The stabilizer for n-qubit symmetric states

The stabilizer group for an n-qubit state|φ> is the set of all invertible local operators (ILO) g=g₁⊗ g₂⊗ …⊗ g_n, g_i∈ GL(2,C) such that|φ>=g|φ>. Recently, Gour et al.[Gour G, Kraus B and Wallach N R 2017 J. Math. Phys. 58 092204] presented that almost all n-qubit states|Ψ〉 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|Ψ> 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|φ> 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.