The Bell Test

The entanglement page shows two qubits whose answers always match. That is striking, but it is not yet an argument: a pair of gloves posted to opposite ends of the country also always match, and nothing spooky happened to the gloves. To turn correlation into a claim about reality you have to measure the pairs along different axes and see whether the answers hang together more tightly than any pre-agreed plan could manage.

That is what this page runs. Alice has two analyzer angles, a and a′; Bob has b and b′. On every pair, each of them picks one of their two at random and records +1 or −1. Four correlators come out, and they are combined into a single number:

S = E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′)

Press play and watch two things race: the violet curve is the entangled source, the rose curve is a classical source doing its best. Watch where each one stalls.

The rose source is not a scripted animation with a cap. It is a simulation of exactly the kind of world Einstein wanted: at the moment the pair is created, a hidden variable λ is drawn — a shared secret polarization angle, plus a private coin for each side — and from then on each detector’s output is a function of its own angle and λ. Neither side can see the other’s setting. That is locality, and in the code it is enforced by the function signature: the outcome function is handed one angle, never two.

Naive realist is the model a nineteenth-century physicist would write: each photon really does carry a definite polarization and passes its analyzer with the Malus-law probability cos²(α − λ). It produces exactly half the quantum correlation and tops out at S = √2 ≈ 1.41. Best conspiracy is the provably optimal local strategy, the one that squeezes every last drop out of the shared secret. It climbs to 2.000 and stops dead. Switch between them and watch: cleverness buys the classical source a lot, and then buys it nothing at all.

The reason is four lines of arithmetic. Give one hidden state λ, ask it all four questions at once, and form the same combination: A(a)[B(b) − B(b′)] + A(a′)[B(b) + B(b′)]. Every outcome is ±1, so one bracket is always zero and the other is always ±2. Every single pair contributes exactly ±2, and an average of ±2 cannot leave the interval [−2, 2].

Two honest caveats about that rose curve. It is an estimate, so with the conspiracy strategy at the optimal angles — where the true value is exactly 2 — it will jitter a hundredth or so either side of the line, and sometimes sit fractionally above it. That is finite-sample noise, not a violation; it shrinks as 1/√N, and it is the reason real Bell tests are quoted in standard deviations rather than in decimal places. The audit above, by contrast, cannot jitter past 2 at all, because it averages values that are each exactly ±2.

Notice what the audit above required: asking one λ what it would have said at an angle that was never used. For a classical source that question is free — λ is a number in memory, and the code can evaluate it four times. For an entangled pair it is not a question at all. Measure Alice’s photon at a and the counterfactual value at a′ does not exist to be read, not even in principle.

That, and nothing more exotic, is what Bell’s theorem rules out. Not faster-than-light signalling — check the marginals on the entanglement page, they stay at a dead 50/50 no matter what Bob does, which is why no message can ride this channel. What fails is the assumption that the unasked question had an answer.

Set the optimal angles and let the violet curve settle. It converges on 2√2 ≈ 2.828, which is Tsirelson’s bound — quantum mechanics has a ceiling of its own, and it is not infinity. A hypothetical theory that could reach 4 would let you signal. Nature picked a number strictly between the classical limit and the paradoxical one.

You measured a number that a whole class of theories is arithmetically forbidden to produce, and the forbidden number came out anyway — repeatedly, from a source that follows the ordinary rules of quantum mechanics with no special pleading.

Drag the angles away from the preset and watch the violation evaporate: at aligned settings the entangled source is perfectly correlated and S collapses to 2 or below, which is why the matching-outcomes demo on its own proves nothing. The violation lives in the mismatch — in measuring the two halves of a pair along axes that disagree by 22.5°, where the classical best-fit to a cosine is a triangle wave and falls short by exactly the margin you can see on the chart.

Real laboratories have been running this experiment since 1972, closing one loophole at a time — detection efficiency, locality of the setting choice, the freedom to choose at all. The 2015 loophole-free tests settled it, and the 2022 Nobel Prize in Physics went to Clauser, Aspect and Zeilinger for the campaign. The number on your screen is the number they measured.