GHZ Paradox: Three-Qubit Entanglement Beats Local Realism (2D)

Measurement Setting

Batch trials 100

Live Readouts

QM predicts

Last measured

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Trials run / match rate

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Local Hidden-Variable Model

Fix a predetermined value for every qubit's X and Y outcome, then check it against all four settings at once.

Best possible LHV match

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How it works

Three qubits are prepared in the GHZ state |GHZ⟩ = (|000⟩ + |111⟩)/√2. On each qubit you can measure either Pauli X or Pauli Y. For the four combinations used here, the GHZ state is an exact eigenstate of the joint operator — the product of the three individual outcomes is fixed with certainty, not just statistically biased:

X⊗X⊗X |GHZ⟩ = +1 |GHZ⟩
X⊗Y⊗Y |GHZ⟩ = −1 |GHZ⟩
Y⊗X⊗Y |GHZ⟩ = −1 |GHZ⟩
Y⊗Y⊗X |GHZ⟩ = −1 |GHZ⟩

Each individual qubit's own outcome is still a random coin flip (50/50) — only the three-way product is pinned down. The simulator samples two outcomes uniformly and derives the third from the required product, exactly reproducing this joint distribution.

The paradox: suppose each qubit secretly carries predetermined values x_i, y_i ∈ {+1,−1} (local realism) that measurement merely reveals. The first three lines force:

x1·y2·y3 = −1
y1·x2·y3 = −1
y1·y2·x3 = −1
Multiply all three (yᵢ² = 1):
x1·x2·x3 = −1

But quantum mechanics says X⊗X⊗X = +1, always. No assignment of x_i, y_i can satisfy all four constraints — the "Find Best" search always caps out at 3 of 4. Unlike Bell/CHSH, which needs statistics over many runs to show a violation, GHZ delivers a flat logical contradiction from a single set of deterministic predictions — the strongest form of the entanglement argument against local realism (Greenberger, Horne & Zeilinger, 1989; Mermin, 1990).

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X-axis measurement Y-axis measurement Entanglement link