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Monogamy relations and upper bounds for the generalized $W$-class states using Rényi-$α$ entropy

We investigate monogamy relations and upper bounds for generalized $W$-class states related to the Rényi-$α$ entropy. First, we present an analytical formula on Rényi-$α$ entanglement (R$α$E) and Rényi-$α$ entanglement of assistance (REoA) of a reduced density matrix for a generalized $W$-class states. According to the analytical formula, we show monogamy and polygamy relations for generalized $W$-class states in terms of R$α$E and REoA. Then we give the upper bounds for generalized $W$-class states in terms of R$α$E. Next, we provide tighter monogamy relations for generalized $W$-class states in terms of concurrence and convex-roof extended negativity and obtain the monogamy relations for R$α$E by the analytical expression between R$α$E and concurrence. Finally, we apply our results into quantum games and present a new bound of the nonclassicality of quantum games restricting to generalized $W$-class states.