AB}}_{N-1}}))\\ \qquad +\,\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{N-1}}))\right]}^{\eta }\}\\ \quad =\,{E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{1}})+\left({2}^{\eta }-1\right){E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{2}})\\ \qquad +\,\cdots +{\left({2}^{\eta }-1\right)}^{N-4}{E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{N-3}})\\ \qquad +\,{\left({2}^{\eta }-1\right)}^{N-3}\left\{{E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{N-2}})\right.\\ \qquad +\,\displaystyle \frac{\eta }{2}{E}_{\alpha }^{\eta -1}({\rho }_{{{AB}}_{N-2}}){E}_{\alpha }({\rho }_{{{AB}}_{N-1}})\\ \qquad \left.+\,\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{N-1}})\right\},\end{array}\end{eqnarray}$
where we have utilized the monogamy inequality (Let us now consider an N-qubit mixed state ${\rho }_{A| {B}_{1}\cdots {B}_{N-1}}$. Assume that ${\rho }_{A| {B}_{1}\cdots {B}_{N-1}}={\sum }_{k}{p}_{k}| {\varphi }_{k}{\rangle }_{A| {B}_{1}\cdots {B}_{N-1}}\langle {\varphi }_{k}| \in {{ \mathcal H }}_{A}\,\otimes {{ \mathcal H }}_{{B}_{1}}\otimes \cdots \otimes {{ \mathcal H }}_{{B}_{N-1}}$ is the optimal pure-state decomposition for ${E}_{\alpha }({\rho }_{A| {B}_{1}\cdots {B}_{N-1}})$. Thus, we can deduce
$ \begin{eqnarray}\begin{array}{rcl}{E}_{\alpha }({\rho }_{A| {B}_{1}\cdots {B}_{N-1}}) & = & \sum _{k}{p}_{k}{E}_{\alpha }(| {\varphi }_{k}{\rangle }_{A| {B}_{1}\cdots {B}_{N-1}})\\ & = & \sum _{k}{p}_{k}{g}_{\alpha }(C(| {\varphi }_{k}{\rangle }_{A| {B}_{1}\cdots {B}_{N-1}}))\\ & \geqslant & {g}_{\alpha }\left(\sum _{k}{p}_{k}C(| {\varphi }_{k}{\rangle }_{A| {B}_{1}\cdots {B}_{N-1}}\right)\\ & \geqslant & {g}_{\alpha }\left(\sum _{l}{p}_{l}C(| {\chi }_{l}{\rangle }_{A| {B}_{1}\cdots {B}_{N-1}}\right)\\ & = & {g}_{\alpha }(C({\rho }_{A| {B}_{1}\cdots {B}_{N-1}})),\end{array}\end{eqnarray}$
where the first inequality follows from the convex property of the function ${g}_{\alpha }(x)$, the second equality is satisfied because $\{{p}_{l},| {\chi }_{l}{\rangle }_{A| {B}_{1}\cdots {B}_{N-1}}\}$ is the optimal pure-state decomposition for $C({\rho }_{A| {B}_{1}\cdots {B}_{N-1}})$.Consequently we can write5 ) and the monotonically increasing property of the function ${g}_{\alpha }(x)$. Iterative use of inequality (14 ), we have the third inequality. We also use the fact that $C({\rho }_{{{AB}}_{i}})\,\geqslant C({\rho }_{A| {B}_{i+1}\cdots {B}_{N-1}})\geqslant \sqrt{{C}^{2}({\rho }_{{{AB}}_{i+1}})\,+\cdots +\,{C}^{2}({\rho }_{{{AB}}_{N-1}})}$, $i=1,2,...,\,N-2$ and $1+\tfrac{\eta }{2}x+\left({2}^{\eta }-\tfrac{\eta }{2}-1\right){x}^{\eta }\geqslant 1+\left({2}^{\eta }-1\right){x}^{\eta }$ for $\eta \geqslant 1$. Because when $\alpha \geqslant \tfrac{\sqrt{7}-1}{2},{E}_{\alpha }({\rho }_{{AB}})={g}_{\alpha }(C({\rho }_{{AB}}))$ for any two-qubit state ${\rho }_{{AB}}$, one gets the last equality. Combining (16 ) and (17 ) completes the proof of inequality (11 ).
