85%: two players who can't talk beat every crib sheet ever written

Here's a quiz score that no plan made in advance can reach in the long run: 85%.

The quiz is tiny. Two players sit in separate rooms. In each round, each of them is asked one of two questions, picked by a coin toss, and each answers 0 or 1. They can't talk once the rounds begin.

Before the quiz, they can agree on anything they like. Write the answers on a crib sheet. Roll shared dice. Toss coins on the spot. Whatever they do, in the long run they win at most 3 rounds in 4: 75%.

Hand them entangled particles instead (pairs of particles linked in a way chapter 4 explains), and quantum physics lets them win up to 85%. Real experiments have beaten 75%. 🤯

By the end of this story you'll know why 75% is a wall, why 85% is quantum physics' own limit, why the game still can't carry a message, and why at least one everyday idea about the world has to give (physicists still argue about which).

Chapter 01

Eighty-five per cent

Any plan written down in advance (this page calls it a crib sheet) wins at most 75% of rounds in the long run. Chapter 3 shows why, with a pencil.

Players who share entangled particles, and let a measurement on each one choose the answer, can win 85.4% of rounds. That's the 85% in the title, and it's a ceiling: in 1980 Boris Tsirelson proved that no quantum strategy wins more often.

No experiment with the two main loopholes closed (ways a crib sheet could still sneak a win: chapter 8) has reached 85.4%, and none needs to. What matters is beating 75%, and since 2015 experiments have. Delft's 2015 test won 196 of 245 rounds of its own version of the quiz: 80%. Munich's in 2017 scored about 78%, and Zurich's in 2023 about 76% (both this page's arithmetic from the published results).

Where the scores sit

75%any crib sheet,
long run 85.4%quantum ceiling 70% 90% ~76%Zurich ~78%Munich 80%Delft

Dots: three tests with both main loopholes closed. Munich's and Zurich's scores are this page's arithmetic.

And the game can't carry a message. Whatever one player is asked, the other's answers come out 0 or 1 half the time each. The extra matching shows only when the two lists are put side by side.

So what gives? Not quantum physics, which predicted it all. What fails is a picture of the world: that each particle carries its answers with it, like a crib sheet in a pocket, and that only nearby things matter. As long as each measurement has one result and the questions are picked freely, at least one of those two ideas has to go (chapter 12).

Chapter 02

The quiz

Call the players Alice and Bob. In each round a referee tosses a fair coin for each of them. Heads, you're asked question 0; tails, question 1. Each player then answers 0 or 1.

The rule for winning fits on a sticky note:

if at least one of them got question 0, their answers must match;
if both got question 1, their answers must differ.

The four kinds of round
	Bob asked 0	Bob asked 1
Alice asked 0	match	match
Alice asked 1	match	differ

So there are four kinds of round, one for each way the coins can fall: three want matching answers, one wants different answers. Alice and Bob may agree any plan beforehand. Once the rounds start, no talking, no notes, no signals.

This is the CHSH game, named after John Clauser, Michael Horne, Abner Shimony and Richard Holt, who in 1969 turned John Bell's theorem (a mathematical proof, chapter 7) into a test that real equipment could run.

In physics, a crib sheet means the particles carry properties from their source that fix each result, or its odds, before the questions are known. The comparison breaks in one place: particles have no intentions, so a "plan" just means whatever properties fix the outcome.

Chapter 03

Every crib sheet loses a round

A crib sheet for one player is just a rule that turns a question into an answer. With two questions and two answers, there are only four rules: always say 0; always say 1; copy the question; flip the question.

A team needs one rule each, so there are 16 possible crib sheets. Check them all with a pencil:

8 of the 16 win three of the four kinds of round, and lose the fourth every time;
the other 8 win only one kind;
none wins all four.

There's a neat reason. Each of Alice's two answers turns up in two kinds of round, and so does each of Bob's, so the number of kinds of round where their answers differ is always even. The quiz wants exactly one: an odd number. Something always fails.

Every kind of round comes up a quarter of the time, so the best a crib sheet can do is 3 rounds in 4. The obvious plan, "both always say 0", gets there already: it loses only when both are asked 1.

Randomness doesn't help. Shared dice just pick one of the 16 sheets each round, and a mix of sheets that each win at most 75% wins at most 75%. A coin tossed at answer time is no different from one tossed the night before and written down: in 1982 Arthur Fine showed that any such use of chance works out as a mix of fixed sheets.

Write a crib sheet for each player and deal some rounds. Press Try all 16 sheets, or let entangled photons (particles of light, chapter 4) answer 👇

One warning: 75% is a long-run limit, and short quizzes can be lucky. Over 20 rounds, a best crib sheet scores 85% or better about one time in five; over 100 rounds, only about 1% of the time (this page's arithmetic). Over a few hundred rounds, luck stops being an excuse.

Look at the grid, too. A crib sheet loses at least one kind of round every time. Entangled photons lose about one round in seven of every kind, and no kind every time.

Chapter 04

A curve beats a line

Now the quantum trick. Before the quiz, Alice and Bob share many pairs of entangled photons, particles of light. Quantum physics gives neither photon a polarisation (a direction for its light to wiggle in) of its own, yet each pair has a definite relationship.

In each round, each player sends a photon through a polarising filter turned to an angle set by their question, and answers 1 if it gets through, 0 if not.

On its own, each photon gets through half the time, at any angle. What depends on the angle is whether the two answers match. Lined up, they always match. Tilt the filters apart, and matching falls away along a smooth curve.

Here are the winning settings. Alice turns her filter to 0° for question 0 and 45° for question 1. Bob turns his to 22.5° for question 0 and to 22.5° the other way for question 1.

Three kinds of round put the filters 22.5° apart and want matching answers. They match about 85% of the time.
The fourth kind puts them 67.5° apart and wants different answers. They differ about 85% of the time.

So every kind of round is won 85.4% of the time.

John Bell's own 1964 paper had a crib-sheet model, in which each pair carries a hidden direction. Its matching falls in a straight line as the tilt grows. The line and the curve agree at 0°, 45° and 90°. In between, the curve bulges above the line where matching helps the team, and dips below it where differing helps. Tilt the dials and watch 👇

Lined up, the photons always match, but so can a crib sheet: perfect matches can be faked. It's the in-between tilts that no crib sheet can copy.

Real pairs are never perfect. In the simplest model, imperfect pairs blur the pattern: every lean towards matching, or towards differing, shrinks by the same fraction, so each result drifts towards a coin toss. Pairs 90% clean win about 82%, and pairs less than about 71% clean lose to a crib sheet (this page's arithmetic). Chapter 3's quiz has a slider for it.

Chapter 05

No message gets through

An 85% score looks like cheating, as if the photons were passing notes. The numbers alone can't rule that out: a team passing notes could win every round and still give answers that are half 0s. What rules notes out is distance: in the best tests the stations are too far apart for any note to arrive in time, even at the speed of light (chapters 8 and 9).

What the statistics show is narrower: neither player can use the game to send the other a message.

Look at Alice's answers on their own. About half are 0s and half are 1s, whatever she's asked and whatever Bob is asked. Philippe Eberhard proved in 1978, and Giancarlo Ghirardi, Alberto Rimini and Tullio Weber in 1980, that part of this always holds in quantum physics, whatever the pairs: the odds of one player's answers don't depend on the other player's question. The exact half-and-half comes from these perfectly linked pairs.

The matching is real, but it lives in the two lists together. Alice sees it only with Bob's notebook beside her own, and getting it to her takes an ordinary phone call, which can't beat light either.

First look at Alice's two bars. Then pick Send a message: Bob tries to send Alice a secret anyway, by choosing his own questions. Can she read it from her notebook? 👇

Alice's two bars differ only by luck, and her guesses at Bob's secret are right half the time in the long run. Only the pretend leaky box gives it away.

