Bell Inequalities & CHSH

Bell Inequalities & the CHSH Test

Bell's theorem (1964) proves that no theory of local hidden variables can reproduce all predictions of quantum mechanics. The CHSH inequality (Clauser–Horne–Shimony–Holt) provides a practical experimental test.

Setup

Alice and Bob each receive one qubit of a shared Bell pair. They independently choose measurement angles from sets $\{a, a'\}$ and $\{b, b'\}$ respectively. Each measurement outcome is $\pm 1$.

The correlation function for angles $a$ and $b$ is:

$E(a, b) = \langle A(a)\, B(b) \rangle$

where $A(a), B(b) \in \{+1, -1\}$ are the measurement results.

The CHSH Inequality

Any local hidden variable (LHV) theory must satisfy:

$|S| = |E(a,b) - E(a,b') + E(a',b) + E(a',b')| \leq 2$

This is the classical bound. If $|S| > 2$, the correlations cannot come from any LHV theory.

Quantum Prediction

For a maximally entangled Bell pair, the quantum correlation function is:

$E(a, b) = -\cos\!\bigl(2(a - b)\bigr)$

The Tsirelson bound gives the maximum quantum value:

$|S|_{\text{max}} = 2\sqrt{2} \approx 2.828$

This is achieved with the optimal angle settings:

$a = 0,\quad b = \frac{\pi}{8},\quad a' = \frac{\pi}{4},\quad b' = \frac{3\pi}{8}$

Verifying the Optimal CHSH Value

At these angles:

$E\!\left(0, \tfrac{\pi}{8}\right) = -\cos\!\left(-\tfrac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}$

$E\!\left(0, \tfrac{3\pi}{8}\right) = -\cos\!\left(-\tfrac{3\pi}{4}\right) = +\frac{1}{\sqrt{2}}$

$E\!\left(\tfrac{\pi}{4}, \tfrac{\pi}{8}\right) = -\cos\!\left(\tfrac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}$

$E\!\left(\tfrac{\pi}{4}, \tfrac{3\pi}{8}\right) = -\cos\!\left(-\tfrac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}$

$S = \left(-\frac{1}{\sqrt{2}}\right) - \left(+\frac{1}{\sqrt{2}}\right) + \left(-\frac{1}{\sqrt{2}}\right) + \left(-\frac{1}{\sqrt{2}}\right) = -2\sqrt{2}$

$|S| = 2\sqrt{2} \approx 2.8284 \quad \checkmark$

Physical Significance

Experiments (Aspect 1982, and many since) consistently measure $|S| \approx 2.8$, ruling out local hidden variables. Quantum entanglement is not just a computational trick — it is a fundamental feature of nature that cannot be explained classically.

Your Task

Implement:

1. bell_correlation(a, b) — return $E(a, b) = -\cos(2(a - b))$
2. chsh_value(a, b, a_prime, b_prime) — return $S = E(a,b) - E(a,b') + E(a',b) + E(a',b')$