Bell Inequalities and the CHSH Game

Bell Inequalities and the CHSH Game

Every entanglement-based protocol you are about to meet rests on one statistical test. Before E91 and device-independent QKD can use it, you need to see how it works: what the CHSH quantity S is, why classical physics caps it at 2, and why entangled pairs push past that cap.

The Instruction-Set Picture

Entangled particles produce correlated outcomes even when measured on opposite sides of a continent. The common-sense explanation, and the one Einstein championed, is that the outcomes were decided in advance. When the pair was created, each particle carried away a copy of the same instruction list: "if measured at angle X, answer +1; if measured at angle Y, answer −1." The particles never need to communicate; they just read from the shared list. Physicists call this a local hidden variable model: local because each particle answers using only its own list, hidden because nobody ever sees the list directly.

Instruction sets explain ordinary correlations perfectly well. Mail a pair of gloves in two separate boxes: finding the left glove in one box tells you instantly that the other holds the right one, because the answer was packed in from the start. The question John Bell asked in 1964, which Clauser, Horne, Shimony, and Holt (CHSH) sharpened into a practical test in 1969, is whether any instruction list, however elaborate, can reproduce what entangled pairs actually do.

The CHSH Setup

A source sends one particle to Alice and one to Bob. In each round:

After many rounds they compare notes and compute, for each of the four setting combinations, the correlator E(x, y): the average of the product of their outcomes. E = +1 means they always agree, E = −1 means they always disagree, E = 0 means no correlation at all. Then they combine the four correlators into a single number: