GHZ state

In plain terms

Bell's argument has one feature a determined sceptic can pick at, which is that it's statistical. No single run of a Bell experiment is impossible on the common-sense picture, in which every measurement has an answer fixed in advance, as the runs are only collectively improbable. You need thousands of runs and error bars, and someone can always mutter about statistics.

Greenberger, Horne and Zeilinger found a way to remove the error bars. With three particles instead of two, prepared together in what's now called a GHZ state, there's a set of measurements for which the common-sense picture predicts one definite answer and quantum mechanics predicts the opposite definite answer, so the two predictions aren't different averages but opposite certainties.

One run settles the matter, in principle, and the conflict stops being about probability and becomes arithmetic. Each measurement gives plus one or minus one, and if you multiply out the answers the common-sense picture demands the product must be plus one, while quantum mechanics says minus one. Something has to give, and experiment says it's the answer sheet.

In CSD

The corpus carries the three-party obstruction alongside the two-party one for a specific defensive reason, which is that the two have different logical shapes and an account built on an underlying substrate has to survive both. You could imagine a mechanism that reproduces the statistical Bell violations through some averaging trick but fails on a deterministic contradiction, and carrying the GHZ result closes that escape.

The proof quantifies over arbitrary spaces of hidden states rather than examining one model, which is what makes it a constraint on the whole class of accounts in which the hidden description splits into a separate part for each particle, including naive rewrites of this programme, and that's the point.

Mathematically

No product partition of a shared hidden-state space reproduces the GHZ correlations, proved over an arbitrary measurable state space.

The contradiction is algebraic. The four predicted outcome-sign products can't be consistently assigned, as multiplying the local predictions forces plus one where the quantum state forces minus one. No inequality is involved and no bound is approached, so the two predictions simply differ, which is why a single ideal run distinguishes them.

The name

Daniel Greenberger of City College New York, Michael Horne (1943-2019) of Stonehill College, and Anton Zeilinger of Vienna and Innsbruck introduced the argument in 1989. Horne, as Shimony's graduate student at Boston University, had already co- authored the CHSH inequality.

The publication story is odd. The result appeared in the proceedings of a conference at Erice rather than in a journal, and it spread mostly by word of mouth until Mermin wrote it up and made it famous. Zeilinger shared the 2022 Nobel Prize for entanglement experiments, and Horne had died three years earlier.

Related entries

• Bell and CHSHNo pre-written answer sheet for all four measurement settings can match what a singlet pair of particles does, and the corpus proves this as a theorem about every possible answer sheet.
• Mermin inequalityAdd more particles and the conflict between quantum mechanics and common sense grows exponentially, so it doesn't average out but compounds.
• ContextualityFor some sets of quantum questions no consistent answer sheet exists at all, and CSD's response is to stop demanding one answer per question and to say instead what the answer depends on.
• Hidden variablesThe no-go theorems don't kill hidden variables, as they price them, and CSD reads the price tag and pays it.
• No-signallingNothing done to the far wing of an entangled pair changes the statistics seen at the near wing, and CSD proves this exactly, for every state, entangled ones included.