GHZ Argument

The Greenberger–Horne–Zeilinger (GHZ) argument uses perfect three-party correlations to contradict a Bell-local assignment of outcomes without first averaging an inequality. Its experimental use still requires a statistical witness when correlations are imperfect. The ideal contradiction and the noisy test are complementary descriptions of the same model obstruction.

Helpful background. Local Hidden Variables defines local response models and setting independence.

Four compatible measurement settings

Prepare three spatially separated qubits in

$$|G\rangle=\frac{|000\rangle+|111\rangle}{\sqrt2}.$$

Each party chooses a local Pauli observable $X$ or $Y$ with outcomes $\pm1$. Using $Y|0\rangle=i|1\rangle$ and $Y|1\rangle=-i|0\rangle$ gives

$$XXX|G\rangle=|G\rangle, \qquad XYY|G\rangle=YXY|G\rangle=YYX|G\rangle=-|G\rangle.$$

Each product refers to one joint setting, with one observable measured by each party. The notation does not ask a party to measure $X$ and $Y$ simultaneously.

Why local signs cannot reproduce the predictions

Let $x_j,y_j\in\{-1,+1\}$ denote local predetermined responses for party $j$. The three settings containing two $Y$ measurements demand

$$x_1y_2y_3=-1,\quad y_1x_2y_3=-1,\quad y_1y_2x_3=-1.$$

Multiplication cancels each $y_j^2$ and yields $x_1x_2x_3=-1$, contrary to the quantum prediction for $XXX$. Reusing the same response for a given local setting across remote choices is the locality premise. Measurement independence supplies a common hidden-variable ensemble for the different setting choices.

Allowing stochastic local responses does not rescue the model: in a finite scenario local randomness can be included in an enlarged shared-variable description, or the perfect correlations can be shown to force deterministic responses on the relevant support. The local-model treatment explains that reduction.

The Mermin witness tolerates imperfect correlations

Define

$$M=\langle XXX\rangle-\langle XYY\rangle -\langle YXY\rangle-\langle YYX\rangle.$$

For any assignment of six local signs, the four signed terms have product $-1$. Their sum is therefore $+2$ or $-2$. Convex mixtures give $|M|\le2$. The GHZ state yields $M=4$.

For the explicitly specified noise model $\rho_v=v|G\rangle\langle G|+(1-v)I/8$, each Pauli product is traceless, so $M=4v$. This witness violates the local bound for $v>1/2$. Failure below that threshold means this witness no longer certifies nonlocality; it does not establish separability of the state or a universal local model for every measurement.

Scope of the conclusion

The contradiction is with a Bell-local model under the stated setting assumptions, not with probability theory or every hidden-variable theory. A nonlocal contextual account need not assign the six responses in this form. The measured correlations also remain compatible with no-signaling marginals.

An ordinary Mermin violation above two excludes fully local correlations. Stronger statements about genuine tripartite nonlocality or device-independent certification need the appropriate model class and bound. The word “three-party” alone does not supply them.

Exercise: a saturated local strategy

Give six local signs that achieve $M=2$, and explain why averaging different saturated strategies cannot give $M>2$.

Solution

Take all $x_j=+1$, $y_1=-1$, $y_2=y_3=+1$. The four signed contributions are $1,-1,1,1$, summing to two. An average of values bounded by two remains bounded by two, which is the convex extension from deterministic to stochastic local models.

References

• Greenberger, Daniel M., Michael A. Horne, Abner Shimony, and Anton Zeilinger, “Bell’s theorem without inequalities,” American Journal of Physics 58, 1131–1143 (1990), doi:10.1119/1.16243.
• Mermin, N. David, “Extreme quantum entanglement in a superposition of macroscopically distinct states,” Physical Review Letters 65, 1838–1840 (1990), doi:10.1103/PhysRevLett.65.1838.