---
title: "GHZ correlations: four settings, no consistent instruction table"
description: "Why does an ideal certainty argument still require a careful ledger of imperfect detections?"
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[Quantum physics: reading the evidence](https://sandboxphysics.com/stories/quantum-evidence/)

Nobel A · Evidence and control

# GHZ correlations: four settings, no consistent instruction table

Sandbox Physics · 2026-10-18

Why does an ideal certainty argument still require a careful ledger of imperfect detections?

## Three entangled photons, four registered photons

Bouwmeester and colleagues’ 1999 experiment combined two photon pairs and selected a trigger together with detections in three other arms. The three-photon GHZ evidence was thus conditioned on a fourfold coincidence. A trigger click alone did not certify that an independently usable triple had already been delivered.

The paper tested the desired polarization components and coherence between them. Those questions differ: an incoherent mixture can reproduce the component populations. Pan and colleagues’ 2000 experiment then examined the GHZ nonlocality argument through four polarization settings. Preparing a state and testing a particular correlation contradiction are related achievements with different records.

Read the evidence: D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter and A. Zeilinger (1999), Observation of Three-Photon Greenberger-Horne-Zeilinger Entanglement, PRL 82, 1345–1349 J.-W. Pan et al. (2000), Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement, Nature 403, 515–519 — abstract and figure captions

## Fix the phase before multiplying signs

For our calculation, zero denotes horizontal and one vertical polarization in each arm. Use an equal, positive superposition of all-zero and all-one components. The first analyzer, called X, distinguishes the two diagonal linear polarizations. The second, Y, distinguishes their equal-amplitude counterparts with opposite imaginary relative phases; its positive outcome has positive imaginary vertical amplitude.

This defines the outcome signs without relying on an ambiguous circular-handedness label. The 1999 paper instead writes components with the third polarization reversed; a local bit flip converts that convention into ours. Changing a state phase or relabeling an analyzer outcome changes the corresponding parity signs. Keep those changes together.

$$
\begin{gathered}|G_+\rangle=\frac{|000\rangle+|111\rangle}{\sqrt{2}},\\ \langle XXX\rangle=1,\quad\langle XYY\rangle=-1,\\ \langle YXY\rangle=\langle YYX\rangle=-1\end{gathered}
$$

Read the evidence: D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter and A. Zeilinger (1999), Observation of Three-Photon Greenberger-Horne-Zeilinger Entanglement, PRL 82, 1345–1349 D. M. Greenberger, M. A. Horne and A. Zeilinger (1989), Going Beyond Bell’s Theorem — original contribution made available as arXiv:0712.0921 in 2007

## The contradiction concerns a shared instruction table

Suppose a local deterministic description gives each station an answer for either analyzer choice, independent of the distant choices. There are six binary answers and therefore 64 complete instruction tables. In our convention, each context containing two Y choices demands a negative product of the three recorded outcomes.

Multiplying those three demands makes every assigned Y answer appear twice, leaving the product of the three X answers negative. Quantum theory predicts it positive. Exhaustive enumeration finds eight tables satisfying the first three demands, and every one fails the fourth. No table meets all four; mixing tables cannot supply four perfect correlations.

Read the evidence: D. M. Greenberger, M. A. Horne and A. Zeilinger (1989), Going Beyond Bell’s Theorem — original contribution made available as arXiv:0712.0921 in 2007 J.-W. Pan et al. (2000), Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement, Nature 403, 515–519 — abstract and figure captions

## Compatible products are not simultaneous local answers

Each context measures one analyzer on each of three separate photons, so its three outcomes can be multiplied. The four global product observables also commute. This does not make X and Y jointly measurable on one photon. Different contexts are acquired on different members of the prepared ensemble.

Our independent calculation sums complex amplitudes for all eight outcomes in each context. It finds four allowed outcomes, each with probability one quarter, and zero probability for the wrong parity. Individual outcomes remain balanced. The certainty belongs to a product, not to a locally predictable photon result or a controllable distant message.

Read the evidence: D. M. Greenberger, M. A. Horne and A. Zeilinger (1989), Going Beyond Bell’s Theorem — original contribution made available as arXiv:0712.0921 in 2007

## Finite counts replace perfect promises

Real records include wrong parities, drift and uncertainty. Our enumeration also gives a maximum of three successful contexts in four for local instructions with uniformly chosen contexts. Equivalently, the signed correlation sum is bounded by two; our ideal quantum state gives four. This finite-score form is our calculation, not a historical fit.

For a declared white-noise teaching model containing sixty percent ideal GHZ state, each requested parity succeeds eighty percent of the time and that sum becomes 2.4. This number alone supplies no experimental significance. Trial definitions, setting independence, stopping rules and actual outcome counts determine the applicable statistical test.

Read the evidence: D. M. Greenberger, M. A. Horne and A. Zeilinger (1989), Going Beyond Bell’s Theorem — original contribution made available as arXiv:0712.0921 in 2007 J.-W. Pan et al. (2000), Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement, Nature 403, 515–519 — abstract and figure captions

## The missing events stay in the conclusion

Fourfold selection retains only a small detected subset of the source attempts. Applying a common local-model bound to that selected record needs a sampling assumption, such as fair sampling, or a trial design that explicitly includes nondetections. A trigger does not by itself close the detection loophole; locality also needs a separate spacetime and setting-choice audit.

Selected parity correlations do not reconstruct every entry of a three-qubit density matrix. The linked two-photon Bell and single-qubit tomography Labs illustrate separate questions, and neither reproduces this apparatus. The achievement is a multiparticle correlation test under stated acquisition assumptions, not an unconditional source, complete state characterization or fully loophole-free experiment.

Read the evidence: D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter and A. Zeilinger (1999), Observation of Three-Photon Greenberger-Horne-Zeilinger Entanglement, PRL 82, 1345–1349 J.-W. Pan et al. (2000), Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement, Nature 403, 515–519 — abstract and figure captions

## Continue with the models

These are existing teaching Labs. Their simulated records are separate from the historical observations discussed above.

- [Compare a separate two-photon Bell score](https://sandboxphysics.com/labs/bell-chsh/)
- [Distinguish a parity test from state reconstruction](https://sandboxphysics.com/labs/quantum-state-process-tomography/)

## Sources and reading notes

We read the five-page 1999 PRL and the original GHZ contribution available as arXiv:0712.0921. For the 2000 Nature experiment, only the publisher’s abstract and figure captions were accessible; its full text was subscription-limited. The positive-phase three-qubit convention, exhaustive instruction tables and complex-amplitude checks below are our own teaching calculations, not a reconstruction of historical counts.

1. [D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter and A. Zeilinger (1999), Observation of Three-Photon Greenberger-Horne-Zeilinger Entanglement, PRL 82, 1345–1349](https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.82.1345)
2. [J.-W. Pan et al. (2000), Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement, Nature 403, 515–519 — abstract and figure captions](https://www.nature.com/articles/35000514)
3. [D. M. Greenberger, M. A. Horne and A. Zeilinger (1989), Going Beyond Bell’s Theorem — original contribution made available as arXiv:0712.0921 in 2007](https://arxiv.org/pdf/0712.0921)

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