$ \begin{eqnarray}\begin{array}{l}{E}_{\alpha }^{\eta }({\rho }_{A| {B}_{1}\cdots {B}_{N-1}})\\ \quad \geqslant \,{\left[{g}_{\alpha }(C({\rho }_{A| {B}_{1}\cdots {B}_{N-1}}))\right]}^{\eta }\\ \quad \geqslant \,{\left[{g}_{\alpha }\left(\sqrt{{C}^{2}({\rho }_{{{AB}}_{1}})+\cdots +{C}^{2}({\rho }_{{{AB}}_{N-1}})}\right)\right]}^{\eta }\\ \quad \geqslant \,{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{1}}))\right]}^{\eta }+\left({2}^{\eta }-1\right){\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{2}}))\right]}^{\eta }+\cdots \\ \qquad +\,{\left({2}^{\eta }-1\right)}^{N-4}{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{N-3}}))\right]}^{\eta }\\ \qquad +\,{\left({2}^{\eta }-1\right)}^{N-3}\{{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{N-2}}))\right]}^{\eta }\\ \qquad +\,\displaystyle \frac{\eta }{2}{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{N-2}}))\right]}^{\eta -1}{g}_{\alpha }(C({\rho }_{{{AB}}_{N-1}}))\\ \qquad +\,\left.\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{N-1}}))\right]}^{\eta }\right\}\\ \quad =\,{E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{1}})+\left({2}^{\eta }-1\right){E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{2}})\\ \qquad +\,\cdots +{\left({2}^{\eta }-1\right)}^{N-4}{E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{N-3}})\\ \qquad +\,{\left({2}^{\eta }-1\right)}^{N-3}\left\{\Space{0ex}{3.10ex}{0ex}{E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{N-2}})\right.\\ \qquad +\,\displaystyle \frac{\eta }{2}{E}_{\alpha }^{\eta -1}({\rho }_{{{AB}}_{N-2}}){E}_{\alpha }({\rho }_{{{AB}}_{N-1}})\\ \qquad +\,\left.\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{N-1}})\right\},\end{array}\end{eqnarray}$
here in the second inequality we have used the monogamy inequality (The proof of inequality (12 ) is very similar to that of the inequality (11 ). By the inequality (9 ), for $2\gt \alpha \geqslant \tfrac{\sqrt{7}-1}{2}$, $\gamma \geqslant 2,t=\tfrac{\gamma }{2}\geqslant 1$, we can obtain
$ \begin{eqnarray}{\left[{g}_{\alpha }\left(\sqrt{{x}^{2}+{y}^{2}}\right)\right]}^{\gamma }\geqslant {\left\{{\left[{g}_{\alpha }(x)\right]}^{2}+{\left[{g}_{\alpha }(y)\right]}^{2}\right\}}^{t}.\end{eqnarray}$
Again, without loss of generality one may assume $x\geqslant y$, by the inequality (10 ) of lemma 1 , one finds19 ), following a similar procedure as above, we can obtain the inequality (12 ). The proof of lemma 2 is completed.
$ \begin{eqnarray}\begin{array}{l}{\left[{g}_{\alpha }\left(\sqrt{{x}^{2}+{y}^{2}}\right)\right]}^{\gamma }\geqslant {\left[{g}_{\alpha }(x)\right]}^{\gamma }+\displaystyle \frac{t}{2}{\left[{g}_{\alpha }(x)\right]}^{\gamma -2}{\left[{g}_{\alpha }(y)\right]}^{2}\\ \qquad +\,\left({2}^{t}-\displaystyle \frac{t}{2}-1\right){\left[{g}_{\alpha }(y)\right]}^{\gamma }.\end{array}\end{eqnarray}$
Now, using the inequality (In particular, we consider the case N = 3. Note that when α ≥ 2 and the power η ≥ 1, if ${E}_{\alpha }({\rho }_{{{AB}}_{1}})\geqslant {E}_{\alpha }({\rho }_{{{AB}}_{2}})$, then we arrive at
$ \begin{eqnarray}\begin{array}{rcl}{E}_{\alpha }^{\eta }({\rho }_{A| {B}_{1}{B}_{2}}) & \geqslant & {E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{1}})+\displaystyle \frac{\eta }{2}{E}_{\alpha }^{\eta -1}({\rho }_{{{AB}}_{1}}){E}_{\alpha }({\rho }_{{{AB}}_{2}})\\ & & +\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{2}}).\end{array}\end{eqnarray}$