The careful wording matters: it's the odds of each player's answers that ignore the other's question. The rule says nothing about single results. In one theory, David Bohm's (chapter 7), a single result does depend on the far setting, but in a way no one can use to send a message.

Chapter 06

Why stop at 85?

Could a cleverer kind of pair, or cleverer settings, beat 85.4%? No. Boris Tsirelson proved in 1980 that nothing quantum physics allows does better.

You might guess the ceiling is there because anything higher would let players send messages. It isn't. In 1994 Sandu Popescu and Daniel Rohrlich imagined a box that wins every round. Alice's answer is a coin toss; Bob's matches hers, except when both are asked 1, when it's the opposite. Alice's answers are still half 0s whatever Bob is asked, so the box sends no message either.

No known physics makes such a box. But it shows there are three ceilings, not two:

75% for any crib sheet;
85.4% for quantum physics;
100% for anything that merely can't send messages.

So "no messages" isn't what stops quantum physics at 85%. Why nature stops there is an open question. One proposed answer, put forward by Marcin Pawłowski and colleagues in 2009, is a rule about how much you can learn from a partner's messages, but it's one idea among several, not the accepted answer.

Experiments have come within a hair's breadth of the ceiling. A 2015 test in Singapore scored 85.3%, against the ceiling's 85.4%. But it counted only the pairs it caught, and it left the loopholes open.

Chapter 07

Einstein's worry, Bell's limit

1935

The story starts with a worry. In 1935 Albert Einstein, Boris Podolsky and Nathan Rosen imagined two particles whose results always match when the same thing is measured on both. Measure one, and you know the other's result without touching it. If nothing here can disturb what's over there, they argued, the far particle had that value all along. Quantum physics doesn't give it one, so the theory must be missing something.

The same year, Erwin Schrödinger gave the effect its name: entanglement.

1947

Einstein stayed uneasy. In a 1947 letter to Max Born he called the distant influence the theory seemed to imply "spooky action at a distance".

1952

Could quantum physics be completed with crib sheets? In 1952 David Bohm showed that a theory in which every particle has a definite position can reproduce all of quantum physics. But it openly lets far-away things matter: what happens to one particle depends on what's done to the other, however far apart.

1964

Bell asked whether that was the price of any such theory. In 1964 Bell, a theorist at CERN, Europe's particle-physics laboratory, then on leave in the United States, proved that it is. No crib-sheet theory in which only nearby things matter can match all of quantum physics' predictions.

1969

His version needed perfect matching, which no real apparatus gives; in 1969 Clauser, Horne, Shimony and Holt rewrote it as chapter 2's quiz, which forgives imperfect equipment.

Bell was born in Belfast in 1928 and died in Geneva in 1990.

Chapter 08

The first tests

Stuart Freedman and John Clauser ran the first test at Berkeley, published in 1972, with photon pairs from calcium atoms. They beat the crib-sheet limit by about six standard deviations, as quantum physics predicted. A standard deviation is the usual size of luck's wobble, and six of them is far more than luck gives. But their settings were fixed.

A loophole is a way a crib sheet could still fake a win. Two mattered most.

Locality.

If a setting is chosen early enough, a signal at the speed of light could tell the other station in time, and a crib sheet that read such signals could beat 75%.

The fix: choose each setting at random, and finish each measurement, too quickly for light to cross between the stations.

Detection.

Most photons in the early tests were never caught, so the tests counted only the rounds where both detectors fired, assuming those were a fair sample. A crib sheet allowed to say "don't click" can fake a win among the caught pairs.

The fix: catch more than about 83% of the photons (or two-thirds, with a different scorecard).

In 1982 Alain Aspect's group at Orsay, near Paris, scored about 84%, counting only the pairs both detectors caught. A second experiment switched the settings while the photons were in flight, though not at random: the first attack on the locality loophole, not its closure.

In 1998 Anton Zeilinger's group in Innsbruck shut the locality loophole, with stations 400 metres apart and random settings: about 84%, among the pairs caught. In 2001 a group in Boulder, Colorado, shut the detection loophole instead, with two trapped ions (charged atoms held still by electric fields) read out every round: about 78%. But the ions sat micrometres apart.

One loophole or the other, never both. Not until 2015.

Chapter 09

2015: both loopholes shut

The decisive tests came in 2015, from three groups. At Delft, in the Netherlands, Bas Hensen, Ronald Hanson and colleagues used tiny flaws in diamond, in two labs 1.3 km apart. Each flaw has an electron spin (a tiny built-in magnet) that can be measured the way a photon meets a filter.

Each spin was entangled with a photon, and the photons met at a station between the labs. A joint click there announced that the two distant spins were now entangled.

196of 245

rounds of its own version of the quiz, won at Delft in 2015 with both loopholes shut: 80%

Every announced round counted, and both settings were picked and both spins read out before light could cross between the labs. That shut both loopholes at once. Over 220 hours they got 245 rounds, and won 196 of them, in the experiment's own version of the quiz: 80%.

Teams in Vienna and Boulder followed in December 2015, using photons and a different scorecard. Their papers report only how unlikely luck was, not a number this page can turn into a quiz score, so it gives them none. In 2017 a Munich team linked two atoms in labs 398 metres apart and scored about 78%. The Nobel committee names these four groups as the ones that shut both loopholes together.

More have followed. In 2023 a Zurich team joined superconducting circuits (tiny electric circuits, chilled until they behave as quantum objects) through a 30-metre chilled link and scored about 76% over more than a million rounds. Where a score can be worked out, the tests that shut both loopholes scored from just above 75% to about 80%.

Step through fifty years of tests, and switch the view to see how far apart the stations were 👇

Why 80% at most, and not 85.4%? Imperfect pairs lower the score (chapter 4), and this page blames no shortfall on any one cause. The point was never to reach 85.4%. It was to beat 75% with neither of the two main loopholes left open for a crib sheet.

Chapter 10

One time in 25

Delft's 196 wins out of 245 come with a number: a crib-sheet world would produce a result this strong about one time in 25. Physicists call this a p-value.

Guess first

"About one time in 25": what is that number?

See the answer

The second one. Read on for why.

It is not the chance that crib sheets rule the world. It's worked out by assuming they do, so it can't also say how likely that is. It answers a narrower question: if crib sheets ruled, how often would luck alone do this well?

You can check it on a spreadsheet. A best crib sheet wins each round 3 times in 4; if each round is a fresh start, untouched by the ones before, this page's arithmetic gives the same one time in 25 (Delft's own analysis doesn't assume that).

How long must a perfect quantum team play to stand well clear of luck? On average about 440 rounds puts its score five standard deviations above 75%, a wide margin, but a real run has got there by then only about half the time. To be 95% sure of getting there, it needs about 700 (this page's arithmetic).

Pick a run, or set the wins yourself, and watch the crib-sheet team's lucky tail 👇

Drag the rounds up at Delft's 80% and the tail shrinks: the same score over more rounds makes luck far less likely. Zurich's 75.9% beats 75% by less than one point, but over more than a million rounds the paper puts the chance of luck doing as well at less than one in a 1 followed by 108 zeros.

One caution: a tiny p-value says how surprising a result would be for a crib sheet, not how well the loopholes were shut.

Chapter 11

Who picks the questions?

There's one last worry. Every test assumes the questions are picked independently of whatever fixes the answers. If the machines picking the settings were secretly in step with the particles, a crib sheet could know its questions in advance. Whatever picks a setting has a past, so this worry can only be pushed back, never removed.

