Bell's Theorem and Local Hidden Variables

For decades the open question was not whether nature violates a Bell inequality — experiments since Freedman and Clauser in 1972 had seen it — but whether one experiment could do so while closing every escape hatch at once. Two loopholes mattered most. The locality loophole asks whether a subluminal signal could have coordinated the two wings; closing it demands that each party's setting be chosen and the outcome recorded within the light-cone gap, so the settings are effectively space-like separated. The detection loophole asks whether the detected events are an unrepresentative sample; closing it demands detection efficiency above a threshold near $2/3$ for CHSH. In 2015 three groups closed both together: a Delft experiment using entanglement swapping between nitrogen-vacancy centres $1.3$ km apart, and photonic experiments at NIST and in Vienna reaching very high efficiency with fast random-number generators setting the analyzers. Local realism, as a description of nature, is finished. The technology built on top is now the point. Because $S>2$ certifies genuine entanglement and intrinsic randomness without trusting the devices' internals, Bell violations underwrite device-independent quantum key distribution and certified random-number generation — security and randomness guaranteed by the observed statistics alone, even against hardware you did not build. The whole enterprise reached its symbolic capstone with the 2022 Nobel Prize in Physics to Alain Aspect, John Clauser and Anton Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science."

Picture each particle leaving the source with a sealed instruction sheet $\lambda$. The sheet specifies, in advance and for every dial setting the experimenter might select, exactly what that detector will read: $+1$ or $-1$. Locality is the demand that Alice's sheet mention only Alice's dial, never Bob's, and vice versa — the two sheets were written together at the source but neither can consult the other's setting once the particles fly apart. Bell's question is disarmingly simple: across the four ways of pairing two settings each, how strongly can two such pre-written sheets be made to agree? The answer is a hard cap. Averaged over any distribution of sheets, the CHSH combination of four correlations cannot exceed $2$. Entangled particles break the cap. They are not carrying sheets; the joint outcome is decided only when both measurements happen, and the correlation they produce, $E(\theta)=-\cos\theta$ for the singlet, bends more sharply near $\theta=0$ than any convex mixture of straight-line local strategies can imitate. The first figure lays out the four optimal measurement axes and totals the correlators to the quantum maximum; the second places the three regimes — local, quantum, super-quantum — on a single axis of $S$.

Determinism is not the crux, though it is the easiest picture. Stochastic local models — where the sheet gives only probabilities $P(a|x,\lambda)$ rather than fixed answers — obey exactly the same bound, because one can always fold the local coin-flips into an enlarged hidden variable and recover a deterministic response. What genuinely matters is that the joint distribution factorizes, $P(a,b|x,y)=\int\mathrm{d}\lambda\,\rho(\lambda)\,P(a|x,\lambda)\,P(b|y,\lambda)$. Geometrically the set of all such local correlations is a convex polytope — the local polytope — and Bell inequalities are literally its bounding faces. Quantum correlations form a strictly larger, curved convex set capped by Tsirelson's bound; the no-signalling correlations form a still larger polytope whose extreme points are the hypothetical PR boxes. The second figure is that nested ladder made quantitative.

The starting point is the mathematical translation of "locality plus realism." A complete state $\lambda$ is shared at the source with density $\rho(\lambda)$; given $\lambda$, each party's outcome distribution depends only on its own setting. The joint probability then factorizes.

where: $a,b$ — outcomes of Alice's and Bob's measurements, taken as $\pm 1$; $x,y$ — their freely chosen settings; $\lambda$ — shared hidden variable, the common cause or instruction set; $\rho(\lambda)$ — probability density, $\int\mathrm{d}\lambda\,\rho(\lambda)=1$; $P(a\mid x,\lambda)$ — Alice's local response, crucially independent of Bob's setting $y$. Reading it: all correlation between the wings must be routed through $\lambda$; nothing else links them. For deterministic strategies the response collapses to a definite $\pm1$ function, and the correlation function is the average of the product of outcomes.

