Superior monogamy and polygamy relations and estimates of concurrence

Abstract

It is well known that any well-defined bipartite entanglement measure \(\mathcal {E}\) obeys \(\gamma \)th-monogamy relations Eq. (1.1) and assisted measure \(\mathcal {E}_{a}\) obeys \(\delta \)th-polygamy relations Eq. (1.2). Recently, we presented a class of tighter parameterized monogamy relation for the \(\alpha \)th \((\alpha \ge \gamma )\) 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:

1. (i)

   tighter monogamy relation for the \(\alpha \)th (\(0\le \alpha \le \gamma \)) power of any bipartite entanglement measure \(\mathcal {E}\) based on Eq. (1.1);

2. (ii)

   tighter polygamy relation for the \(\beta \)th (\( \beta \ge \delta \)) power of any bipartite assisted entanglement measure \(\mathcal {E}_{a}\) based on Eq. (1.2);

3. (iii)

   tighter polygamy relation for the \(\omega \)th (\(0\le \omega \le \delta \)) 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 \(\omega \)th (\(0\le \omega \le 2\)) power of CoA, we obtain good estimates or bounds for the \(\omega \)th (\(0\le \omega \le 2\)) power of concurrence for any N-qubit pure states \(|\psi \rangle _{AB_{1}\cdots B_{N-1}}\) 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.