If ${E}_{\alpha }({\rho }_{{{AB}}_{1}})\leqslant {E}_{\alpha }({\rho }_{{{AB}}_{2}})$, then $ \begin{eqnarray}\begin{array}{rcl}{E}_{\alpha }^{\eta }({\rho }_{A| {B}_{1}{B}_{2}}) & \geqslant & {E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{2}})+\displaystyle \frac{\eta }{2}{E}_{\alpha }^{\eta -1}({\rho }_{{{AB}}_{2}}){E}_{\alpha }({\rho }_{{{AB}}_{1}})\\ & & +\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{1}}).\end{array}\end{eqnarray}$
Also, note that when $2\gt \alpha \geqslant \tfrac{\sqrt{7}-1}{2}$ and the power γ ≥ 2, if ${E}_{\alpha }({\rho }_{{{AB}}_{1}})\geqslant {E}_{\alpha }({\rho }_{{{AB}}_{2}})$, we can write
$ \begin{eqnarray}\begin{array}{rcl}{E}_{\alpha }^{\gamma }({\rho }_{A| {B}_{1}{B}_{2}}) & \geqslant & {E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{1}})+\displaystyle \frac{\gamma }{4}{E}_{\alpha }^{\gamma -2}({\rho }_{{{AB}}_{1}}){E}_{\alpha }^{2}({\rho }_{{{AB}}_{2}})\\ & & +\left({2}^{\tfrac{\gamma }{2}}-\displaystyle \frac{\gamma }{4}-1\right){E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{2}}).\end{array}\end{eqnarray}$
If ${E}_{\alpha }({\rho }_{{{AB}}_{1}})\leqslant {E}_{\alpha }({\rho }_{{{AB}}_{2}})$, then
$ \begin{eqnarray}\begin{array}{rcl}{E}_{\alpha }^{\gamma }({\rho }_{A| {B}_{1}{B}_{2}}) & \geqslant & {E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{2}})+\displaystyle \frac{\gamma }{4}{E}_{\alpha }^{\gamma -2}({\rho }_{{{AB}}_{2}}){E}_{\alpha }^{2}({\rho }_{{{AB}}_{1}})\\ & & +\left({2}^{\tfrac{\gamma }{2}}-\displaystyle \frac{\gamma }{4}-1\right){E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{1}}).\end{array}\end{eqnarray}$
Moreover, based on lemma 2 , if $C({\rho }_{{{AB}}_{i}})\,\geqslant C({\rho }_{A| {B}_{i+1}\cdots {B}_{N-1}})$ for i = 1, 2, ⋯, m and $C({\rho }_{{{AB}}_{j}})\,\leqslant C({\rho }_{A| {B}_{j+1}\cdots {B}_{N-1}})$ for $j=m+1,\cdots ,N-2,\forall 1\,\leqslant m\leqslant N\,-3,N\geqslant 4$, we have the following lemma.
For an $N$-qubit state ${\rho }_{{{AB}}_{1}\cdots {B}_{N-1}}\in {{ \mathcal H }}_{A}\otimes {{ \mathcal H }}_{{B}_{1}}\,\otimes \cdots \otimes {{ \mathcal H }}_{{B}_{N-1}}$, if $C({\rho }_{{{AB}}_{i}})\geqslant C({\rho }_{A| {B}_{i+1}\cdots {B}_{N-1}})$ for $i=1,2,\cdots ,m$, and $C({\rho }_{{{AB}}_{j}})\leqslant C({\rho }_{A| {B}_{j+1}\cdots {B}_{N-1}})$ for $j=m+1,\cdots ,N-2,\forall 1\leqslant m\leqslant N-3,N\geqslant 4$, then
$ \begin{eqnarray}\begin{array}{l}{E}_{\alpha }^{\eta }({\rho }_{A| {B}_{1}\cdots {B}_{N-1}})\\ \quad \geqslant \,{E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{1}})+\left({2}^{\eta }-1\right){E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{2}})\\ \qquad +\,\cdots +\,{\left({2}^{\eta }-1\right)}^{m-1}{E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{m}})\\ \qquad +\,{\left({2}^{\eta }-1\right)}^{m+1}[{E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{m+1}})\,+\cdots +\,{E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{N-3}})]\\ \qquad +\,{\left({2}^{\eta }-1\right)}^{m}\left\{\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){E}_{\alpha }^{\eta }({\rho }_{{{AB}}_{N-2}})\right.\\ \qquad +\left.\,\displaystyle \frac{\eta }{2}{E}_{\alpha }\left({\rho }_{{{AB}}_{N-2}}\right){E}_{\alpha }^{\eta -1}\left({\rho }_{{{AB}}_{N-1}}\right)+{E}_{\alpha }^{\eta }\left({\rho }_{{{AB}}_{N-1}}\right)\right\},\end{array}\end{eqnarray}$
