Recap: Bell Nonlocality

Bell nonlocality is the sharpest break between quantum theory and any classical worldview built on local hidden variables. This module develops it into a working tool — nonlocal games, Tsirelson's bound, self-testing, certified randomness — but first we recall the theorem precisely, because every later result is a quantitative refinement of it.

The setup

Two separated parties, Alice and Bob, each receive a system from a common source. Alice freely chooses a measurement setting $x$ and records an outcome $a$ ; Bob chooses $y$ and records $b$ . After many rounds they tabulate the joint conditional distribution $p(a,b\mid x,y)$ . Crucially, the choices $x$ and $y$ are made after the systems separate, so no signal travelling at or below the speed of light can inform one party's apparatus of the other's setting.

The locality assumption

A local hidden-variable (LHV) model posits a shared variable $\lambda$ , distributed as $q(\lambda)$ , that was fixed at the source. Given $\lambda$ , each party's outcome depends only on its own setting:

p(a,b\mid x,y) = \int d\lambda\; q(\lambda)\, p_A(a\mid x,\lambda)\, p_B(b\mid y,\lambda).

Two physical principles are encoded here. Locality (factorisation): once $\lambda$ is known, Alice's statistics are independent of Bob's setting $y$ and vice versa. Measurement independence: the distribution $q(\lambda)$ does not depend on the freely chosen settings $x,y$ . Any theory respecting these — including all of classical physics — produces correlations of exactly this form.

The CHSH inequality

With binary settings $x,y\in\{0,1\}$ and binary outcomes $a,b\in\{+1,-1\}$ , define the correlator $E(x,y)=\langle ab\rangle$ and the CHSH quantity

S = E(0,0) + E(0,1) + E(1,0) - E(1,1).

For any LHV model each outcome is a definite function $a_x,b_y\in\{\pm1\}$ of $\lambda$ , so

a_0 b_0 + a_0 b_1 + a_1 b_0 - a_1 b_1 = a_0(b_0+b_1) + a_1(b_0-b_1).

Exactly one of $b_0+b_1$ and $b_0-b_1$ is $\pm2$ while the other is $0$ , so the right-hand side is $\pm2$ for every $\lambda$ . Averaging over $q(\lambda)$ cannot enlarge a quantity bounded by $2$ in magnitude, giving the CHSH inequality

|S| \le 2 \qquad \text{(all local hidden-variable theories).}

What quantum mechanics does

Take the singlet $|\psi^-\rangle = \tfrac{1}{\sqrt2}(|01\rangle - |10\rangle)$ and measure spin along directions separated by angle $\theta$ . The quantum correlator is $E = -\cos\theta$ . Choosing Alice's settings at $0^\circ, 90^\circ$ and Bob's at $45^\circ, 135^\circ$ makes each of the four terms equal in magnitude to $\tfrac{1}{\sqrt2}$ , and

S = 2\sqrt2 \approx 2.828.

This violates the classical bound. No assignment of pre-existing values $\lambda$ can reproduce quantum correlations: this is Bell's theorem. Nature is not describable by any local hidden-variable model.

Why this matters for the module

Bell's theorem is usually presented as a no-go result about realism. The modern view, developed in this module, is constructive: a Bell violation is an operational resource. Because $S>2$ is impossible classically, observing it certifies — without trusting the internal workings of any device — that genuine quantum behaviour is present. That single idea drives device-independent certification, randomness expansion, and the security proofs we build next.

The takeaway

A local hidden-variable model forces $|S|\le 2$ ; quantum mechanics reaches $S=2\sqrt2$ on a maximally entangled pair. The gap between these numbers is not a curiosity — it is the quantitative budget that the rest of this module spends.