A COMPREHENSIVE PROOF OF THE GREENBERGER-HORNE-ZEILINGER THEOREM FOR THE FOUR-QUBIT SYSTEM

• Published in: Acta Mathematica Scientia Vol. 27; no. 4; pp. 753 - 776
• Main Author: 唐莉 陈泽乾 钟杰 任耀峰 詹明生
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.10.2007; Graduate School, Chinese Academy of Sciences, Wuhan 430071, China%Department of Mathematics, The Naval University of Engineering, Wuhan 430033, China%State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China; State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China; Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China; Graduat School,Chinese Academy of Sciences Wuhan 430071,China%Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China%Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
• Subjects: GHZ state; GHZ theorem; GHZ定理; GHZ状态; multi-qubit system; 单式转化; 多量子位系统; 本征函数; GHZ state; GHZ theorem; multi-qubit system
• ISSN: 0252-9602; 1572-9087; 1003-3998
• DOI: 10.1016/S0252-9602(07)60073-3

Greenberger-Horne-Zeilinger （GHZ） theorem asserts that there is a set of mutually commuting nonlocal observables with a common eigenstate on which those observables assume values that refute the attempt to assign values only required to have them by the local realism of Einstein, Podolsky, and Rosen （EPR）. It is known that for a three-qubit system, there is only one form of the GHZ-Mermin-like argument with equivalence up to a local unitary transformation, which is exactly Mermin＇s version of the GHZ theorem. This article for a four-qubit system, which was originally studied by GHZ, the authors show that there are nine distinct forms of the GHZ-Mermin-like argument. The proof is obtained using certain geometric invariants to characterize the sets of mutually commuting nonlocal spin observables on the four-qubit system. It is proved that there are at most nine elements （except for a different sign） in a set of mutually commuting nonlocal spin observables in the four-qubit system, and each GHZ-Mermin-like argument involves a set of at least five mutually commuting four-qubit nonlocal spin observables witha GHZ state as a common eigenstate in GHZ＇s theorem. Therefore, we present a complete construction of the GHZ theorem for the four-qubit system.