Mermin inequality

In plain terms

A natural guess is that quantum strangeness is a small-systems phenomenon, and that as you add particles it washes out into classical behaviour as averages of averages dilute the strangeness.

Mermin's inequalities say the opposite happens. Start from the Bell experiment, in which two separated particles are measured and their results compared, and add parties, three particles, four, ten. For any number of parties there's a bound every common-sense account must respect, meaning any account in which each particle carries its own definite answers and nothing influences anything at a distance. Quantum mechanics predicts correlations that exceed the bound, and the gap between what it delivers and what the bound allows doesn't shrink as parties are added, as it grows exponentially with the number of particles, doubling roughly every two parties added.

So the conflict with local realism, the combination of definite local properties and no action at a distance, isn't a fragile corner case of tiny systems. It's a structural feature that becomes more pronounced with size, which is the reverse of what the wash-out intuition predicted.

In CSD

CSD's machine-checked corpus carries the multi-party inequalities so that its results on contextuality, the dependence of measurement outcomes on what else is measured alongside them, aren't hostage to the smallest case, and the exponential growth is the actual point of doing so. A substrate mechanism that happened to reproduce the two-party violation by some special trick would be exposed here, as whatever produces the correlations has to keep producing them, and more strongly, as parties are added.

Together with the GHZ argument, which reaches the same conflict through a single outright contradiction rather than through statistics, this establishes the conflict at every size and in both logical shapes, statistical and deterministic. That's the standard a substrate account must meet, and the corpus states the standard before claiming to meet it.

Mathematically

A family of multipartite Bell inequalities whose classical bound is exceeded by quantum mechanics by a factor growing exponentially in the number of parties, 2 to the power (n-1)/2 for n parties.

The corpus proves the quantum value and the classical bound separately, which keeps the exponential gap a theorem rather than a plot.

The name

David Mermin, born in 1935, took his degrees at Harvard and spent his career at Cornell, much of it directing the Laboratory of Atomic and Solid State Physics. With Ashcroft he wrote the solid-state textbook that trained a generation.

He's that rare thing, a physicist whose prose is part of his scientific reputation. The phrase shut up and calculate is his, coined in 1989 as a description of an attitude he was criticising, then promptly attributed to Feynman and quoted with approval, a fate he has written about with resignation. The multi-party inequalities date from 1990, in a paper short enough to read over coffee, which turned the GHZ conference argument into something an experiment could score.

Related entries

• GHZ stateThree particles can contradict local realism with certainty in a single run, with no statistics, no inequality and no margin of error.
• 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.
• CGLMP inequalityBell's inequality is built for measurements with two outcomes, so once you go beyond two you need a new inequality, and the conflict with local realism grows stronger rather than weaker.
• 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.