Three observers — Alice, Bob, Carol — each receive one qubit from a shared GHZ state (|000⟩ + |111⟩)/√2. Each independently chooses to measure their qubit in basis X or Y. The product of their three results is not just statistically different in the quantum and classical worlds — it's deterministically opposite. No probability, no inequality, no statistical loophole. Just contradiction.
| scenario | quantum predicts | any local hidden-variable predicts |
|---|---|---|
| XXX (all measure X) | +1 | −1 (impossible to satisfy) |
| XYY | −1 | ±1 needed |
| YXY | −1 | ±1 needed |
| YYX | −1 | ±1 needed |
(|000⟩ + |111⟩)/√2, quantum mechanics predicts with certainty:
A_x · B_y · C_y = −1 every timeA_y · B_x · C_y = −1 every timeA_y · B_y · C_x = −1 every timeA_x · B_x · C_x = +1 every time
(α · β' · γ') · (α' · β · γ') · (α' · β' · γ) = α α' β β' γ γ' · (α'β'γ'). After cancellation, this equals α' β' γ' · (α' β' γ') wait let me redo. The trick is: multiplying the LHS of all three "−1 results" gives α β'γ' · α'β γ' · α'β'γ = α·α'·α' · β'·β·β' · γ'·γ'·γ = α · β · γ (since each prime appears twice and squares to 1).
α β γ = (−1) × (−1) × (−1) = −1. But the fourth measurement A_x B_x C_x = α β γ is predicted by quantum to be +1. Contradiction. No probability needed. This is the Mermin-Greenberger-Horne-Zeilinger paradox in its sharpest form: local realism predicts the opposite of what quantum mechanics — and experiment — confirm.