So physicists handed the choice to the sky. 🌌 In April 2016 a team in Vienna let the colour of light from stars in our galaxy pick the settings. Any plot to fix the questions, travelling no faster than light, must then have started at least about 600 years earlier, when that light left the nearer star: before Gutenberg's printing press. Counting the pairs both stations caught, they scored about 80% and 81% in two runs.

In 2018 the same group used light from quasars, the blazing hearts of distant galaxies. In the first run that light had travelled for nearly 8 billion years or more; in the second, for about 3 billion years or more. Among the pairs caught, they scored about 83%.

How old the light that picked the questions was

Shanghai, 2018 · starlightboth loopholes shutat least 11 years

Vienna, 2016 · starlightcaught pairs onlyat least about 600 years

Vienna, 2018 · quasar, second runcaught pairs onlyabout 3 billion years or more

Vienna, 2018 · quasar, first runcaught pairs onlynearly 8 billion years or more

Bar lengths on a log scale: each tenth of a bar is ten times older. Solid: both loopholes shut. Striped: only the pairs caught were counted.

Both cosmic tests counted only the pairs they caught, so neither was loophole-free. But a 2018 test in Shanghai was: it shut both loopholes, with starlight at least 11 years old picking the questions, using a different scorecard.

And on 30 November 2016, about 100,000 volunteers playing an online game, the BIG Bell Test, made over 97 million choices that set the questions for 13 experiments on five continents. People are far too slow to shut the locality loophole, so as far as the choices go none was loophole-free. Most experiments still found strong results against crib sheets.

Chapter 9's figure lines up the three tests that used starlight or quasars: pick How old the questions.

Chapter 12

Which idea goes?

So what have the experiments ruled out? Answers fixed in advance, carried by the particles, with only nearby things mattering, when the questions are picked freely. All of that assumes each measurement has just one result.

Something has to give: either the answers were not settled by anything the particles carried with them, or something far away matters. Physicists still disagree about which.

Give up answers fixed in advance

No signal crosses the gap: the odds at each end ignore the far setting. But this is no easy escape. Answers tossed up on the spot at each end are just another crib sheet, capped at 75% (chapter 3). So the matching has no explanation that stays on each side, even though no message travels. Some, such as Tim Maudlin and Travis Norsen, argue this means nature itself is nonlocal: that far-away things matter. Others disagree.

Give up "only nearby things matter"

Keep real properties that fix the results, and let something far away matter. Bohm's theory (chapter 7) does exactly this, openly, and it predicts the same 85.4% as quantum physics. So a theory with extra properties that fix each result, beyond what quantum physics describes, isn't impossible, and the experiments didn't kill the idea of a real world with real properties.

Where does that leave Einstein? The local picture he hoped for can't be kept, if each measurement has one outcome and the questions are chosen freely. His other claim, that quantum physics is incomplete, is a separate question the experiments don't settle.

There are other escape routes, minority views with serious critics, set out in the ∫ Full version: every answer happens, in separate branches; the questions were never really free; or influences run backwards in time.

Either way, the game can't carry a message.

Chapter 13

From puzzle to tool

In 2022 Alain Aspect, John Clauser and Anton Zeilinger shared the Nobel Prize in Physics, "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science". Bell had died in 1990, and the prize isn't given after death.

Beating 75% has turned out to be useful.

Beat 75% If two devices beat it with the loopholes shut, their answers can't have been written down in advance, by the maker or anyone else.
Genuine randomness Given freely picked settings, the answers must hold genuine randomness, and that guarantee comes from the score itself, not from trusting how the devices were built.
Put to work In 2018 two experiments, one at Boulder and one in China, produced random numbers certified by loophole-free Bell tests.

Later a beacon in Colorado posted random numbers for anyone to use, certified the same way. That certified record has entries from October 2023 to 22 August 2025 (as of 25 October 2026).

In 2022 an Oxford experiment with two trapped ions about 2 metres apart made the first secret key (a string of random digits that two people share to lock messages) certified by a Bell test without trusting the hardware, as far as we can find. Its devices were shielded from each other rather than far apart. In 2026 a team in China tested the idea with two atoms in one lab, joined by coils of fibre up to 100 km long. 🔑

Chapter 14

No crib sheet

What's settled: no advance plan beats 75% in the long run; quantum physics allows up to 85.4%, and no more; since 2015, tests that shut the locality and detection loopholes together have beaten 75%, scoring from just above 75% to about 80% where a score can be worked out; and none has shown one player's odds following the other's question, which quantum physics says none can.

What's open: which idea gives way, and why nature stops at 85.4% rather than 75% or 100%.

John Bell found the limit in 1964. Clauser, Horne, Shimony and Holt turned it into a test in 1969. Boris Tsirelson found the quantum ceiling in 1980. Clauser (1972), Aspect (1982) and Zeilinger's group (from 1998) led the experiments, and Clauser, Aspect and Zeilinger shared the 2022 Nobel Prize.

Final round · three questions

Any crib sheet, in the long run: the best score?

See the answer

75%: no advance plan beats it in the long run.

Tests with both loopholes shut, since 2015: what did they score?

See the answer

From just above 75% to about 80%, where a score can be worked out. None has reached 85.4%, and none needs to.

Can the game carry a message?

See the answer

No. Whatever one player is asked, the other's answers come out 0 or 1 half the time each.

Let's add it all up. 📋

Report cardThe quiz · Nº 20

Best crib sheetlong run75%

Crib sheets winning 3 rounds in 4out of 168

Quantum ceilingTsirelson, 198085.4%

Imaginary boxsends no message either100%

Delft, 2015196 of 245 rounds of its own version80%

Zurich, 2023over a million rounds75.9%

Starlight that set the questionsVienna, 2016~600 years old

Satellite stations apartMicius, 20171,203 km

Rounds to be 95% sureperfect team, of a wide margin~700

VolunteersBIG Bell Test, 2016~100,000

Messages the game can carrywhatever Bob is asked0

no crib sheet ✦ no message ✦ something has to give

Two stations in the dark, a long way apart.

A photon arrives at each. Neither station knows the other's question.

Each notebook, on its own, looks like a list of coin tosses.

Side by side, they match more often than any plan allows.

And still no message travelled.

No crib sheet beats 75%. Entangled particles have, and no message travelled.

Part IA quiz no crib sheet can beat

Chapter 1Eighty-five per cent

Put two players in separate rooms. In each round a referee tosses a fair coin for each room and asks that player question 0 or question 1 accordingly; each player answers 0 or 1. The answers must agree, except when both were asked question 1, when they must differ. The players may agree any plan beforehand, but once the rounds begin they cannot communicate. This is the CHSH game, named after the four physicists who in 1969 turned John Bell's theorem into a test that real equipment could run.¹

No plan written down in advance, no crib sheet, wins more than 75% of rounds in the long run. Every fixed crib sheet loses at least one of the four kinds of round, one for each way the two questions can fall (Chapter 3). But if the players share entangled photons, pairs prepared in one joint quantum state (Chapter 4), and each lets a measurement on their own photon choose the answer, quantum mechanics predicts that they win 85.36% of rounds, written 85.4% on this page. That is the headline number, and it is a ceiling: in 1980 Boris Tsirelson proved that no quantum strategy can win more often.²

No test with the two main loopholes closed has reached 85.4%, and none needs to: what matters is beating 75%, and since 2015 experiments have done that (Chapters 8 and 9). Delft's 2015 test counted 196 wins in 245 rounds of its own version of the game, 80.0%.³ Munich's 2017 test scored 77.8%,⁴ and an ETH Zurich test of 2023 scored 75.9% over more than a million rounds,⁵ both on this page's arithmetic from the published CHSH values. So the scores of these tests lie between about 76% and 80%, under a ceiling of 85.4% that is a bound, not a result.

