Monogamy relations and upper bounds for the generalized \(W\)-class states using R\'{e}nyi-\(\alpha\) entropy

Abstract

We investigate monogamy relations and upper bounds for generalized \(W\)-class states related to the R\'{e}nyi-\(\alpha\) entropy. First, we present an analytical formula on R\'{e}nyi-\(\alpha\) entanglement (R\(\alpha\)E) and R\'{e}nyi-\(\alpha\) 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\(\alpha\)E and REoA. Then we give the upper bounds for generalized \(W\)-class states in terms of R\(\alpha\)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\(\alpha\)E by the analytical expression between R\(\alpha\)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.