Reflexia

Reflexia's halves never message each other. Their answers line up anyway, beyond what any messageless plan allows.

Send two particles in the singlet state far apart. At each end an experimenter picks one of two measurement settings and records +1 or −1. Repeat, computing the correlation E for each setting pair. The first formula on this page combines four of those correlations into one number, S.

In 1964 John Bell showed that if each particle carries its own instructions from the start, and nothing done at one end can affect the other, the correlations are limited. The version here is the 1969 form by Clauser, Horne, Shimony and Holt, made for real experiments: S can be at most 2 in size. Quantum mechanics predicts E equal to minus the cosine of the angle between the settings, the third formula. Choose 0° and 90° at one end, 45° and 135° at the other, and the size of S comes out 2√2, about 2.83. Boris Tsirelson proved in 1980 that 2√2 is the most quantum mechanics can ever give.

Experiments side with quantum theory: Freedman and Clauser in 1972; Alain Aspect's group in 1982, switching settings while the photons were in flight; and in 2015 Bas Hensen and colleagues at Delft, with electron spins 1.3 kilometres apart, closing the main loopholes in a single experiment. Aspect, Clauser and Anton Zeilinger shared the 2022 Nobel Prize.

None of this lets anyone send a signal. Each side alone sees random +1s and −1s, whatever the other does. The correlation shows up only when the two records are brought together by ordinary means, no faster than light. Whether something nonlocal links the pair, or another assumption fails, depends on interpretation and is still argued.

Reflexia's halves never talk. They just never disagree like strangers.