45°
spread angle θ
the ground / stratum · Physical
Two players, kept apart, forbidden to talk. A referee asks each a private yes-or-no question; each must answer. There is a simple rule for when they win — and a hard wall at three-quarters that no plan, no shared coin, no cleverness can climb over. Then you hand them a pair of entangled particles, and they climb over it. The gap is not a trick. It is the cleanest evidence we have that the world is not made the way it looks.
Here is the whole game. A referee flips two fair coins in private and sends one bit to Alice — call it x — and one bit to Bob — call it y. Alice sees only her bit; Bob sees only his. Each must immediately answer with a single bit of their own: Alice says a, Bob says b. They have agreed on a strategy beforehand, but once the round starts they cannot communicate — they are in sealed rooms, or on opposite sides of the solar system.
They win the round if their answers obey one rule:
In words: if at least one of the two questions was a 0, Alice and Bob must give the same answer. Only when both questions are 1 must they give different answers. Three of the four question-pairs ask for agreement; the fourth, the rare double-one, demands a disagreement — and the players never know which case they are in, because neither can see the other's question.
That single twist is the whole difficulty. To win the three "agree" cases, the obvious move is for both to always answer the same thing. But then they can never produce the disagreement the fourth case needs. To win the fourth case, they must somehow coordinate a disagreement — using information neither of them has. Play it below and find the wall yourself.
Pick what each player answers for each question they might receive. A deterministic strategy is just four bits.
All sixteen deterministic strategies, sorted by win rate. Eight win exactly 75%; their bit-flipped twins win 25%. Nothing reaches higher. And mixing strategies with a shared coin only averages these bars — it can never exceed the best one. Three-quarters is a wall.
The best a classical strategy can do is win three of the four cases and sacrifice the fourth. "Both always say 0" wins whenever the answer should be agree — that's three cases out of four — and loses only the double-one. Win rate: 75%. The enumeration above shows eight different strategies tie at exactly that, and not one does better.
What about randomness? Let the players share a coin flipped before they separate, and switch strategies on its outcome. This is the most general thing two separated players can do — physicists call it a local hidden variable: some shared information, fixed in advance, that both can consult. But a random mixture of strategies just wins at the average of their rates, and an average of numbers that are all ≤ 75% is itself ≤ 75%. Shared randomness cannot break a ceiling that none of its ingredients can reach.
This is Bell's inequality, in its sharpest modern form. In 1964 the physicist John Stewart Bell proved that any theory in which each particle carries its answers with it, set in advance and uninfluenced by the distant measurement, must obey a bound like this one. For this game the bound is exactly 75%. It is not a statement about cleverness or technology. It is a theorem about a whole class of possible universes — the ones that are locally real, where things have definite properties and nothing travels faster than light. In every such universe, this game is won at most three-quarters of the time.
So if anything in the real world wins it more often than 75%, that whole class of universes is ruled out. Ours included — unless ours isn't in the class.
Before they separate, give Alice and Bob each one particle of an entangled pair — two photons, say, prepared together so their polarizations are perfectly correlated, then carried to opposite rooms. Neither particle has a definite polarization of its own; only the pair has a joint state. This is not a shared coin. A shared coin is a value already written down; an entangled pair is a value that doesn't exist yet for either particle alone.
The new strategy: when Alice gets her question, she measures her particle's polarization along an angle that depends on x; Bob measures his along an angle that depends on y. Each reads off a 0 or 1 and answers with it. The angles are chosen once, in advance — four fixed angles, two for each player. That's the entire plan. Play it.
The four measurement angles
Run it long enough and the win rate settles, stubbornly, on 85.36% — well above the classical wall, and not a hair above a second line, cos²(π/8) = (2+√2)/4. The players are winning a game that, in any locally-real universe, they could win at most three-quarters of the time. They do it with no communication: nothing in the apparatus sends a signal from Alice's room to Bob's. This is what it means to say nature is not locally real. Something has to give — either the answers weren't fixed in advance (no hidden variables), or distant things are connected in a way that ignores the space between them (no locality), or both. Bell's theorem says you cannot keep all of it.
The first worry is that this lets you send messages instantly. It does not. Look only at Alice's stream of answers, ignoring Bob: it is a perfectly fair, random sequence of 0s and 1s, no matter what Bob does or which angle he chooses. The same is true of Bob's. The correlation only shows up when you bring the two records together and compare them — and that comparison needs a classical channel, a phone call, a meeting, something that crawls along at the speed of light like everything else. So entanglement breaks local realism without breaking relativity: you cannot use it to signal, only to share a correlation stronger than any pre-arranged plan could be. Physicists call this peaceful coexistence the no-signalling principle, and the quantum world obeys it exactly.
The quantum win rate isn't arbitrary. Drag the angle below and watch the advantage appear, peak, and vanish. The win probability as you rotate the measurements by an angle θ traces a smooth curve — and it bulges above the flat classical wall only in a window around 45°, touching its maximum of exactly (2+√2)/4 there. That maximum is the Tsirelson bound (Boris Tsirelson, 1980): quantum mechanics can push the CHSH quantity S up to 2√2 ≈ 2.828, and not one bit further.
The picture is the whole argument in one shape: a flat red wall the classical world cannot cross, and a cosine curve from the quantum world that bulges over it. The bulge is the violation. Its height — that 2√2 instead of 2 — is the exact amount by which the universe declines to be locally real.
And the wall on the other side is just as interesting. Why does quantum stop at 2√2 and not go all the way to a perfect 100%? A hypothetical "super-quantum" correlation that won every round — the Popescu–Rohrlich box (1994) — would still send no signals and break no relativity, yet no quantum state achieves it. The reason quantum mechanics sits exactly where it does, between the classical 75% and the no-signalling 100%, is still not fully understood; one candidate principle, information causality (Pawłowski et al., 2009), singles out 2√2 almost exactly. The wall we beat hides a second wall we don't yet understand.
Bell's 1964 paper was, for years, treated as philosophy. The breakthrough was turning it into something you could build: in 1969 Clauser, Horne, Shimony, and Holt recast Bell's inequality into the experiment-ready form played here. John Clauser, with Stuart Freedman, ran the first test in 1972; Alain Aspect, in a famous series at Orsay in 1981–82, took the decisive step toward closing the "locality loophole" by switching the measurement angles while the photons were already in flight — though his switching was periodic rather than truly random, so the loophole was only fully closed later (Weihs and colleagues, 1998, with random fast switching; and conclusively in 2015). The last loopholes — that the detectors missed too many photons, or that the two stations weren't truly independent — were finally closed together in 2015, in three "loophole-free" experiments (Hensen and colleagues in Delft using electron spins in diamond; Giustina and colleagues in Vienna and Shalm and colleagues at NIST using photons). The verdict every time: the wall is broken, by just about 2√2. In 2022 the Nobel Prize in Physics went to Aspect, Clauser, and Zeilinger for exactly this — for proving, in the laboratory, that the world is not locally real.
"Simulation," carefully. The game above samples the statistics the Born rule predicts for a real entangled pair — the same numbers a photon experiment produces. It is not modelling a hidden mechanism inside the particles, because Bell's whole point is that no local mechanism can produce these numbers. What's faithful is the distribution; what's deliberately absent is any "how."
What actually fails is "local realism." The popular phrase "the universe is not locally real" packs together two assumptions — locality (no faster-than-light influence) and realism (measurements reveal pre-existing values). Bell's theorem says at least one must go; it does not, by itself, tell you which. Some interpretations keep realism and drop locality; others keep locality and drop definite values. The experiments rule out keeping both.
The remaining logical escape. One assumption survives: that the referee's questions are freely chosen, statistically independent of the particles. A universe in which the choices and the particles were correlated in advance — superdeterminism — could fake the violation while staying locally real. Almost no physicist takes this route (it would make all of science's reliance on independent variables suspect), and experiments have pushed the correlation back in time using starlight and even light from quasars billions of years old — but it cannot be fully closed by any experiment, and the page does not claim otherwise.
The thing worth sitting with is that you can run this. The classical wall is not an opinion; enumerate the sixteen strategies and it is simply there, at 75%, with nothing above it. The quantum bulge is not a metaphor; play ten thousand rounds and the rate lands on 85.36%, and you can read off the four correlations climbing to ±0.707 and their sum to 2√2. Two numbers, 2 and 2√2, separated by a gap you can measure on a slider — and that gap is the most direct answer physics has ever given to the oldest question about the world: is it really there when you're not looking? The honest answer, won in this little game ten thousand times over, is not in the way you think.
J. S. Bell, "On the Einstein Podolsky Rosen Paradox," Physics 1, 195 (1964).
J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, "Proposed Experiment to Test Local Hidden-Variable Theories," Phys. Rev. Lett. 23, 880 (1969).
B. S. Cirel'son (Tsirelson), "Quantum generalizations of Bell's inequality," Lett. Math. Phys. 4, 93 (1980) — the bound S ≤ 2√2.
R. Cleve, P. Høyer, B. Toner, J. Watrous, "Consequences and Limits of Nonlocal Strategies" (2004) — the inequality recast as this game.
A. Aspect, J. Dalibard, G. Roger, "Experimental Test of Bell's Inequalities Using Time-Varying Analyzers," Phys. Rev. Lett. 49, 1804 (1982).
Hensen et al., Nature 526, 682 (2015); Giustina et al. and Shalm et al., Phys. Rev. Lett. 115, 250401 & 250402 (2015) — loophole-free.
S. Popescu, D. Rohrlich, "Quantum nonlocality as an axiom," Found. Phys. 24, 379 (1994) — the PR box.
The Nobel Prize in Physics 2022 — Aspect, Clauser, Zeilinger.
Every number on this page is recomputed in your browser and was checked offline first, in the same double-precision arithmetic, in /research/chsh-bell/ (17 checks: the 3/4 ceiling by brute force over all 16 strategies; the 0.8536 optimum three independent ways; the 2√2 Tsirelson sweep; the slider's closed form). The Monte Carlo you ran is the same sampler, seeded there for reproducibility.