= GHZ theorem
{c}
The <GHZ theorem> contradicts predetermined local outcomes using perfect <quantum correlations>. For the <GHZ state> $(|000\rangle+|111\rangle)/\sqrt2$, the products of <Pauli operators> obey
$$
X_1Y_2Y_3=-1,\qquad Y_1X_2Y_3=-1,\qquad Y_1Y_2X_3=-1
$$
with certainty, while $X_1X_2X_3=+1$ with certainty.
Suppose each site has predetermined outcomes $x_i,y_i\in\{-1,+1\}$, independent of which observable is chosen at the other sites. Multiplying the first three constraints cancels every $y_i^2$, forcing $x_1x_2x_3=-1$, in contradiction with the <quantum measurement> prediction. The <Pauli operators> evade that classical multiplication rule because $XY=-YX$ at the same site. The displayed signs use the plus-<GHZ state>; changing its relative sign changes the individual correlations consistently. <Quantum mysteries revisited by N. David Mermin>[Section III and equation 5; direct Pauli calculation here uses the plus-GHZ convention]{ref}
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