CGLMP inequality

In plain terms

A Bell inequality is a bound on how strongly the results of measurements on two separated systems can be correlated if every result is fixed in advance and no influence travels between the systems, a picture called local realism. Quantum mechanics breaks the bound, and experiments agree with quantum mechanics. Bell's argument, as usually stated, assumes each measurement has exactly two possible results, while real systems often have three, or five, or any number, so you can ask whether the conflict with local realism survives when outcomes multiply or whether it's an artefact of the two-outcome case.

It survives, and it takes a new inequality to say so, the CGLMP inequality, one for each number of outcomes. The surprise is the direction of travel, as the violation grows with the number of outcomes, so higher-dimensional systems disagree with local realism more strongly and not less.

There was a second surprise in the same paper. The state that violates the inequality most isn't the maximally entangled one, so the state that's most entangled by the standard bookkeeping isn't the state most at odds with common sense, which is a warning that entanglement isn't a single number.

In CSD

The corpus carries the inequality so that its contextuality results don't lean on two-level systems, which are special in ways that keep surprising people. There's also a structural echo worth noticing. Three outcomes is precisely where the corpus's own fibre analysis finds the natural base-only constructions stop working, which is why it places contextual content in the fibre above the base, and that placement is a judgement against a tight chain of constraints and not a proved dichotomy.

So the dimension where this inequality becomes necessary is the dimension where the programme's geometry changes character. That's either a coincidence or a hint, and the honest label for now is coincidence.

Mathematically

A Bell-type inequality for bipartite systems with d outcomes per side, violated by quantum mechanics for every d at least two, with the maximal violation increasing in d.

The maximising state isn't maximally entangled, which remains one of the sharpest reminders that entanglement measures and non-locality measures order states differently.

The name

Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar and Sandu Popescu published the inequality in Physical Review Letters in 2002, with Gisin at Geneva, Popescu and Linden at Bristol, Massar in Brussels, and Collins a Bristol student. Five authors across four institutions for four pages of paper is a snapshot of how the European quantum information community actually worked in those years, by email and short visits, with the inequality named for the initials of whoever was on the manuscript when it converged.

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.
• Tsirelson boundQuantum mechanics breaks the classical limit on correlations and then stops well short of what logic alone would allow, and why it stops exactly there is a genuinely open question.
• Mermin inequalityAdd more particles and the conflict between quantum mechanics and common sense grows exponentially, so it doesn't average out but compounds.
• Gisin's theoremEvery entangled pure state, however slightly entangled, violates a Bell inequality, so there's no quiet zone.
• 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.