Tightening monogamy and polygamy relations of unified entanglement in multipartite systems

Abstract

We study the monogamy and polygamy inequalities of unified entanglement in multipartite quantum systems. We first derive the monogamy inequality of unified-$(q,s)$ entanglement for multi-qubit states under arbitrary bipartition, and then obtain the monogamy inequalities of the $\alpha$th ($0\le \alpha \le \frac{r}{2},r\ge \sqrt{2}$) power of entanglement of formation for tripartite states and their generalizations in multi-qubit quantum states. We also generalize the polygamy inequalities of unified-$(q, s)$ entanglement for multi-qubit states under arbitrary bipartition. Moreover, we investigate polygamy inequalities of the $\beta$th ($\beta\geq \max\{1, s\}, 0\leq s\leq s_0, 0\leq s_0\leq\sqrt{2}$) power of the entanglement of formation for $2\otimes 2\otimes 2$ and $n$-qubit quantum systems. Finally, using detailed examples, we show that the results are tighter than previous studies.