for $\alpha \geqslant 2$ and the power $\eta \geqslant 1$; and $ \begin{eqnarray}\begin{array}{rcl} & & {E}_{\alpha }^{\gamma }({\rho }_{A| {B}_{1}\cdots {B}_{N-1}})\\ & \geqslant & {E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{1}})+({2}^{t}-1){E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{2}})\\ & & +\cdots +\,{\left({2}^{t}-1\right)}^{m-1}{E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{m}})\\ & & +{\left({2}^{t}-1\right)}^{m+1}[{E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{m+1}})\,+\cdots +\,{E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{N-3}})]\\ & & +{\left({2}^{t}-1\right)}^{m}\left\{\left({2}^{t}-\displaystyle \frac{t}{2}-1\right){E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{N-2}})\right.\\ & & +\displaystyle \frac{t}{2}{E}_{\alpha }^{2}({\rho }_{{{AB}}_{N-2}}){E}_{\alpha }^{\gamma -2}({\rho }_{{{AB}}_{N-1}})\\ & & +{E}_{\alpha }^{\gamma }({\rho }_{{{AB}}_{N-1}})\Space{0ex}{3.10ex}{0ex}\},\end{array}\end{eqnarray}$
for $2\gt \alpha \geqslant \tfrac{\sqrt{7}-1}{2}$ and the power $\gamma \geqslant 2$, where $t=\tfrac{\gamma }{2}$. For α ≥ 2, from lemma
$ \begin{eqnarray}\begin{array}{l}{E}_{\alpha }^{\eta }({\rho }_{A| {B}_{1}\cdots {B}_{N-1}})\\ \quad \geqslant \,{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{1}}))\right]}^{\eta }+\left({2}^{\eta }-1\right){\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{2}}))\right]}^{\eta }\\ \qquad +\,\cdots +{\left({2}^{\eta }-1\right)}^{m-2}{[{g}_{\alpha }(C({\rho }_{{{AB}}_{m-1}}))]}^{\eta }\\ \qquad +\,{\left({2}^{\eta }-1\right)}^{m-1}\{{[{g}_{\alpha }(C({\rho }_{{{AB}}_{m}}))]}^{\eta }\\ \qquad +\,\displaystyle \frac{\eta }{2}{[{g}_{\alpha }(C({\rho }_{{{AB}}_{m}}))]}^{\eta -1}{g}_{\alpha }(C({\rho }_{A| {B}_{m+1}\cdots {B}_{N-1}}))\\ \qquad +\,\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){\left[{g}_{\alpha }(C({\rho }_{A| {B}_{m+1}\cdots {B}_{N-1}}))\right]}^{\eta }\}\\ \quad \geqslant \,{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{1}}))\right]}^{\eta }+\left({2}^{\eta }-1\right){\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{2}}))\right]}^{\eta }\\ \qquad +\,\cdots +{\left({2}^{\eta }-1\right)}^{m-1}{[{g}_{\alpha }(C({\rho }_{{{AB}}_{m}}))]}^{\eta }\\ \qquad +\,{\left({2}^{\eta }-1\right)}^{m}{\left[{g}_{\alpha }(C({\rho }_{A| {B}_{m+1}\cdots {B}_{N-1}}))\right]}^{\eta }.\end{array}\end{eqnarray}$
When $C({\rho }_{{{AB}}_{j}})\leqslant C({\rho }_{A| {B}_{j+1}\cdots {B}_{N-1}})$ for $j=m+1,\cdots ,\,N-2$, applying the preceding procedure, one finds $ \begin{eqnarray}\begin{array}{l}{\left[{g}_{\alpha }(C({\rho }_{A| {B}_{m+1}\cdots {B}_{N-1}}))\right]}^{\eta }\\ \quad \geqslant \,\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{m+1}}))\right]}^{\eta }\\ \qquad +\,\displaystyle \frac{\eta }{2}{g}_{\alpha }(C({\rho }_{{{AB}}_{m+1}})){\left[{g}_{\alpha }(C({\rho }_{A| {B}_{m+2}\cdots {B}_{N-1}}))\right]}^{\eta -1}\\ \qquad +\,{\left[{g}_{\alpha }(C({\rho }_{A| {B}_{m+2}\cdots {B}_{N-1}}))\right]}^{\eta }\\ \quad \geqslant \,\left({2}^{\eta }-1\right)\left\{{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{m+1}}))\right]}^{\eta }\,+\cdots +\,{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{N-3}}))\right]}^{\eta }\right\}\\ \qquad +\,\left({2}^{\eta }-\displaystyle \frac{\eta }{2}-1\right){\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{N-2}}))\right]}^{\eta }\\ \qquad +\,\displaystyle \frac{\eta }{2}{g}_{\alpha }(C({\rho }_{{{AB}}_{N-2}})){\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{N-1}}))\right]}^{\eta -1}\\ \qquad +\,{\left[{g}_{\alpha }(C({\rho }_{{{AB}}_{N-1}}))\right]}^{\eta }.\end{array}\end{eqnarray}$
Combining inequalities (To see the tightness of our monogamy relations of multiqubit entanglement, we give an example below.
Under local unitary operations, the three-qubit pure state can be written as [31]
$ \begin{eqnarray}\begin{array}{rcl}| \varphi {\rangle }_{{ABC}} & = & {\lambda }_{0}| 000\rangle +{\lambda }_{1}{{\rm{e}}}^{{\rm{i}}\phi }| 100\rangle \\ & & +{\lambda }_{2}| 101\rangle +{\lambda }_{3}| 110\rangle +{\lambda }_{4}| 111\rangle ,\end{array}\end{eqnarray}$
where $0\leqslant \phi \leqslant \pi ,{\lambda }_{s}\geqslant 0$, $s=0,1,2,3,4$, and ${\sum }_{s=0}^{4}{\lambda }_{s}^{2}\,=1$. Suppose that ${\lambda }_{0}=\tfrac{1}{2},{\lambda }_{1}=\tfrac{\sqrt{47}}{14}$, ${\lambda }_{2}=\tfrac{4}{7},{\lambda }_{3}=\tfrac{3}{7},{\lambda }_{4}=0$.Straightforward calculation of the Rényi-α entanglement shows that ${E}_{2}(| \varphi { \rangle }_{A| {BC}})=0.424\,89$, ${E}_{2}\left({\rho }_{A| B}\right)=0.138\,98$, ${E}_{2}({\rho }_{A| C})=0.257\,16$, for α = 2. One can explicitly see that our lower bound is larger than the results in [9, 28, 29], as shown in figure 1.
Figure 1. The y axis is the Rényi-α entanglement of $| \varphi \rangle $ with α = 2 and its lower bound. The (red solid) line a represents the Rényi-2 entanglement of $| \varphi {\rangle }_{A| {BC}}$ in Example. The (green dashed) line b denotes the lower bound given by inequality ( |
Figure 2. The y axis is the Rényi-α entanglement of $| \varphi \rangle $ with α = 1.5 and its lower bound. The (red solid) line e represents the Rényi-1.5 entanglement of $| \varphi {\rangle }_{A| {BC}}$ in Example. The (green dashed) line f denotes the lower bound given by inequality ( |
4. Conclusion
In this paper we have investigated the tight monogamy relations in terms of Rényi-α entanglement. By using the power of the Rényi-α entanglement, we have provided a class of tight monogamy relations for α ≥ 2, the power η > 1 and $2\gt \alpha \geqslant \tfrac{\sqrt{7}-1}{2}$, the power η > 2, respectively. We have also shown that these new monogamy relations of multiparty entanglement with larger lower bounds than the former results [9, 28, 29].
Multipartite entanglement can be regarded as a fundamental problem in the theory of quantum entanglement. It has attracted much attention over the past two decades. Our results provide a finer characterization of multiqubit entanglement sharing and distribution based on the Rényi-α entanglement. The framework can also be applied to other entanglement measures [4–10]. Our results cannot only provide a useful methodology to study further the monogamous property of multipartite quantum entanglement, but also may contribute to a fully understanding of the multipartite quantum entanglement.