The game cannot carry a message between the players: whatever question one gets, the other's answers come out 0 or 1 half the time each.^6,7 The extra agreement appears only when the two lists of answers are compared (Chapter 5).

So what gives? Not quantum mechanics, which predicted all of it. What fails is a picture of the world: that each particle carries its answers with it, like a crib sheet in a pocket, and that only nearby things affect a measurement. With single outcomes and freely chosen questions, at least one of those two ideas must go; Part III sets out who gives up which, and why physicists still disagree.

Chapter 2A quiz for two players who cannot talk

Call the players Alice and Bob. In each round Alice receives a question x and Bob a question y, each 0 or 1, from independent fair coins. Alice answers a and Bob b, each 0 or 1. They win when

a ⊕ b = x ∧ y⊕ is addition modulo 2 (it is 1 when the answers differ); ∧ is logical AND (it is 1 only when both were asked question 1). Three of the four question pairs ask for agreement; the pair (1, 1) asks for disagreement.

This is the CHSH game in the nonlocal-game form given by Richard Cleve, Peter Høyer, Ben Toner and John Watrous in 2004.⁸ A nonlocal game is one played by partners who cannot communicate. The "questions" are not questions about the particles. In a laboratory they are measurement settings, such as the angle of a polariser; the "answers" are the outcomes, which detector clicked.

A plan agreed in advance is a crib sheet, a card in each player's pocket saying what to answer to each question. In physics it is a local hidden-variable theory: the particles carry properties from their source that fix the result of each possible measurement, or its odds. The analogy fits where it matters, because the answers are settled before the questions are known and each depends only on the player's own question and pocket. It breaks in one place: particles have no intentions, so a "strategy" means only "whatever physical properties fix the outcome".

Physicists usually write the same thing as an inequality. Let E(x, y) be the correlation between the answers for the question pair (x, y): the fraction of rounds in which they agree minus the fraction in which they differ, from +1 (always agree) to −1 (always differ). Clauser, Horne, Shimony and Holt combined the four correlations:¹

S = E(0, 0) + E(0, 1) + E(1, 0) − E(1, 1), |S| ≤ 2 for any local hidden-variable theoryThe minus sign marks the question pair for which the game asks for different answers. The first argument is Alice's question, the second Bob's.

The game and the inequality are the same thing. The chance of winning a "match" pair is (1 + E)/2, and of winning the "differ" pair (1 − E)/2. When the four question pairs come up equally often, averaging the four chances gives

P_win = 1/2 + S/8So S = 2 is a 75% win rate, S = 2√2 ≈ 2.83 is 85.36%, and S = 4 would win every round. A standard identity; see the review by Brunner and colleagues.

That identity turns an experiment's published S into a score, always this page's arithmetic, and only for tests that report S.⁹

Chapter 3Why no crib sheet beats 75%

A crib sheet for one player is a rule that turns a question into an answer. With two possible questions and two possible answers there are exactly four rules: always answer 0; always answer 1; copy the question (a = x); flip it (a = 1 − x). A team's plan is a pair of rules, one for each player, so there are 4 × 4 = 16 plans. Checking all of them takes a few minutes with a pencil (this page did it, and Figure 1 does it on demand): 8 of the 16 win three of the four question pairs and lose the fourth, and the other 8 win only one. None wins all four.⁸

There is a one-line reason. Write Alice's rule as the two answers a₀ and a₁ she would give to questions 0 and 1, and Bob's as b₀ and b₁. Winning all four question pairs would need

a₀ ⊕ b₀ = 0, a₀ ⊕ b₁ = 0, a₁ ⊕ b₀ = 0, a₁ ⊕ b₁ = 1Add the four left-hand sides modulo 2: each of a₀, a₁, b₀ and b₁ appears twice, so the sum is 0. The right-hand sides add up to 1. At least one of the four equations must fail.

Every crib sheet therefore loses at least one kind of round, and since each kind comes up a quarter of the time, no crib sheet wins more than 3/4 of rounds in the long run. The obvious plan, "both always answer 0", already reaches the limit: it wins every question pair except (1, 1), where it should have disagreed. Some natural first tries do worse. "Both copy the question" wins only when both are asked 0: a 25% plan, with S = −2.

Randomness does not help. Shared dice that choose a plan each round just pick among the 16 plans, and an average of plans that each win at most 75% wins at most 75%. A coin tossed at answer time is no different from one tossed the night before and written on the sheet. Arthur Fine showed in 1982 that local models whose outcomes are random rather than fixed can always be rewritten as mixtures of deterministic ones, which obey the same bound.¹⁰ Figure 1's coin option, a fresh coin for every answer with nothing shared, settles at 50%, well inside the limit.

Figure 1 is the game. Write a crib sheet for each player, or let entangled photons answer, or try an imaginary box (Chapter 6), and deal rounds.

Figure 1🎲 Try it

The quiz no crib sheet can beat

Write a crib sheet, then deal

Alice's rule

Bob's rule

Rounds

Won

Score so far

Long-run win rate (S)Where it settles

The CHSH game in the nonlocal-game form given by Cleve, Høyer, Toner and Watrous (2004): questions x, y come from independent fair coins, and the team wins when a ⊕ b = x ∧ y. A crib sheet gives each player one of four rules; each of the 16 pairs loses at least one question pair, so none exceeds 3/4. Coins tossed on the spot are a mixture of the four rules (Fine 1982). The quantum strategy measures (|HH⟩ + |VV⟩)/√2 photon pairs at 0° or 45° (Alice) and ±22.5° (Bob), winning each question pair with probability 1/2 + √2V/4: 85.36% at V = 1, Tsirelson's bound. The box is the hypothetical Popescu–Rohrlich box. Rounds follow a fixed pseudo-random sequence; the luck line is this page's arithmetic.A two-question quiz for two players who can't talk. Give each a crib sheet and deal rounds: the best sheets win 3 rounds in 4 in the long run, and none wins them all. Coins tossed on the spot don't help. Perfectly clean entangled photons win 85.4% of every kind of round in the long run. The box that wins everything is imaginary.

The bound is a statement about the long run, and short quizzes can be lucky. Over 20 rounds, a team using a best crib sheet wins 17 or more, a score of 85% or better, with probability 22.5%, about one time in five; over 100 rounds it wins 85 or more with probability 1.1% (this page's arithmetic: exact binomial, win chance 3/4 per round, independent rounds). With a crib sheet or coins, the luck line under Figure 1 gives the chance for whatever you have dealt.

Every fixed crib sheet loses at least one kind of round every time, and Figure 1's grid shows which. The entangled photons of the next chapter lose about one round in seven of every kind, and no kind every time; which rounds they lost shows only when the two lists are compared.

Chapter 4A cosine beats a straight line

Now the quantum strategy. Before the game the players share many pairs of photons from a source that prepares each pair in the polarisation state

|Φ⁺⟩ = (|HH⟩ + |VV⟩)/√2H and V are horizontal and vertical polarisation; the first letter is Alice's photon, the second Bob's.

In words: each pair is in a superposition of "both horizontal" and "both vertical", with neither chosen. Quantum mechanics gives neither photon a polarisation of its own, yet the pair has a definite relationship: measured with aligned polarisers, at any angle, the two always agree. That is entanglement, a joint state of two photons that cannot be split into a state for each, and this pair is maximally entangled: on its own each photon is completely random, and everything definite about the pair lies in the relationship.

Each player sends one photon of a pair through a polariser set by their question and answers 1 if it passes, 0 if not. For this state, quantum mechanics predicts that the two answers agree with probability

P_same(Δ) = cos²Δ, E(Δ) = cos 2ΔΔ is the angle between the two polarisers. On its own, each photon passes with probability 1/2 at any angle; only the agreement depends on Δ. The correlation E of Chapter 2, agreement minus disagreement, is cos²Δ − sin²Δ = cos 2Δ.

The winning settings are 0° or 45° for Alice (questions 0 and 1) and 22.5° or −22.5° for Bob.^9,11 Three of the four question pairs then have polarisers 22.5° apart and want agreement, which happens with probability cos²22.5° ≈ 0.854. The fourth, (1, 1), puts Alice at 45° and Bob at −22.5°, 67.5° apart, and wants disagreement, which happens with probability sin²67.5°: the same 0.854. Every question pair is won with probability

cos²22.5° = (2 + √2)/4 = 0.853553…, S = 4 cos 45° = 2√2 ≈ 2.828Each correlation has magnitude cos 45° = 1/√2, with the sign the game wants. Then 1/2 + 2√2/8 = 85.36%, written 85.4% on this page.

Bell's 1964 paper contained the comparison that Figure 2 draws. Beside his theorem he wrote down a simple local model, a crib sheet in which each pair carries a hidden direction (his eq. (10), written for spins).¹² Where quantum mechanics gives a cosine, its correlation falls in a straight line as the angle grows. Rewritten for polarisers, the line is P_same = 1 − Δ/90° between 0° and 90°, and it meets the cosine at 0°, 45° and 90°. In between, the cosine bulges above the line below 45°, where agreement helps the team, and dips below it beyond 45°, where disagreement helps. That is the whole of the 85%: settings spaced θ apart put three question pairs at Δ = θ, in the bulge, and the fourth at 3θ, in the dip. Adding the four quantum correlations, cos 2θ three times and minus cos 6θ once, gives S for any spacing; Bell's line gives a flat 2 up to 30°:

S_quantum(θ) = 3 cos 2θ − cos 6θ, S_line(θ) = 2 (0 ≤ θ ≤ 30°)Settings 0 and 2θ for Alice, θ and −θ for Bob. The quantum value peaks at 2√2 when θ = 22.5°. Bell's line stays at exactly 2 up to 30°, then falls to 0 at 45°. A crib sheet that ignores the angles altogether, such as "both always answer 0", scores S = 2, 75%, at every spacing.

Figure 2📐 Try it

A cosine beats a straight lineA curve beats a line

Turn the dials

Dial spacing θTilt between the dials

Spacing

Agree at θ: cos²θMatch rate, entangled

Agree at θ: Bell's lineMatch rate, Bell's crib-sheet line

S (quantum / line)Quiz score, crib-sheet line

Game score, quantumQuiz score, entangled

Chance that polariser measurements on a (|HH⟩ + |VV⟩)/√2 photon pair agree, against the angle Δ between the polarisers. Quantum mechanics gives cos²Δ; Bell's own local model (his 1964 eq. (10), rewritten for polarisers) gives a straight line, 1 − Δ/90° up to 90° (a triangle wave beyond). Markers: the four CHSH setting pairs, three at Δ = θ (should agree) and one at 3θ (should differ). S is 3 cos 2θ − cos 6θ for quantum mechanics (2√2 at 22.5°) and 2 for the line up to θ = 30°; every local model has S ≤ 2. The spin switch shows spin-½ singlets: angles doubled, signs flipped (E = −cos φ), so Bob reports the opposite of each result. This page's arithmetic.How often two entangled photons give the same answer as you tilt the detectors apart. The curve is what quantum physics predicts; the straight line is one crib-sheet model, John Bell's own example (no crib sheet of any kind scores more than 75%). Lined up, both always match. In between, the curve pulls ahead, and at a 22.5° tilt it is worth the 85% quiz score.

Perfect correlations can be faked: with the dials lined up, θ = 0°, the photons always agree, and so does a crib sheet, with S = 2 for both. That is why the perfect correlations of Einstein, Podolsky and Rosen (Chapter 7) settle nothing on their own.

Real photon pairs are never perfect. In the simplest model a source of imperfect pairs multiplies every correlation by a visibility V between 0 and 1, so that E = V cos 2Δ and

S = 2√2 V, P_win = 1/2 + √2 V/4, P_win > 3/4 only if V > 1/√2 ≈ 70.7%This page's arithmetic. At V = 0.90 the team wins 81.8%; at V = 0.70, 74.7%, just short of the best crib sheet. Figure 1's slider sets V.

Many experiments measure spins rather than polarisations: electrons in diamond, trapped atoms and ions. For spin-½ pairs in the singlet state the angles double and the signs flip: directions φ apart give E = −cos φ, the best directions are 0° and 90° for Alice and ±45° for Bob, and the raw value is S = −2√2. It is the same game with one player reporting the opposite of each result: |S| = 2.83, a score of 85.4% (this page's arithmetic), as Figure 2's spin switch shows.⁹

Chapter 5No message gets through

An 85% score looks like cheating, as if the photons were passing notes. The statistics alone cannot rule that out: a team that secretly passed Alice's question to Bob could win every round while each player's answers still came out half 0s. What rules out notes is that the stations are too far apart for one to arrive in time (Chapters 8 and 9). The statistics show something narrower: neither player can use the game to send the other a message. Look at Alice's answers alone. For the state of Chapter 4, each of her answers is 0 or 1 with probability exactly 1/2, whatever her question and whatever Bob's:

P(a | x, y) = P(a | x) = 1/2The probability of Alice's answer a, given her question x and Bob's question y; the same holds for Bob. For any shared state, quantum mechanics makes the odds of each player's answers independent of the other's setting; the value 1/2 is special to this maximally entangled state.

This is the no-signalling theorem, proved by Philippe Eberhard in 1978 and by Giancarlo Ghirardi, Alberto Rimini and Tullio Weber in 1980: in quantum mechanics, the odds of one party's outcomes do not depend on the other party's choice of measurement, so choosing measurements cannot be used to send a message.^6,7 The correlation is real, but it lives in the joint statistics, and Alice can see it only with Bob's list beside her own, which takes an ordinary signal, at or below the speed of light, to reach her.

In Figure 3 Bob tries to break the rule. In "Send a message" he ignores the referee's coin and picks his own question, 1 in every round for YES and 0 for NO, and Alice guesses the secret from her own answers. A made-up "leaky box" shows what a real signal would look like.

Figure 3📵 Try it

Try to send a message

Bob tries to signal

Rounds per run

P(a = 0 | Bob asked 0)Alice's 0s when Bob got 0

P(a = 0 | Bob asked 1)Alice's 0s when Bob got 1

Gap

Chance allows the gap (2σ)Luck allows

The no-signalling check, with the photon strategy of Figure 1 (V = 1). In "Play the game" the bars are Alice's fractions of 0 answers when Bob is asked 0 and when he is asked 1, each pooled over her own question; the shaded band is two standard deviations of their gap, ±2/√N. For any state, quantum mechanics makes P(a | x, y) independent of y (Eberhard 1978; Ghirardi, Rimini and Weber 1980); for these photon pairs it is 1/2 for every pair. In "Send a message" Alice guesses YES when fewer than half of her N answers are 0. The joint statistics do depend on Bob's question, but only someone holding both lists can see that. This page's model and arithmetic.Bob tries to send Alice a message by choosing his questions. Alice's notebook alone stays at about half 0s whatever he does, so she can't read it. The matching only shows when the two notebooks are put side by side, and that needs an ordinary phone call. The 'leaky box' is pretend: it shows what a real signal would look like.

Figure 3's two bars are Alice's shares of 0s when Bob is asked 0 and when he is asked 1. With entangled photons they differ only by chance. Over N rounds their gap has a standard deviation, the typical size of its chance wobble, of 1/√N; after 1,000 rounds two standard deviations is ±6.3 points, and the gap falls inside that allowance about 19 runs in 20 (this page's arithmetic). Her guesses at Bob's secret are right half the time in the long run; the leaky box is read almost every time.

