Superior monogamy and polygamy relations and estimates of concurrence

Abstract

It is well known that any well-defined bipartite entanglement measure $\mathcal{E}$ obeys $γ$th-monogamy relations Eq. (1.1) and assisted measure $\mathcal{E}_{a}$ obeys $δ$th-polygamy relations Eq. (1.2). Recently, we presented a class of tighter parameterized monogamy relation for the $α$th $(α\geqγ)$ power based on Eq. (1.1). This study provides a family of tighter lower (resp. upper) bounds of the monogamy (resp. polygamy) relations in a unified manner. In the first part of the paper, the following three basic problems are focused: (i) tighter monogamy relation for the $α$th ($0\leq α\leq γ$) power of any bipartite entanglement measure $\mathcal{E}$ based on Eq. (1.1); (ii) tighter polygamy relation for the $β$th ($ β\geq δ$) power of any bipartite assisted entanglement measure $\mathcal{E}_{a}$ based on Eq. (1.2); (iii) tighter polygamy relation for the $ω$th ($0\leq ω\leq δ$) power of any bipartite assisted entanglement measure $\mathcal{E}_{a}$ based on Eq. (1.2). In the second part, using the tighter polygamy relation for the $ω$th ($0\leq ω\leq 2$) power of CoA, we obtain good estimates or bounds for the $ω$th ($0\leq ω\leq 2$) power of concurrence for any $N$-qubit pure states $|ψ\rangle_{AB_{1}\cdots B_{N-1}}$ under the partition $AB_{1}$ and $B_{2}\cdots B_{N-1}$. Detailed examples are given to illustrate that our findings exhibit greater strength across all the region.