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Chin. Phys. B, 2018, Vol. 27(10): 100311    DOI: 10.1088/1674-1056/27/10/100311
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The stabilizer for n-qubit symmetric states

Xian Shi(石现)1,2,3
1 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
2 University of Chinese Academy of Sciences, Beijing 100049, China;
3 UTS-AMSS Joint Research Laboratory for Quantum Computation and Quantum Information Processing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Abstract  

The stabilizer group for an n-qubit state|φ> is the set of all invertible local operators (ILO) g=g1g2⊗ …⊗ gn, giGL(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.

Keywords:  symmetric states      stabilizer group  
Received:  30 April 2018      Revised:  01 July 2018      Accepted manuscript online: 
PACS:  03.67.Mn (Entanglement measures, witnesses, and other characterizations)  
  03.65.Ud (Entanglement and quantum nonlocality)  
Fund: 

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.

Corresponding Authors:  Xian Shi     E-mail:  shixian01@gmail.com

Cite this article: 

Xian Shi(石现) The stabilizer for n-qubit symmetric states 2018 Chin. Phys. B 27 100311

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