Bell's Theorem & the End of Local Realism

In Chapter 20 you met photon pairs whose polarisations are entangled: measure both along the same axis and the results always agree, although each result alone is a fair coin toss. In 1935 Einstein, Boris Podolsky and Nathan Rosen argued that perfect correlations of this kind, in their case between positions and momenta, exposed a gap in quantum mechanics. If you can predict Bob's result with certainty by looking only at Alice's, without disturbing Bob's photon, then Bob's result must have been fixed all along by something his photon carried from the source. Quantum mechanics, which assigns no such value before measurement, would then be incomplete.

That picture is called local realism, and it rests on two pillars. Realism, in this technical sense, says that the outcome of any measurement you might choose is fixed in advance by properties of the system, conventionally written as a hidden variable λ. Locality says that no influence travels faster than light, so Alice's choice of setting cannot alter Bob's result when the two measurements are too far apart for a light signal to link them. A third assumption usually goes unspoken: measurement independence, or free choice, meaning the settings are chosen independently of λ. Picture every photon pair leaving the source with a card listing an answer for every question a detector could ask.

Ordinary correlations work this way. Seal a pair of gloves into two boxes and mail them to opposite ends of the Earth; open one, find a left glove, and you know at once that the other is a right glove. Nothing travelled between the boxes and nothing was undecided. For nearly three decades the dispute looked like a matter of taste, since both pictures predicted the same perfect agreement whenever the detectors were set alike. Wolfgang Pauli, writing to Max Born in 1954, likened such worries to asking how many angels can sit on the point of a needle.

John Stewart Bell, born in Belfast in 1928, spent most of his career on particle theory at CERN and pursued the foundations of quantum mechanics on the side. He had been struck by David Bohm's 1952 hidden-variable theory, which reproduces every quantum prediction using particles with definite positions, but at a price: in Bohm's theory the arrangement of a distant apparatus instantly affects what happens here. Bell asked whether that nonlocality was a quirk of Bohm's construction or unavoidable. His answer, published in 1964 as 'On the Einstein Podolsky Rosen paradox', is now called Bell's theorem.

Bell's theorem states that any local hidden-variable theory must obey an inequality, a ceiling on how strongly results at two distant detectors can be correlated when they are set at different angles. Quantum mechanics predicts correlations that break through that ceiling. The genius lies in the mismatched settings. When the detectors are aligned, a card of pre-written answers easily guarantees agreement. But a card must also carry answers for settings that are never chosen, and a card answering several questions must be internally consistent. That consistency limits the statistics. Quantum mechanics carries no card, so it is not bound by the limit.

Bell's original inequality assumed perfect correlations, which no real detector delivers. In 1969 John Clauser, Michael Horne, Abner Shimony and Richard Holt reworked it into a form that real experiments with imperfect equipment could test, now called the CHSH inequality after their initials. A philosophical dispute became a measurement with photons, polarisers and coincidence counters.

In the CHSH experiment a source sends one photon of an entangled pair to Alice and the other to Bob. Alice sets her polariser at angle α₀ or α₁, Bob at β₀ or β₁, each choosing at random for every pair. A photon either passes, recorded as +1, or is blocked, recorded as −1. For each combination of settings the experimenters compute a correlation E: matching results minus mismatched results, divided by the total. E = +1 means perfect agreement, −1 perfect disagreement and 0 no relationship. The four correlations combine into one number: S = E(α₀, β₀) + E(α₀, β₁) + E(α₁, β₀) − E(α₁, β₁).

Now let local realism speak. Each pair carries a card of four pre-set answers: A₀ and A₁, the results Alice's photon would give at her two angles, and B₀ and B₁ for Bob's, each +1 or −1. For one pair the CHSH combination is A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁, which factors as A₀(B₀ + B₁) + A₁(B₀ − B₁). If B₀ and B₁ are equal, the first bracket is ±2 and the second is zero; if they differ, the roles swap. Every card scores exactly +2 or −2, so any average lies between −2 and +2. Free choice ensures each setting pair samples the same cards, and Bell showed in 1971 that the bound survives even if hidden variables fix only probabilities.

For the entangled photons of Chapter 20 the probability of matching results is cos²(α − β), so E = cos 2(α − β), the cosine of twice the angle between the polarisers. Choose α₀ = 0°, α₁ = 45°, β₀ = 22.5° and β₁ = −22.5°. Three angle differences are 22.5°, each giving E = cos 45° ≈ 0.707; the fourth is 67.5°, giving cos 135° ≈ −0.707, which enters with a minus sign. So S = 4 × 0.707 ≈ 2.83, twice √2. In 1980 Boris Tsirelson proved that no quantum state and no choice of measurements can exceed this value, now called the Tsirelson bound.

Computer scientists recast the CHSH inequality as a cooperative game, the form you will play in today's lab. Alice and Bob sit in separate sealed rooms. Each round, a referee hands each of them a random question, 0 or 1, and each must reply at once with +1 or −1. They win if their answers match, except when both questions were 1, when the answers must differ. They may agree on any strategy beforehand, but once play begins they cannot communicate.

Try to win classically. A fixed strategy is a card listing each player's answers. Winning all four question pairs would need Alice's answer to 0 to equal Bob's answer to 0, which must equal Alice's answer to 1, which must equal Bob's answer to 1, yet the fourth rule demands that the two answers to 1 differ. The chain contradicts itself, so every card loses at least one of four equally likely question pairs: at most 75 percent wins, achieved by the lazy strategy of always answering +1. Shared random numbers merely choose among cards. This is the CHSH inequality in disguise: the win probability equals one half plus S divided by 8.

