Three-photon GHZ correlations defeat a shared answer sheet

Three entangled photons, four detected

Three distant observers each measure a photon and obtain plus or minus one. Quantum theory can make the product of their three answers certain even though each observer’s own answer is random. This is the striking feature of a Greenberger–Horne–Zeilinger, or GHZ, state. Bouwmeester and colleagues prepared three-photon entanglement in 1999 by combining two pairs and selecting a trigger click together with detections in the other three arms.

The four detections were the condition for verifying the state; the trigger alone did not announce a freely available triple. The experiment also had to distinguish a coherent superposition from an ordinary mixture, which could give the same counts in the original polarization directions.

The paper checked both the intended polarization components and the coherence between them. These are different questions, because an incoherent mixture can reproduce the component populations. Pan and colleagues’ 2000 experiment then tested the GHZ nonlocality argument with four polarization settings. Preparing the state and testing the contradiction are related achievements, and each needs its own data.

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

Two polarization choices at each station

Each station chooses between two polarization measurements, called X and Y. X distinguishes the two diagonal directions of linear polarization: picture light oscillating along one diagonal or the other. Y distinguishes two circular polarizations: the corresponding polarization pictures rotate in opposite senses. Each detector result is labeled plus or minus one; the precise sign convention is given in the visible source-reading note below.

Zero in the formula means horizontal polarization and one means vertical. The ideal GHZ state adds the all-horizontal and all-vertical possibilities with equal positive amplitude, preserving their relative phase. The bracketed expressions give the average product of the three results. With three X measurements that product is positive; with one X and two Y measurements it is negative. These are the four predictions the experiment compares.

\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

No set of local instructions fits all four settings

Suppose a local deterministic model gives each station a preset answer for either analyzer choice, independent of what the distant stations choose. That is six binary answers, so there are 64 complete instruction tables. In our convention, each setting with two Y analyzers requires the product of the three outcomes to be negative.

Multiply those three requirements together. Each preset Y answer appears twice and drops out, which leaves the product of the three X answers negative. Quantum theory predicts it is positive. Checking every table finds eight that meet the first three requirements, and every one of them fails the fourth. Since no table meets all four, no mixture of tables can produce 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

Each trial measures one polarization choice per photon

A trial uses one analyzer setting on each of three separate photons. To compare all four combinations, the experiment prepares new triples and changes the analyzer choices between trials. The answers at different settings therefore come from different photon samples. The local instruction-table argument asks whether one prearranged list could account for all those samples.

For the ideal state, each setting allows four joint outcomes, each with probability one quarter. Their products have the same sign, while either result at any one station is still equally likely. The certainty belongs to the product of three random answers. None of the stations can use its own result to send a controllable message to another.

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

Imperfect data need a score with a local limit

Real data include wrong parities, drift and uncertainty. Our enumeration also shows that, with the four settings chosen uniformly, local instructions can succeed in at most three settings out of four. Equivalently, the signed sum of correlations is at most two, while the ideal quantum state gives four. This score is our own calculation, not a fit to historical data.

In a teaching model where the state is sixty percent ideal GHZ and the rest white noise, each required parity comes out right eighty percent of the time, and the sum becomes 2.4. That number alone does not establish experimental significance. The right statistical test depends on how trials are defined, whether settings are chosen independently, when data-taking stops and the actual count of each outcome.

Undetected events limit the conclusion

Requiring a fourfold coincidence keeps only a small fraction of the source’s attempts. Applying the usual local-model bound to that selected subset needs an assumption such as fair sampling, or a trial design that counts nondetections explicitly. A trigger alone does not close the detection loophole, and locality needs its own check of spacetime separation and setting choices.

The selected parity correlations test multiparticle predictions without determining the whole three-photon state. The historical achievement is conditional multiparticle entanglement and a test of its nonlocal correlations under the stated detection and timing assumptions.

Sources and reading notes

We read the five-page 1999 PRL and the original GHZ paper, available as arXiv:0712.0921. For the 2000 Nature experiment we could see only the publisher’s abstract and figure captions; the full text requires a subscription. The positive-phase three-qubit convention, the full list of instruction tables and the complex-amplitude checks described here are our own teaching calculations; they do not reconstruct the historical counts. Exact convention for the displayed formula: zero is horizontal and one vertical polarization. The state is the equal, positive superposition of all-zero and all-one. The positive X outcome is the equal, same-phase horizontal–vertical combination; the positive Y outcome has a positive real horizontal amplitude and a positive imaginary vertical amplitude of the same magnitude. The negative Y outcome has the opposite imaginary relative phase. This defines the signs without choosing ambiguous left- and right-handed names. The 1999 paper reverses the third photon’s polarization in its two components; a local bit flip on that arm gives the convention used here. Changing the state phase or an outcome label changes the corresponding parity signs and must be handled consistently. The four global three-photon product observables commute, but X and Y on one photon are incompatible; their global commutation does not assign simultaneous local outcomes. Our amplitude calculation sums all eight joint alternatives for each setting: four have probability one quarter each, and the wrong parity has zero probability. The four selected parity correlations do not determine every entry of the three-qubit density matrix.