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 Rényi-$\alpha$ entropy. First, we present analytical formulas of Rényi-$\alpha$ entanglement $\left(\mathrm{R}\alpha \mathrm{E}\right)$ and Rényi-$\alpha$ entanglement of assistance $\left(\mathrm{R}\alpha \mathrm{EoA}\right)$ 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 $\mathrm{R}\alpha \mathrm{E}$ and $\mathrm{R}\alpha \mathrm{EoA}$. Next a reciprocal relation of $\mathrm{R}\alpha \mathrm{EoA}$ 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.

• Received 14 May 2020
• Revised 6 December 2020
• Accepted 7 December 2020

DOI:https://doi.org/10.1103/PhysRevA.102.062428