The stabilizer for n-qubit symmetric states
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The stabilizer group for an n-qubit state |ϕ〉 is the set of all invertible local operators (ILO) g=g_1⊗ g_2⊗...⊗ g_n, g_i∈GL(2,C) such that |ϕ〉=g|ϕ〉. Recently, G. Gour et al. GKW presented that almost all n-qubit state |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 the trivial stabilizer group. At last, 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 GKW partly.
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