The careful wording matters. The theorem guarantees only that the odds of each player's answers ignore the other's question. It says nothing about individual results, and interpretations differ about those. In Bohm's theory (Chapter 7) an individual result does depend on the distant setting; only the statistics do not, and that is enough to make the dependence useless for sending messages.¹³

Experiments check this too, since dependence on the far setting would point to a leak between stations, and the checks are not entirely quiet. In 2017 Adam Bednorz reanalysed the data of the first loophole-free tests, which closed the locality and detection loopholes together (Chapter 9): Delft's two runs and the tests at Vienna, Munich and the US National Institute of Standards and Technology (NIST). He found small anomalies in the Delft and NIST data, of a size chance alone would produce about one time in 20 once he allowed for the number of places he had looked; in one of the two they bore on the independence of the setting choices rather than on no-signalling. He described the result as moderately suggesting a violation of no-signalling, but wrote that the data were too weak to refute no-signalling and that any signalling might be only apparent, from a synchronisation error for example. He called for further experiments.¹⁴

Chapter 6Why does quantum physics stop at 85%?

Could a cleverer state or cleverer measurements beat 85.4%? No. Boris Tsirelson proved in 1980 that no quantum state and no choice of measurements makes S larger than 2√2.² A textbook argument shows why. Write each player's two measurements as observables, the quantities a measurement reads out, here with the answers 0 and 1 written as +1 and −1, so that each squares to 1. Alice's commute with Bob's: the order in which the two players measure makes no difference. Then the CHSH combination Ĉ = Â₀(B̂₀ + B̂₁) + Â₁(B̂₀ − B̂₁) satisfies

Ĉ² = 4 − [Â₀, Â₁][B̂₀, B̂₁], |⟨Ĉ⟩| ≤ 2√2A commutator such as [Â₀, Â₁] = Â₀Â₁ − Â₁Â₀ measures how much the order of one player's two measurements matters; each has norm (largest possible size) at most 2, so Ĉ² is at most 8 and S = ⟨Ĉ⟩, the average of Ĉ that quantum mechanics predicts, at most 2√2. If either player's two observables commuted, as they would for answers fixed in advance, S could not exceed 2.

Nor is 85.4% the most that a theory without signalling could allow. In 1994 Sandu Popescu and Daniel Rohrlich described an imaginary device, now called a PR box, that wins every round: Alice's answer is a fair coin, and Bob's equals hers except when both questions are 1, when it is the opposite.¹⁵ It gives S = 4, and Alice's answers are still half 0s whatever Bob is asked, so it sends no message. Forbidding messages allows anything up to 100%:

local: S ≤ 2 (75%) < quantum: S ≤ 2√2 (85.4%) < no-signalling: S ≤ 4 (100%)The three ceilings of the CHSH game. The PR box is hypothetical: no known physics makes one. Figure 1's fourth mode plays it.

So no-signalling does not explain the 85%; why nature stops there is open. One proposed answer is information causality, put forward by Marcin Pawłowski and colleagues in 2009: receiving m classical bits from a partner can never give you more than m bits of information about the partner's data, whatever correlations the two of you share in advance. Correlations stronger than quantum mechanics allows, the PR box among them, would break it.¹⁶ It is one proposal among several, not the accepted answer.

Experiments have come close to the quantum ceiling. A 2015 photon experiment in Singapore measured S = 2.82759 ± 0.00051, within a thousandth of 2√2: 85.34% among detected pairs, on this page's arithmetic, with no loophole closed.¹⁷ It is the most precise approach to the ceiling we found, and its authors call its distance from the bound the smallest "ever reported". Another 2015 experiment, by Bradley Christensen and colleagues, lists S = 2.828, so which is "closest" depends on whether one ranks by central value or by error bar; it calls the results of its "chained" inequalities, longer versions with more settings per player, "the most nonlocal … ever reported".¹⁸

Part I leaves two ceilings: 75% for a crib sheet, 85.4% for quantum mechanics. Part II asks whether real experiments beat the first.

Part IIPlaying it for real

Chapter 7From Einstein's worry to Bell's limit

The story starts with a worry about completeness. In 1935 Albert Einstein, Boris Podolsky and Nathan Rosen considered two particles prepared so that measuring the same quantity on both gives perfectly correlated results. Measure one, and you can predict the result for the other with certainty without touching it. If nothing done here can disturb what is over there, they argued, the far particle must have had that value all along, and since quantum mechanics does not assign it one, the theory must be incomplete.¹⁹ The same year Erwin Schrödinger gave the phenomenon its name, entanglement.²⁰ Einstein's discomfort lasted. In a 1947 letter to Max Born he called the distant influence the theory seemed to imply "spooky action at a distance".²¹

Correlation on its own is not mysterious. Bell's own example was the physicist Reinhold Bertlmann, who liked to wear socks of different colours: see that one of his socks is pink and you know at once that the other is not, with nothing travelling from one foot to the other.²² The colours were settled when he dressed. EPR's perfect correlations have this kind of explanation available, a crib sheet carried from the source.

Could quantum mechanics be completed with such instructions? In 1952 David Bohm showed that a theory with hidden variables can reproduce all of quantum mechanics' predictions: in his theory every particle has a definite position at every moment. But the theory is explicitly nonlocal: what happens to one particle depends on what is done to the other, however far apart they are.¹³

John Bell asked whether that nonlocality was the price of any completion. In 1964 Bell, a CERN theorist then on leave in the United States, proved that it is: no local hidden-variable theory can reproduce all of quantum mechanics' predictions.¹²

Bell's derivation relied on perfect correlation at equal settings, which no real apparatus delivers. In 1969 John Clauser, Michael Horne, Abner Shimony and Richard Holt rewrote the limit as the CHSH inequality of Chapter 2, which tolerates imperfect equipment and so could be tested.¹

Bell was born in Belfast on 28 July 1928, worked at CERN from 1960, and died in Geneva on 1 October 1990.²³

One more distinction matters for Part III. In 1964 Bell worked with outcomes fixed in advance by the hidden variables, which he argued followed from locality and the perfect correlations, as in EPR's argument; in 1976 he derived the limit from an assumption he called local causality, which allows the outcomes to be random. Howard Wiseman has called these "the two Bell's theorems", and they are read differently by different camps (Chapter 12).²⁴

Chapter 8The first tests and their two loopholes

Stuart Freedman and John Clauser at Berkeley ran the first test, published in 1972. Their photon pairs came from calcium atoms, each giving off two photons in quick succession, and the detectors were about 5 m apart. They tested a form of the limit written as δ ≤ 0 and found δ = 0.050 ± 0.008, a violation of about six standard deviations, in agreement with quantum mechanics.²⁵ The settings were static, so no loophole was closed. Two loopholes mattered most.

Locality. If one station's setting is chosen early enough, a signal at or below the speed of light could tell the other station what it is before the other measurement ends, and a crib sheet that read such signals could beat 75%. Closing the loophole means choosing each setting at random, and finishing each measurement, inside a window too short for light to cross between the stations: the choice at one end and the measurement at the other must be space-like separated.

Detection. Most photons in the early experiments were never detected, and the inequality was evaluated on the rounds in which both detectors fired, assuming those were a fair sample of all the rounds: the fair-sampling assumption. A crib sheet allowed to say "don't click" can fake a violation in the detected subset. Closing the loophole needs efficient detection:^26,27

η > 2/(1 + √2) ≈ 82.8% (CHSH, maximally entangled state) η > 2/3 (CH–Eberhard inequality, unbalanced states)η is each station's overall detection efficiency. Below the threshold, a local model that uses non-detections can reproduce the detected statistics (Garg and Mermin 1987; Eberhard 1993).

