No-signalling

In plain terms

Entangled particles are pairs prepared together so that measurements on them show correlations no local account explains. The first thing every newcomer asks is whether those correlations let someone send a message faster than light, that is, whether a choice made at one particle can be read off at the other. Quantum mechanics says no, the principle has a name, no-signalling, and a programme that reconstructs quantum mechanics should say no as a theorem rather than as an apology.

In CSD it's a theorem, twice over. The first version is about composite arenas, the state spaces CSD assigns to two systems considered together. Disturb one part however you like, by any operation confined to that part, and every measurable quantity of the other part keeps exactly the same value, so the change cancels identically and not approximately. The second version is about the Bell model, as the explicit contextual model that reproduces the singlet statistics has remote marginals, the outcome frequencies seen on one side alone, that provably don't depend on what's done to the other side.

Correlation without signalling is therefore not a tension to be managed. It follows from what an operation confined to one part can and can't touch.

In CSD

In CSD, no-signalling is a fact about where operations act and not a constraint placed on probabilities by hand. An operation on the right-hand part of a composite arena is built from the right-hand modes only (a mode is one of the elementary degrees of freedom from which the arena is assembled), and every observable of the left-hand part is built from the left-hand modes only. The two act independently of each other, so applying the operation and then reading the left-hand observable gives the same answer as reading it alone, and because the argument never mentions which state the composite is in, it holds for every state, entangled states included.

The version for the Bell model matters separately. Contextual models, in which outcomes depend on the measurement context, are exactly the kind of model critics suspect of hidden signalling, so the corpus proves the operational no-signalling conditions for its exhibited model rather than gesturing at them.

Mathematically

The theorem composite_no_signalling states that on the composite field arena, conjugating by any unitary supported on one sector leaves every observable of the other sector's subalgebra exactly invariant, for every composite state, as the kick commutes past every observable supported on the other sector.

The contextual-model version proves remote marginal invariance for both wings, with one shared measure across contexts, which is measurement independence stated in- module. Both results are foundational-triple-only.

Related entries

• 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.
• Bell state is not a joinThe arena CSD assigns to two systems together is strictly larger than the two arenas simply paired, and the Bell state is a point of the excess, so entanglement is located exactly.
• IsolationBeing isolated isn't scene-setting, as it's the variable the whole theory turns on, and quantum mechanics is what isolated pieces of the arena look like from outside.
• Field arenaTake finitely many field modes, cap the number of quanta each can hold, and the whole toolbox of quantum field theory, propagators, light cones, Wick's theorem and thermal states, reappears as exact theorems on one finite arena.
• 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.