Now give Alice and Bob an entangled pair. Each turns the question into a polariser angle, Alice using 0° or 45°, Bob 22.5° or −22.5°, and answers with the measured result. For three question pairs the angles differ by 22.5° and a match is wanted; for the 1-and-1 pair they differ by 67.5° and a mismatch is wanted. In every case the chance of winning is cos²(22.5°), about 0.854. Entangled players win 85.4 percent of rounds, the most any quantum strategy allows, thanks to correlations that no card could have prepared.

The first ballots were counted at the University of California, Berkeley. In 1972 the doctoral student Stuart Freedman and the postdoctoral researcher John Clauser excited calcium atoms that relaxed in two quick steps, emitting a green photon at 551.3 nm and a violet one at 422.7 nm with correlated polarisations. The faint source demanded more than 200 hours of counting. The result violated a Bell inequality by about six standard deviations, a standard deviation being the typical size of random scatter, and matched the quantum curve. A rival Harvard result favouring local realism did not survive Clauser's repeat.

Clauser's polarisers stood still, so the settings were fixed long before each pair was emitted, and some unknown signal travelling no faster than light could in principle have coordinated the two ends. This is the locality loophole. At the Institut d'Optique in Orsay, near Paris, Alain Aspect, with Philippe Grangier, Gérard Roger and Jean Dalibard, attacked it in 1981 and 1982. Laser excitation made the source far brighter, and two-channel polarisers counted both outcomes directly, giving S = 2.697 ± 0.015: more than 40 standard deviations above the classical ceiling, in line with the quantum prediction of 2.70.

The celebrated third experiment changed the settings while the photons were in flight. Each photon met an acousto-optic switch, where an ultrasonic standing wave in water steered it toward one of two polarisers at different angles, flipping about every 10 nanoseconds. The switches were roughly a dozen metres apart, so a light-speed signal between them needed about 40 nanoseconds. The inequality was still violated, by five standard deviations. Two weaknesses remained: the switches were driven periodically rather than randomly, and most photons went undetected.

Every test before 2015 left at least one loophole through which a local-realist model could still escape. The locality loophole closes only if each station picks its setting at random and finishes measuring before any light-speed signal from the other could arrive. The detection, or fair-sampling, loophole arises because detectors miss particles, so a local model could fake a violation by controlling which ones get counted. For a CHSH test with maximally entangled pairs, Anupam Garg and N. David Mermin showed in 1987 that detection efficiency must exceed about 82.8 percent. The freedom-of-choice loophole asks whether the settings might themselves be correlated with the hidden variables.

Anton Zeilinger's group at Innsbruck closed the locality loophole in 1998. Gregor Weihs and colleagues placed the stations 400 metres apart, about 1.3 μs of light-travel time, while quantum random number generators chose the settings and switching plus detection took under 100 ns. They measured S = 2.73 ± 0.02. In 2001 David Wineland's group at NIST in Boulder closed the detection loophole with two trapped ions read out at over 90 percent efficiency, but the ions sat only micrometres apart.

In 2015 both doors shut at once. Ronald Hanson's team at Delft University of Technology entangled electron spins in two nitrogen-vacancy centres, atom-sized defects in diamond, 1.3 km apart, with efficient spin read-out. Their 245 trials gave S = 2.42 ± 0.20, a probability of 0.039 that a local-realist model would produce so large a violation. Weeks later two photon experiments with superconducting detectors appeared together: Zeilinger's Vienna group, with Marissa Giustina as first author, reported a probability no larger than 3.74×10⁻³¹, and a NIST team with Krister Shalm as first author, its detectors 184 metres apart, reported probabilities as small as 5.9×10⁻⁹.

Freedom of choice can never be fully tested, only squeezed. In 2016 the BIG Bell Test drew settings from about 100,000 volunteers playing an online game, and cosmic Bell tests by Zeilinger's group with David Kaiser of MIT chose settings using starlight emitted about 600 years earlier (2017), then quasar light that set out roughly 8 and 12 billion years ago (2018). Any conspiracy linking settings to hidden variables would have to predate that light.

Headlines in 2022 declared that the universe is not locally real, or that Bell had proved quantum mechanics nonlocal. Be precise. Bell's theorem is conditional: if locality, pre-set outcomes and free choice all hold, and every measurement has a single result, then S lies between −2 and +2. Experiments find S above 2, so at least one assumption is false, but the theorem does not say which. Local realism, a joint assumption, has been ruled out. Bell himself read the result as a verdict against locality, and some philosophers agree; that reading is debated and interpretation-dependent.

Each surviving option gives up something different. Bohmian mechanics, the pilot-wave theory of Louis de Broglie (1927) and David Bohm (1952), keeps definite particle positions and openly accepts nonlocal hidden variables, while still forbidding usable faster-than-light signals. Hugh Everett's many-worlds interpretation (1957) denies that a measurement has a single outcome, and its defenders argue that it thereby stays local. Superdeterminism, defended by Gerard 't Hooft, Tim Palmer and Sabine Hossenfelder among others, rejects free choice: settings and particles share a correlating past. QBism and relational quantum mechanics question whether outcomes are observer-independent facts at all. As of 2026 no experiment has decided among them.

What no option allows is signalling. In every interpretation Alice's local statistics ignore Bob's choice, so entanglement can never carry a message faster than light. That limitation has become a feature: because no pre-written cards can fake a Bell violation, it certifies that outcomes were genuinely unpredictable and the devices genuinely entangled, even if you distrust the hardware. Artur Ekert's 1991 protocol builds cryptographic keys on this idea, and in 2018 a NIST team used a loophole-free Bell test to generate certified random numbers. Bell's theorem began as philosophy, became physics and is now engineering.