Monogamy and polygamy for multi-qudit generalized $W$-class states based on concurrence of assistance and Tsallis-$q$ entanglement of assistance

Abstract

By analyzing the reduced density matrices derived from a generalized $W$-class state under any partition, we present new analytical monogamy inequalities satisfied by the $\hat{I}\pm$-th ($\hat{I}\pm \ge {\hat{I}}^3,{\hat{I}}^3\ge 2$) power and ${\hat{I}}^2$-th ($0\le {\hat{I}}^2\le \frac{\hat{I}}{³}2,{\hat{I}}^3\ge 2$) power of the concurrence of assistance for multi-qudit generalized $W$-class states, which are demonstrated to be tighter than previous studies through detailed examples. Furthermore, using the Tsallis-$q$ entanglement of assistance, we also establish new monogamy and polygamy relations, which are shown to be valid even for multipartite higher-dimensional states that the CKW inequality is violated.