The Greenberger-Horne-Zeilinger theorem, or GHZ theorem, shows that certain perfect quantum correlations cannot be reproduced by assigning predetermined local outcomes to all of the measurements involved. Unlike a Bell inequality, its contradiction is obtained from exact correlations rather than a bound on their expectation values (Greenberger et al. 1990).
Let 
and 
be the Pauli X matrix and Pauli
Y matrix, and let 
be the Greenberger-Horne-Zeilinger
state with a plus sign between its two terms. Using tensor
products, define
 |
These four operators commute, and their action on the state is
 |
If each location had fixed outcomes 
independent of measurement choices at the
other locations, the last three correlations would require 
. Multiplication gives 
because 
, whereas the first correlation requires 
. Thus no such joint assignment exists. The argument
does not assert that incompatible observables at an individual location can be measured
simultaneously.
