The stabilizer for $n$-qubit symmetric states

Abstract

The stabilizer group for an $n$-qubit state $|\varphi ⟩$ is the set of all invertible local operators (ILO) $g=g_1\otimes g_2\otimes \cdots \otimes g_n,$ $g_i\in \mathcal{G}\mathcal{L}(2,\mathbb{C})$ such that $|\varphi ⟩=g|\varphi ⟩.$ Recently, G. Gour $et$ $al.$ \cite{GKW} presented that almost all $n$-qubit state $|\psi ⟩$ 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 $|\psi ⟩$ is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state $|\varphi ⟩$ is nontrivial when $n\le 4$. Then we present a class of $n$-qubit symmetric states $|\varphi ⟩$ 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.