CHSH inequality

The CHSH inequality is the practical, experiment-ready form of Bell's idea, introduced in 1969 by Clauser, Horne, Shimony, and Holt. Where Bell's original argument needed perfect correlations that real apparatus could never deliver, CHSH works with imperfect, realistic measurements. Each of two distant observers chooses between just two settings and records a yes-or-no result, and a single combined quantity is built from the four resulting correlation averages.

For any local hidden-variable theory, that combined quantity cannot exceed two in size. Quantum mechanics, using a suitable entangled state and well-chosen measurement angles, predicts a value as high as two times the square root of two, about 2.83 — the Tsirelson bound. The gap between these two numbers is exactly what a Bell test measures, turning a deep question about the nature of reality into a single number to compare against the limit of 2.

Because it is robust, simple, and tolerant of real-world noise, CHSH has become the standard workhorse of the field. Essentially every modern Bell test reports a CHSH value, and the inequality also underpins applications such as device-independent randomness generation and cryptography, where a verified CHSH violation certifies that genuine quantum behaviour — and not a hidden classical trick — is at work, without needing to trust the inner workings of the devices.