Leggett-Garg inequality

In plain terms

Bell inequalities need two separated particles, so a sceptic can file the whole conflict with common sense under strange facts about distance. Leggett and Garg asked the question with a single particle and no distance at all, which is whether a system has definite properties in between the moments when you look at it, and whether you can look without disturbing it.

Grant those two assumptions, a definite history and gentle measurement, and the correlations between measurements made at different times, the degree to which one result predicts the next, obey a strict bound. Quantum mechanics breaks the bound, and experiments agree with quantum mechanics.

So one of the two assumptions has to go, as either properties aren't definite between measurements or measurement is never really gentle. The inequality can't tell you which, and every account of quantum mechanics has to choose.

In CSD

This test aims closer to CSD's heart than Bell does, which is exactly why the programme's machine-checked corpus carries it. A theory built on an underlying substrate practically advertises definite properties between measurements, as that's what having a substrate means, so an inequality that targets precisely this assumption probes the programme where it's most exposed.

CSD's escape is through the second assumption, and the escape is principled rather than convenient. Measurements in CSD are physical interactions that produce records, and interactions aren't gentle. The underlying state is definite at every moment, and the act of reading it participates in the dynamics. A definite history with invasive measurement violates nothing in the argument, and that's exactly the combination CSD is built from.

Mathematically

The inequality bounds combinations of two-time correlation functions. It follows from macrorealism plus non-invasive measurability, and quantum mechanics violates it.

Both hypotheses are load-bearing, so an account with definite properties but invasive measurements is untouched, and that's the compartment this programme occupies, stated rather than implied.

The name

Anthony Leggett, born in London in 1938, read Greats, the Oxford classics course, before switching to physics, a route that no longer exists. He took his doctorate under Dirk ter Haar, spent twenty years at Sussex, and moved to the University of Illinois at Urbana-Champaign in 1983. His 2003 Nobel Prize was for superfluid helium-3.

The inequality, with his student Anupam Garg in 1985, wasn't foundations tourism. Leggett has argued seriously for decades that quantum mechanics might actually fail at macroscopic scales, and the inequality was built as the instrument to catch it failing. The experiments keep siding with quantum mechanics, which Leggett, to his credit, keeps reporting.

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.
• 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.
• Kochen-Specker theoremThere's a finite list of directions in space that admits no consistent yes-or-no labelling, so contextuality follows by pure counting, with no experiment, no statistics and no separated particles.
• Single trajectoryThere's one world and it follows one path, so probability lives in how experiments are repeated and not in how the world evolves, and theorems rather than wording keep the two apart.
• Hidden variablesThe no-go theorems don't kill hidden variables, as they price them, and CSD reads the price tag and pays it.