Abstract:

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 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.

Key words: GHZ theorem, GHZ state, multi-qubit system