where: $E(x,y)$ — expectation of the product $\langle ab\rangle$; $A(x,\lambda)$ — Alice's outcome for setting $x$ under hidden state $\lambda$; $B(y,\lambda)$ — Bob's outcome for setting $y$. Reading it: stochastic models lose no generality, since local randomness can be absorbed into $\lambda$ (Derivation 1 uses this). The CHSH combination packages four such correlators — two settings each — into a single testable number, with one term subtracted.

where: $x_1,x_2$ — Alice's two settings; $y_1,y_2$ — Bob's two settings; $S$ — the CHSH quantity. Reading it: the lone minus sign is what makes the bound nontrivial; with four plus signs the maximum would simply be $4$ for everyone. The Bell–CHSH inequality is the ceiling every local hidden-variable model obeys.

where: $2$ — the classical bound, saturated by deterministic local strategies. Reading it: this single inequality, derived next, is the entire empirical content of "local realism" for this experiment. Quantum mechanics computes the correlators from the state. For the spin singlet measured along unit axes $\hat a,\hat b$, the correlation is the negative cosine of the angle between them.

where: $|\psi^-\rangle=\tfrac{1}{\sqrt2}(|01\rangle-|10\rangle)$ — the spin singlet; $\hat a,\hat b$ — measurement axes; $\vec\sigma=(\sigma_x,\sigma_y,\sigma_z)$ — Pauli vector; $\theta_{ab}$ — angle between the axes. Reading it: the correlation depends only on the relative orientation, and it is perfect anti-correlation ($-1$) when the axes align. Feeding (5) into (3) at the optimal angles gives the maximal quantum value.

where: $2\sqrt{2}$ — Tsirelson's bound. Reading it: quantum mechanics beats the classical ceiling by exactly the factor $\sqrt{2}$, and cannot do better (Derivation 3). To bound quantum mechanics itself, promote the correlators to Hermitian observables and form the CHSH operator, whose expectation value is $S$.

where: $A_i,B_j$ — Hermitian observables with eigenvalues $\pm1$ (e.g. $\hat n\cdot\vec\sigma$); $\mathcal{B}$ — the CHSH operator, with $S=\langle\mathcal{B}\rangle$. Reading it: turning a sum of numbers into an operator lets us bound $S$ over all states at once by bounding the operator norm. Finally, none of this permits communication. Marginalizing over one party's outcome must leave the other's statistics untouched.

where: $\sum_a$ — marginal over Alice's outcome; $x,x'$ — any two of Alice's settings. Reading it: Bob's local data cannot reveal which dial Alice turned, so a Bell violation carries correlation but not a signal.

The CHSH value $S$ sorts the three nested correlation sets and tells you which physical regime a given experiment inhabits.

1 · Totalling the singlet to the Tsirelson bound (fully worked)

Take the singlet $|\psi^-\rangle$ and the axes $a_1=0°,\,a_2=90°$ for Alice and $b_1=45°,\,b_2=-45°$ for Bob, all coplanar. Compute all four correlators and the CHSH value $S$, and confirm it exceeds the classical bound.

Step 1. Each correlator is $E=-\cos\theta$ with $\theta=|a_i-b_j|$. The relative angles are $\theta_{11}=|0-45|=45°$, $\theta_{12}=|0-(-45)|=45°$, $\theta_{21}=|90-45|=45°$, and $\theta_{22}=|90-(-45)|=135°$.

Step 2. Evaluate the cosines: $\cos45°=1/\sqrt2\approx0.7071$ and $\cos135°=-1/\sqrt2\approx-0.7071$. So $E_{11}=E_{12}=E_{21}=-0.7071$ and $E_{22}=-\cos135°=+0.7071$.

Step 3. Assemble $S=E_{11}+E_{12}+E_{21}-E_{22}=(-0.7071)+(-0.7071)+(-0.7071)-(+0.7071)=-4(0.7071)=-2.8284$.

Step 4. Compare to the bound: $|S|=2.8284=2\sqrt2$, which is $\sqrt2\approx1.414$ times the classical ceiling of $2$. The singlet gives |S| = 2√2 ≈ 2.828 > 2, a clear violation of local realism, and it sits exactly on the Tsirelson bound. The sign of $S$ is a labelling convention; only $|S|>2$ matters.

2 · How much noise the violation survives (Werner state)

Real sources are imperfect. Model the state as a singlet mixed with white noise, $\rho=p\,|\psi^-\rangle\langle\psi^-|+(1-p)\,\mathbb 1/4$, where the visibility $p\in[0,1]$. For what $p$ does the state still violate CHSH, and what visibility does an experiment need?

Step 1. Correlators are linear in the state. The maximally mixed part $\mathbb 1/4$ has no correlation, $E=0$, while the singlet part gives $-\cos\theta$. So the Werner correlator is $E(\hat a,\hat b)=-p\cos\theta_{ab}$ — the same shape, scaled by $p$.

Step 2. Every correlator in $S$ therefore shrinks by the factor $p$, so at the optimal angles $S_{\max}=2\sqrt2\,p$.

Step 3. Demand a violation, $S_{\max}>2$: this needs $2\sqrt2\,p>2$, i.e. $p>1/\sqrt2\approx0.7071$. A CHSH violation requires visibility above 70.7%. At exactly $p=1/\sqrt2$ the state sits on the classical bound, $S=2$.

Step 4. A caution that ties back to the warning: the Werner state is entangled for all $p>1/3$, and admits a local hidden-variable model for projective measurements up to about $p\approx0.66$. Between $p\approx0.66$ and $p=0.707$ the state is entangled and has a local model yet does not violate this particular inequality — a reminder that CHSH is a sufficient, not necessary, test of nonlocality. Entanglement alone (p > 1/3) does not guarantee a Bell violation; you need p > 0.707.

Bell's theorem is the rare foundational result that began as a "no-go" and became a working instrument. The man who proved it spent his day job designing accelerators at CERN and did quantum foundations on the side.

Texts. Bell, J. S., "On the Einstein Podolsky Rosen Paradox", Physics 1, 195 (1964) · Clauser, Horne, Shimony, Holt, Phys. Rev. Lett. 23, 880 (1969) · Nielsen & Chuang, Quantum Computation and Quantum Information, §2.6 · Preskill, Lecture Notes on Quantum Computation, Ch. 4 · Mermin, "Is the moon there when nobody looks?", Physics Today 38, 38 (1985) · Wilde, Quantum Information Theory, Ch. 4 · Brunner et al., "Bell nonlocality", Rev. Mod. Phys. 86, 419 (2014) · Hensen et al., "Loophole-free Bell inequality violation", Nature 526, 682 (2015).

Provenance. A verifier should reconstruct the $\pm1$ algebraic identity giving $|S|\le2$ (Derivation 1); reproduce $E=-\cos\theta$ for the singlet and confirm the axes $a_1=0°,\,a_2=90°,\,b_1=45°,\,b_2=-45°$ give $|S|=2\sqrt2$ (Derivation 2 and Worked example 1 — note the corrected fourth axis $b_2=-45°$, not $135°$, which is what makes the four correlators add constructively); and verify Tsirelson's operator bound via $\mathcal B^2=4\,\mathbb 1-[A_1,A_2]\otimes[B_1,B_2]$ with commutator norm $\le2$ (Derivation 3). Cross-check against Nielsen & Chuang §2.6, Preskill Ch. 4, and CHSH (1969). The Werner threshold $p>1/\sqrt2\approx0.707$ follows from $S_{\max}=2\sqrt2\,p$; the loophole-free 2015 results are Hensen et al. (Delft), Giustina et al. (Vienna) and Shalm et al. (NIST).

Prerequisites. Bell States and EPR Pairs · Separable versus Entangled States · CHSH Inequality and Its Quantum Violation. Leads to. Tsirelson's Bound · Nonlocality and the No-Signalling Principle · PR Boxes and Super-Quantum Correlations.

Ask the tutor: Ask me to walk through why the fourth measurement axis must be −45° and not 135°, or to show how the same S > 2 that kills local realism also certifies randomness you can trust from a black box.