The stabilizer for n-qubit symmetric states

The stabilizer group for an n-qubit state vertical bar phi > is the set of all invertible local operators (ILO) g = g(1 )circle times g(2) circle times ... g(n), g(i) is an element of L (2,C) such that vertical bar phi > = g vertical bar phi >. 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 vertical bar psi > 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 vertical bar psi > is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state vertical bar phi > is nontrivial when n <= 4. Then we present a class of n-qubit symmetric states vertical bar phi > 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.