Monogamy and polygamy for generalized W -class states using Rényi- α entropy

Monogamy of entanglement is an indispensable feature in multipartite quantum systems. In this paper we investigate monogamy and polygamy relations with respect to any partition for generalized $W$-class states using R\'enyi-$\ensuremath{\alpha}$ entropy. First, we present analytical formulas of R\'enyi-$\ensuremath{\alpha}$ entanglement $(\mathrm{R}\ensuremath{\alpha}\mathrm{E})$ and R\'enyi-$\ensuremath{\alpha}$ entanglement of assistance $(\mathrm{R}\ensuremath{\alpha}\mathrm{EoA})$ for a reduced density matrix of an $n$-qudit pure state in a superposition of generalized $W$-class states and vacuum. Based on the analytical formulas, we show monogamy and polygamy relations in terms of $\mathrm{R}\ensuremath{\alpha}\mathrm{E}$ and $\mathrm{R}\ensuremath{\alpha}\mathrm{EoA}$. Next a reciprocal relation of $\mathrm{R}\ensuremath{\alpha}\mathrm{EoA}$ in an arbitrary three-party quantum system is found. Later, we further develop tighter monogamy relations in terms of concurrence and convex-roof extended negativity than former ones. In order to show the usefulness of our results, two partition-dependent residual entanglements are established to get a comprehensive analysis of entanglement dynamics for generalized $W$-class states. Moreover, we apply our results to an interesting quantum game and find a bound of the difference between the quantum game and the classical game.