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The EPR argument uses correlations of an entangled state and a locality assumption to challenge the completeness of the quantum description. Its Einstein–Podolsky–Rosen criterion of reality identifies quantities predictable with certainty without disturbing a distant system as elements of physical reality.

For the spin singlet state, measurements along the same axis are perfectly anticorrelated. By measuring either spin component of particle $B$, one can therefore predict with certainty the corresponding result for distant particle $A$. The Einstein–Podolsky–Rosen criterion of reality then says that every such spin component of $A$ is an element of physical reality, because the choice at $B$ can be made without disturbing $A$. Since the quantum state assigns no simultaneous sharp values to noncommuting spin components, the EPR argument concludes that the wavefunction is not a complete description, provided this locality premise is accepted.

Let $\lambda$ denote a proposed complete hidden state, and let $a,b$ be the chosen axes.

For two axes on each side, every $\lambda$ obeysMeasurement independence permits averaging the same distribution of $\lambda$ for all four setting pairs, producing the CHSH inequality

The singlet prediction isChoose coplanar unit vectorsThenand thereforeThus the predictions of quantum theory violate the conjunction of outcome determinism, parameter independence, and measurement independence. This is Bell theorem; the calculation alone does not select which premise a deeper theory must abandon.

The Einstein–Podolsky–Rosen criterion of reality says that if a physical quantity can be predicted with certainty without disturbing a system, then an element of physical reality corresponds to that quantity.

Let $X_i,Y_i$ be Pauli measurements on particle $i$. The GHZ state in the question obeyswith certainty. Each local outcome can therefore be predicted by spacelike-separated measurements on the other two particles. The EPR criterion assigns predetermined values $x_i,y_i\in \{-1,1\}$ satisfying those three equations. Multiplying them and using $y_i^2=1$ givesQuantum theory instead predictsso a joint $X$ measurement has product $-1$ with certainty. The EPR elements of reality therefore contradict the quantum prediction, which is the GHZ theorem form of Bell theorem.

The Einstein–Podolsky–Rosen criterion of reality says that if, without disturbing a system, one can predict a physical quantity with certainty, then an element of physical reality corresponds to that quantity.

For two settings on each wing and outcomes in $\{-1,1\}$, the CHSH inequality for every local hidden-variable theory is