The stabilizer for..-qubit symmetric states

The stabilizer group for an n-qubit state ｜Φ〉 is the set of all invertible local operators（ILO） g = g1g2…gn,gi 2 GL（2,C） such that ｜Φ〉= g｜Φ〉. Recently, Gour et al. [Gour G, Kraus B and Wallach N R 2017 J. Math. Phys. 58092204] presented that almost all n-qubit states jyi own a trivial stabilizer group when n≥5. In this article, we consider the case when the stabilizer group of an n-qubit symmetric pure state jyi is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state ｜Φ〉 is nontrivial when n≤4. Then we present a class of n-qubit symmetric states ｜Ψ〉 with a trivial stabilizer group when n≥5. Finally, we propose a conjecture and prove that an n-qubit symmetric pure state owns a trivial stabilizer group when its diversity number is bigger than 5 under the conjecture we make, which confirms the main result of Gour et al. partly.