In words: a CHSH test needs each station to detect more than about 83% of its photons. The CH–Eberhard inequality gets by with more than two-thirds, using deliberately unbalanced pairs, less than maximally entangled (Eberhard 1993).²⁷ It is the CHSH inequality with each non-detection counted as one of the answers, but the tests on this page that use it report a p-value rather than an S, so this page quotes no score for them.⁹

In 1982 Alain Aspect, Philippe Grangier and Gérard Roger at Orsay used two-channel polarisers, which detect the photons a polariser blocks as well as those it passes, and measured S = 2.697 ± 0.015 against a quantum prediction of 2.70 ± 0.05.²⁸ That is about 46 standard deviations above 2, (S − 2)/σ on this page's arithmetic, and a score of 83.7% among detected pairs; the Nobel committee's background says "tens of standard deviations".²⁹ In a second experiment that year Aspect, Jean Dalibard and Roger changed the settings while the photons were in flight, though not at random. Optical switches redirected each photon to one of two polarisers about every 10 ns, with the two sides 12 m apart, 40 ns for light (this page's arithmetic). The switching was quasi-periodic, nearly regular, although the Nobel committee's background describes the settings as changing randomly.^30,29 That test used a different inequality, with a bound of 0, and found S′ = 0.101 ± 0.020, about five standard deviations (this page's arithmetic). It was the first attack on the locality loophole, not its closure.

Anton Zeilinger's group in Innsbruck closed the locality loophole in 1998, with stations 400 m apart (1.33 µs for light, this page's arithmetic) and the settings chosen at random: S = 2.73 ± 0.02 from 14,700 detected pairs, 30 standard deviations.³¹ The detection loophole stayed open: the score is 84.1% among detected pairs. In 2001 a NIST group in Boulder closed the detection loophole instead, with two trapped beryllium ions read out in every trial: S = 2.25 ± 0.03, about 8.3 standard deviations (this page's arithmetic), 78.1% per valid trial.³² The ions were micrometres apart, so locality stayed open. Neither experiment closed both, and none would until 2015.

These scores are not like for like: about 84% among detected pairs where detection was left open, 78% per valid trial where it was closed. Figure 4 marks the difference by shape; its highest scores all belong to tests that left the detection loophole open.

Chapter 92015: both loopholes closed

The decisive experiments came in 2015, from three groups. At Delft, Bas Hensen, Ronald Hanson and colleagues used the electron spins of defects in diamond in two labs 1,280 m apart, 4.27 µs for light (this page's arithmetic). Each spin was entangled with a photon; the photons met at a station between the labs, where a joint detection heralded that the distant spins were entangled. In every heralded trial both spins were read out, closing the detection loophole, and settings and readouts were finished before light could cross between the labs. Over 220 hours they recorded 245 heralded trials, with S = 2.42 ± 0.20.³

Delft counted the wins directly. In the experiment's version of the game, with the answers relabelled to suit the heralded state, the team won 196 of 245 rounds: 80.0% per heralded trial, matching the 80% ± 2.5 points that its S implies (this page's arithmetic). The statistics were modest: p = 0.039, so a world run on crib sheets would produce a result this strong about one time in 25 (Chapter 10). That is the paper's "complete" analysis, which allows for memory between trials and predictability in the settings. A second run in 2016, again with both loopholes closed, won 237 of 300 (79.0%, p = 0.061 in the complete analysis); the two runs together won 433 of 545 (79.4%), with a combined p-value between 0.0026 and 0.017, depending on the analysis.³³

Vienna and Boulder followed in December 2015, with photons and detectors efficient enough for the CH–Eberhard inequality. Marissa Giustina, Anton Zeilinger and colleagues put their stations 58 m apart in the Hofburg, Vienna's former imperial palace, and found p ≤ 3.74 × 10⁻³¹, 11.5 standard deviations.³⁴ Krister Shalm and colleagues at NIST put theirs about 185 m apart and reported p-values as small as 5.9 × 10⁻⁹, or 2.3 × 10⁻⁷ after allowing for setting predictability: about five standard deviations (this page's arithmetic).³⁵ Neither paper reports an S, so neither has a score here (Chapter 8).

Munich completed the set in 2017. Wenjamin Rosenfeld and colleagues entangled two rubidium atoms in labs 398 m apart with an event-ready scheme like Delft's and measured S = 2.221 ± 0.033, with p < 2.57 × 10⁻⁹: about 5.8 standard deviations and 77.8 ± 0.4% per heralded trial (this page's arithmetic).⁴ The Nobel committee's background describes these four groups, in 2015–17, as the ones that closed the locality and detection loopholes together.²⁹

More have followed since, among them a 2018 starlight test in Shanghai (Chapter 11), two 2018 randomness experiments (Chapter 14) and, in 2023, ETH Zurich. Simon Storz, Andreas Wallraff and colleagues joined superconducting circuits through a 30 m cryogenic link, 100 ns for light (this page's arithmetic), and over more than a million trials measured S = 2.0747 ± 0.0033 with p < 10⁻¹⁰⁸.⁵ That is 75.93 ± 0.04% per valid trial (this page's arithmetic): the lowest score in the table below and by far its strongest statistics (Chapter 10).

Test	Apart	Light time	Result	Score
Delft, 2015 (spins in diamond)	1,280 m	4.27 µs	S = 2.42 ± 0.20, p = 0.039	196 of 245 (80.0%)
Vienna, 2015 (photons)	58 m	193 ns	CH–Eberhard, 11.5σ	none
Boulder (NIST), 2015 (photons)	≈185 m	617 ns	CH–Eberhard, about 5σ	none
Munich, 2017 (atoms)	398 m	1.33 µs	S = 2.221 ± 0.033	77.8%
Shanghai, 2018 (photons, starlight settings)	183 m	610 ns	CH–Eberhard, p ≤ 7.873 × 10⁻⁴, about 3.2σ	none
Zurich, 2023 (superconducting circuits)	30 m	100 ns	S = 2.0747 ± 0.0033, p < 10⁻¹⁰⁸	75.93%

Tests that closed the locality and detection loopholes together: examples, not a complete list. Light times, scores and the significances of NIST and Shanghai are this page's arithmetic; scores are per heralded or valid trial, and Delft's is its own count of wins.

Distance is a separate achievement, and it came with the detection loophole open. In 2017 the Micius satellite sent entangled photons to two stations in China, at Delingha and Lijiang, 1,203 km apart, 4.01 ms for light (this page's arithmetic). Juan Yin, Jian-Wei Pan and colleagues recorded 1,167 trials in 1,059 seconds and measured S = 2.37 ± 0.09, four standard deviations, with space-like separated, randomly chosen settings. The locality loophole was closed. But with only about one pair a second getting through, most photons were lost on the way, so the detection loophole stayed open and the test is not loophole-free. Its score is 79.6 ± 1.1% among detected pairs (this page's arithmetic).³⁶

Figure 4 puts the tests together: height is the equivalent game score, shape says which loopholes were closed.

