The stabilizer for $n$-qubit symmetric states

The stabilizer group for an $n$-qubit state $\ketφ$ is the set of all invertible local operators (ILO) $g=g_1\otimes g_2\otimes \cdots\otimes g_n,$ $ g_i\in \mathcal{GL}(2,\mathbb{C})$ such that $\ketφ=g\ketφ.$ Recently, G. Gour $et$ $al.$ \cite{GKW} presented that almost all $n$-qubit state $\ketψ$ own a trivial stabilizer group when $n\ge 5.$ In this article, we consider the case when the stabilizer group of an $n$-qubit symmetric pure state $\ketψ$ is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state $\ketφ$ is nontrivial when $n\le 4$. Then we present a class of $n$-qubit symmetric states $\ketφ$ with the trivial stabilizer group. At last, we prove that an $n$-qubit symmetric pure state owns a trivial stabilizer group when its diversity number is bigger than 5, which confirms the main result of \cite{GKW} partly.