The stabilizer for n-qubit symmetric states**Project partially supported by the National Key Research and Development Program of China (Grant No. 2016YFB1000902), the National Natural Science Foundation of China (Grant Nos. 61232015 and 61621003), the Knowledge Innovation Program of the Chinese Academy of Sciences (CAS), and Institute of Computing Technology of CAS.

Abstract

The stabilizer group for an n-qubit state |ϕ⟩ is the set of all invertible local operators (ILO) g = g1 ⊗ g2 ⊗ ⋯ ⊗ gn, 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.