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The GHZ theorem contradicts predetermined local outcomes using perfect quantum correlations. For the GHZ state $(|000⟩+|111⟩)/\sqrt{2}$, the products of Pauli operators obeywith certainty, while $X_1{X}_2{X}_3=+1$ 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 $x_1{x}_2{x}_3=-1$, 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. ^[ref]

Matthew McKague authored Self-testing graph states, which constructs tests for connected graph states using observed quantum correlations. The aim is to identify the shared quantum state and local quantum measurements, up to the equivalences allowed by the test, through their observable correlations. The paper also establishes robustness: small deviations from the ideal correlations yield controlled deviations in the conclusion.^[ref]

A quantum correlation is a statistical relation among observables of a composite quantum system. For local observables $A$ and $B$, a useful diagnostic is $⟨A\otimes B⟩-⟨A\otimes I⟩⟨I\otimes B⟩$. Product states make this expression vanish for every pair of local observables.

Not every correlation in a quantum state is quantum entanglement. The separable mixture $[|00⟩⟨00|+|11⟩⟨11|]/2$ gives perfectly correlated computational-basis outcomes but has no entanglement. The pure Bell state $|{\Phi }^{+}⟩$ has the same outcomes in that basis while retaining coherent relative-phase information detectable in complementary bases. The measurement choices therefore matter when characterizing the kind of correlation present.