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

Greenberger-Horne-Zeilinger(GHZ)theorem asserts that there is a set of mutually commuting nonlocal observables with a common eigenstate on which those ob- servables 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 equiva- lence 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 with a 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.