and
ΣaΣb = σaz⊗σzb⊗Ic
ΣbΣc = Ia⊗σbz⊗σzc
ΣcΣa = σaz⊗Ib⊗σcz (17.64)The eigenvalue conditions of ΣaΣb, ΣbΣc, and ΣcΣa,on|Φ〉then translate to the fact that in a
basis whereσaz,σzb,σzcare diagonal, the eigenvalues of these three operators mustbe the same. This
allows for the states|+ ++〉and|−−−〉. Separately, these states are not eigenstates of Σa,Σb,
and Σc, however, and only then combination
|Φ〉=1
√
2
(
|+ ++〉−|−−−〉)
(17.65)satisfies the original conditions (17.62). This is an entangled 3-particle state of total spin 3/2. It is
referred to as a GHZ (Greenberger-Horne-Zeilinger) state.Such entangled states have remarkable
properties. They also allow one to exhibit the conflict between the predictions of quantum theory
and of hidden variable theory even more dramatically.
The product of the operators Σa,Σb, and Σcis also an observable,Σ = σax⊗σbx⊗σxc
= −ΣaΣbΣc (17.66)of which|Φ〉is obviously an eigenstate with eigenvalue−1.
Now if the measurements ofσa,b,cx,y really had independent physical reality, as hidden variable
theory proclaims, then we can again make a list of all possible outcomes of measurements by three
observersA,B,C. Denoting these outcomesAx,By,Cyetc, we are led to the following relations in
view of the fact that the state|Φ〉measured here satisfies (17.62),
AxByCy = +1
AyBxCy = +1
AyByCx = +1 (17.67)As a result, the measurement of the observableσax⊗σxb⊗σxcwill yield
AxBxCx= (AxByCy)(AyBxCy)(AyByCx) = +1 (17.68)since we haveA^2 y=By^2 =C^2 y= +1. But this is in blatant contradiction with the eigenvalue of the
operator Σ which is−1.