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The EPR criterion of reality says that if a physical quantity can be predicted with certainty without disturbing the system, then it represents an element of physical reality. For separated particles, the EPR locality assumption regards choices of measurements on the other particles as nondisturbing. The perfect correlations below therefore imply predetermined local values for both $X={\sigma }_x$ and $Y={\sigma }_y$, independent of which distant measurements are chosen.

Put $|0⟩=|\uparrow ⟩$ and $|1⟩=|\downarrow ⟩$. The Pauli matrices satisfyFor three particles, each operator with one $X$ and two $Y$ takes $|000⟩$ to $-|111⟩$ and $|111⟩$ to $-|000⟩$, whereas $X\otimes X\otimes X$ interchanges the two strings without a minus sign. Thus the GHZ state in the question has the certain outcomesEach local $X$ or $Y$ value can be predicted from measurements at the other two sites in one of these contexts. If the EPR criterion of reality and locality assign values $x_j,y_j\in \{\pm 1\}$, the first three equalities requireMultiplying these ordinary numbers and using $y_j^2=1$ yields $x_1{x}_2{x}_3=1$, contradicting the certain quantum outcome $-1$. This derives the GHZ theorem explicitly; the multiplication is of hypothetical local values, not of mutually incompatible local observables.

For general $N$, let $O_j=X_j{⨂}_{k\ne j}{Y}_k$, with the tensor factors ordered by their site labels, and let $X_N=X^{\otimes N}$. Directly,For the difference of the two strings to be an eigenstate, these phases must agree, which happens exactly when $N-1$ is even. For odd $N$ the common phase is $(-1{)}^{(N-1)/2}$, and interchanging the strings negates their difference. ConsequentlyFor even $N$, $X_N$ still has eigenvalue $-1$, but every $O_j$ takes the difference state to a phase times the sum state, which is orthogonal to it. Thus it is not an eigenstate of any $O_j$; those operators each have outcomes $\pm 1$ with equal probabilities. Among the stated $N\ge 4$, simultaneous eigenstates therefore occur precisely at $N=5,7,9,\dots$.

For odd $N$, certainty of each $O_j$ again permits remote prediction of both local $X$ and $Y$. Writing ${\lambda }_N=(-1{)}^{(N+1)/2}$, the local hidden-variable theory would require $x_j{\prod }_{k\ne j}{y}_k={\lambda }_N$ for all $j$. Their product ishence ${\prod }_j{x}_j={\lambda }_N$, since $N-1$ is even and $N$ is odd. Quantum theory requires the same product to be $-1$. The contradiction occurs when ${\lambda }_N=+1$, giving the infinite familyFor $N\equiv 1(\mathrm{mod}4)$, these particular certain product constraints are consistent; for example $x_j=-1$, $y_j=+1$ satisfies them. This distinguishes the GHZ contradiction for N congruent to 3 modulo 4 from the larger set of odd $N$ for which simultaneous eigenstates exist.

A putative simultaneous assignment of classical values ${\tilde{s}}_x^{(i)},{\tilde{s}}_y^{(i)}\in \{\pm a\}$ reproducing the first three certain quantum outcomes must obey, almost surely,Multiply the three equalities. Each $y$ value occurs twice and its square is $a^2$, leaving $a^6{\tilde{s}}_x^{(1)}{\tilde{s}}_x^{(2)}{\tilde{s}}_x^{(3)}=a^9$. Thuscontrary to the quantum prediction $-a^3$.

This is the GHZ theorem against predetermined local measurement values. Experimentally prepare many copies of the three-spin state and choose independently among the four measurement settings on separated subsystems. The three mixed correlations force the classical prediction for the fourth, whereas quantum mechanics predicts the opposite sign. Since $s_x,s_y$ on the same subsystem do not commute, the four settings are measured on separate copies; the classical contradiction relies precisely on the common assignment of values across settings.

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.

For tensor products of Pauli factors from $\{I,X,Y,Z\}$, let $k$ count the sites where both factors are nonidentity and different. Distinct nonidentity Pauli matrices anticommute, and equal or identity factors commute. Reversing the product site by site therefore gives $PQ=(-1{)}^{k}QP$. Thus two Pauli strings commute exactly when $k$ is even. The criterion applies to the perfect-correlation observables in the GHZ theorem.