Bell’s Theorem

In 1964 John Bell asked a precise question: can the local hidden-variable framework reproduce all the correlations of an entangled pair, for every choice of measurement settings? His answer — no — is one of the most important results in the foundations of physics. It converts a philosophical dispute into an experimental one.

The statement

Bell's theorem. No local hidden-variable theory can reproduce all the statistical predictions of quantum mechanics. There exist measurement settings for which any local-realistic model obeys an inequality that quantum mechanics violates.

In other words, local realism is not merely unproven — it is incompatible with quantum mechanics, and the disagreement is quantitative and testable.

The original Bell inequality (singlet correlations)

For the spin singlet, quantum mechanics predicts the correlation between Alice's axis $\hat a$ and Bob's axis $\hat b$ is

E_{\text{QM}}(\hat a, \hat b) = -\,\hat a \cdot \hat b = -\cos\theta_{ab},

where $\theta_{ab}$ is the angle between the axes. Bell showed that any LHV correlation of the form $E(a,b) = \int \rho(\lambda) A(a,\lambda) B(b,\lambda)\,d\lambda$ (with $A,B = \pm 1$ and the perfect anticorrelation $A(a,\lambda) = -B(a,\lambda)$ ) must satisfy, for three settings $a, b, c$ ,

\big|\,E(a, b) - E(a, c)\,\big| \;\le\; 1 + E(b, c).

Where the inequality comes from

Using $B(b,\lambda) = -A(b,\lambda)$ and $A^2 = 1$ , write

E(a,b) - E(a,c) = -\!\int \rho\,A(a)\big[A(b) - A(c)\big]\,d\lambda = -\!\int \rho\,A(a)A(b)\big[1 - A(b)A(c)\big]\,d\lambda,

where the last step inserts $A(b)^2 = 1$ . Since $|A(a)A(b)| = 1$ and the bracket $1 - A(b)A(c) \ge 0$ , taking absolute values gives

\big|E(a,b) - E(a,c)\big| \le \int \rho\big[1 - A(b)A(c)\big]\,d\lambda = 1 + E(b,c),

using $E(b,c) = \int \rho\, A(b)B(c) = -\int \rho\, A(b)A(c)$ . That is the Bell inequality — a hard constraint obeyed by every local-realistic model.

Quantum mechanics breaks it

Pick coplanar axes with $\theta_{ab} = \theta_{bc} = 60^\circ$ and $\theta_{ac} = 120^\circ$ . The quantum correlations are $E(a,b) = -\cos 60^\circ = -\tfrac12$ , $E(b,c) = -\tfrac12$ , and $E(a,c) = -\cos 120^\circ = +\tfrac12$ . Plug in:

\big|E(a,b) - E(a,c)\big| = \big|{-\tfrac12} - \tfrac12\big| = 1, \qquad 1 + E(b,c) = 1 - \tfrac12 = \tfrac12.

The inequality demands $1 \le \tfrac12$ — false. Quantum mechanics violates Bell's inequality. No assignment of predetermined $\pm 1$ values to all three axes can reproduce these numbers.

From principle to experiment

Bell's original inequality assumes perfect anticorrelation, which real detectors never achieve. The CHSH inequality of the next lessons is the experiment-ready descendant: it needs no perfect correlations, uses two settings per side, and is what every modern Bell test actually measures. We turn to it now.