Bell and CHSH

In plain terms

Prepare two particles together in a particular joint state called a singlet, separate them, and measure each one with a choice of two settings. Common sense says every measurement you might make already has an answer waiting, fixed when the pair was created and merely revealed by the apparatus. Bell's insight was that this innocent-sounding assumption has teeth, as it forces a numerical limit on how strongly the results on the two sides can be correlated.

Quantum mechanics predicts correlations above that limit, so common sense and quantum mechanics disagree about an experiment, and the experiment can be run. It has been run for five decades with every loophole closed in turn, and the correlations are there while the answer sheet isn't.

What the corpus proves is the sharp version of this. Take one shared space of hidden states, the imagined complete description of the pair, and no assignment of answers to all four settings at once, however contrived, reproduces the singlet's statistics.

In CSD

CSD is contextual and non-local at the level of its substrate, by design. Contextual means what a measurement reveals can depend on which other measurements are made alongside it, and non-local means the hidden description of the pair doesn't split into a separate part for each particle. That commitment creates a debt, as the programme must show that its own model can't be secretly rewritten as the common- sense picture, or it would be claiming a virtue it hadn't earned. This theorem pays the debt.

The result comes paired. The corpus also exhibits an explicit contextual model that does reproduce the singlet's correlations, so the claim is two-sided, with the impossibility on one hand and the thing that evades it on the other, and the obstruction and the existence are cited together as one unit.

There's a lesson in how the theorem came to be stated. For a long time the formal model gave each measurement setting its own separate space of hidden states, so the Bell question couldn't even be asked, as there was nothing against which to compare a single global assignment of answers. A distinction in the bookkeeping stood where a theorem should have been. Rebuilding the model on one shared space made the question expressible, and the answer turned out to be the theorem above.

Mathematically

Over a single shared measurable state space, no jointly defined assignment of outcomes to the four CHSH settings reproduces the singlet correlations. Companion results rule out a different class rather than restating this one, as no setting- local (product) pair of response functions reproduces the singlet at every setting pair over any shared probability space. Neither result subsumes the other, and the corpus says so. The GHZ analogue for three parties is proved separately.

The shared-space form is the load-bearing one, since it's the shape the contextuality claim actually posits. The corpus pairs the obstruction with an explicit contextual model reproducing the same correlations, so the negative and positive halves are proved together rather than the no-go standing alone.

The name

John Stewart Bell (1928-1990) was born in Belfast to a working-class family and left school at sixteen. He got into physics by working as a laboratory technician at Queen's University, where the staff noticed him and pushed him into a degree. He took his doctorate at Birmingham and spent his career at CERN designing accelerators, and he insisted foundations was a hobby for sabbaticals.

It was some hobby. He wrote the 1964 paper while visiting Stanford, and it appeared in Physics 1, a journal so obscure that it paid authors instead of charging them and which died after a few issues. The paper showed that the hidden-variables question, which physicists had filed under philosophy for thirty years, was experimentally decidable, and Clauser, Horne, Shimony and Holt made it testable in 1969.

Bell died of a stroke in 1990, the year he was nominated for the Nobel Prize, and the 2022 prize went to the experimenters who confirmed him.

Related entries

• 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.
• 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.
• Singlet contextual modelOne machine-checked theorem settles both halves of the Bell question, as an explicit model in which outcomes depend on the measurement context reproduces the singlet pair's statistics in full, and no assignment that ignores the context can.
• Hidden variablesThe no-go theorems don't kill hidden variables, as they price them, and CSD reads the price tag and pays it.
• Does CSD violate Bell's theorem?No. Bell's theorem stands, CSD proves its own versions of it, and CSD escapes its conclusion by the one route the theorem itself leaves open, measurement outcomes that depend on the measurement context, paid for openly.