Figure 4🔬 Try it

Fifty years of Bell tests

Step through fifty years

● both loopholes closed · ○ detection open (among detected pairs) · ■ detection closed only · ◆ ◇ no single CHSH score · dashed ring: preprint● both loopholes shut · ○ only the caught pairs counted · ■ detection loophole shut only · ◆ ◇ no single quiz score · dotted, quasar test: how far back a common cause of both quasars' light would have to liedotted: how old anything that set both quasars' light would have to be

Step through the testsPick a test

Test

Apart (light time)How far apart

Score

SignificanceHow sure

Loopholes closedLoopholes shut

Bell tests from 1972 to 2026 (examples, not a complete list). For CHSH tests the height is the equivalent game score 1/2 + S/8 with error bar ΔS/8, this page's arithmetic: among detected pairs, under fair sampling, for detection-open tests (rings); per valid or heralded trial for event-ready and efficient-detection tests (filled). Two-run tests get two markers. CH–Eberhard tests (Vienna and NIST 2015, Shanghai 2018 and CURBy) and tests with no single CHSH value in the sources used here sit on the strip below, filled where both loopholes were closed. Significances are labelled as the paper's or this page's arithmetic; light times (separation/c) and the quasar test's ≈1.0 km are this page's arithmetic. Orsay 1982's score is its static two-channel test's; the 12 m belongs to its time-varying test. The quantum ceiling, 85.36%, is a bound, not a measurement. CURBy's record and Padova's preprint status are as of October 2026.Every dot is a real experiment (two dots where a test had two runs); higher means a better quiz score. The short line through a dot, and the ± in the Score box, show the typical size of luck's wobble on that score. Filled dots had both main loopholes shut; rings only counted the pairs they caught; the square shut only the detection loophole. Diamonds along the bottom have no single quiz score: five give no number this page can turn into one, and the volunteers' test was 13 experiments with no single score. The dashed line at 85.4% is the most quantum physics allows. No loophole-free test got near it, and none needed to: beating 75% is the point. Switch the view to see how far apart the stations were, or how old the starlight that picked the questions.

Why at most 80%, and not 85.4%? Imperfect pairs lower S (Chapter 4's visibility model, S = 2√2 V), and the papers give their own error budgets; this page blames no shortfall on any one cause. The point was never to reach 85.4%, but to beat 75% with neither main loophole left for a crib sheet to exploit.

Chapter 10One time in 25

Delft's p-value of 0.039 is easy to misread. A p-value answers a narrow question: if the world were local-realist, run on crib sheets (Chapter 12), how often would an experiment like this one produce a result at least this strong? Delft's answer is about one time in 25.³ It is not the probability that local realism is true. A p-value is computed by assuming the hypothesis it tests; it cannot also give that hypothesis's probability.

A simple version takes a few lines on a spreadsheet. A team using a best crib sheet wins each round with probability 3/4; if the rounds are independent, the chance that it wins at least k of N is a binomial tail:

p = P(K ≥ k) = Σ_{j = k}^N C(N, j) (3/4)^j (1/4)^N−j, z = (k/N − 3/4) / √(3/(16N))C(N, j) is a binomial coefficient. z is the number of standard deviations by which the score k/N exceeds 75%, on the crib-sheet team's own spread.

For 196 wins in 245 rounds, this page's arithmetic gives p = 0.039 and z = 1.8: the same, to two figures, as the p-value Delft published. Delft's analysis does not assume independent rounds, so the agreement is reassuring, not a check of its method.

How many rounds does a perfect quantum team need to stand clear of luck? Its expected score of 85.36% sits five standard deviations above 75% after about 440 rounds. That is an average: at that point a perfect team actually reaches five standard deviations only 48% of the time, and it needs about 700 rounds to be 95% sure of doing so (this page's arithmetic, exact binomial).

In Figure 5 the shaded tail is the p-value for the run you pick.

Figure 5🍀 Try it

Luck or physics?

Pick a run, set the wins

Rounds

Wins

Score

Above 75% byPast 75% by

Crib-sheet team does this well: pA crib-sheet team gets this lucky

Perfect quantum team does at least this wellA perfect quantum team gets here

The chance of a lucky run. The distributions of wins over N independent rounds for a team using a best crib sheet (win chance 3/4) and for a perfect quantum team (win chance cos²22.5° = 0.8536). The shaded tail is the p-value for k wins against the local-realist hypothesis, P(K ≥ k), exact binomial; z = (k/N − 0.75)/√(0.1875/N). Real analyses, Delft's among them, also allow for memory between rounds and imperfect setting choices, so their p-values need not equal this simple tail. All values this page's arithmetic; experiment counts from Hensen et al. 2015 and 2016.How often luck alone fakes a good score. One hump, labelled on the plot, is a crib-sheet team playing many quizzes of this length; the other is a perfect quantum team. With a few rounds they overlap and luck can fool you. With hundreds they pull apart.

Drag the rounds up at Delft's 80% and watch the crib-sheet tail shrink: the same score over more rounds makes luck far less likely, which is how the lowest score in Chapter 9's table has the strongest statistics. ETH Zurich's 75.93% beats 75% by less than one point, but over more than a million trials the paper puts the chance of a local-realist world doing as well below 10⁻¹⁰⁸, more than 22 standard deviations (this page's arithmetic from the paper's p-value).⁵ Figure 5 stops at 1,995 rounds, so it cannot show this case.

One caution: a significance measures how surprising the data would be for a crib sheet, not how well the loopholes were closed. A loophole-open test can show enormous significance and still leave a crib sheet a way out.

Chapter 11Who chooses the questions?

Every Bell test assumes one more thing: that the questions are chosen independently of whatever fixes the answers. The assumption is called freedom of choice, or measurement independence. If the random-number generators that pick the settings were correlated with the particles' hidden variables, a crib sheet could know its questions in advance and beat 75%. Whatever picks a setting has a past, which could in principle have influenced the source too, so the assumption can only be pushed back, never closed outright.^29,37

The first step came on La Palma and Tenerife. In 2010 a Vienna-led team, with Thomas Scheidl as first author, closed the locality loophole over 144 km, 480 µs for light (this page's arithmetic), and also made each setting choice space-like separated from the emission of the photon pair. It measured S = 2.37 ± 0.02, more than 16 standard deviations, 79.6 ± 0.3% among detected pairs (this page's arithmetic), relying on fair sampling.³⁸ The paper says it closes the freedom-of-choice loophole under assumptions that include non-deterministic setting choices; this page says only that it addressed the loophole.

Then the settings were handed to the sky. In April 2016 Johannes Handsteiner and colleagues in Vienna let the colour of single photons from Milky Way stars, 577 to 3,624 light-years away, choose the settings. Any local mechanism fixing the questions to match the answers must then have acted no later than about 600 years before the test, when the light left the nearer star of each pair: before Gutenberg's printing press, the paper's own comparison. The two runs gave S = 2.425 and 2.502, at least 7.31 and 11.93 standard deviations: 80.3% and 81.3% among detected pairs, on this page's arithmetic.¹¹

In 2018 the same group (first author Dominik Rauch) used light from quasars, the bright cores of distant galaxies: in the first run, light emitted 7.78 and 12.21 billion years ago; in the second, light at least 3.22 billion years old. The first run excluded local mechanisms acting on the settings from 96% of the experiment's past light cone: the region of space and time from which anything at or below the speed of light could have reached it. Any common cause of both quasars' light would have to date from at least 13.15 billion years ago, less than 650 million years after the big bang. The first pair of quasars gave S = 2.65, 9.3 standard deviations, and a second pair gave 2.63, 7.1 standard deviations: 83.1% and 82.9% among detected pairs, with stations about 1.0 km apart (the scores and the distance are this page's arithmetic, the distance worked out from the paper's coordinates).³⁹

Both cosmic tests assumed fair sampling, so neither was loophole-free. In 2018 Ming-Han Li and colleagues in Jian-Wei Pan's group at the University of Science and Technology of China (USTC), working in Shanghai, closed both main loopholes with photon pairs 183 m apart, the settings chosen by starlight at least 11.46 years old, in the paper's conservative figure, from pairs of stars on opposite sides of the sky. The four runs of 23 March 2018, each with a different pair of stars, together gave p ≤ 7.873 × 10⁻⁴ after allowing for a time-varying setting bias, about 3.2 standard deviations (this page's arithmetic). Like Vienna and NIST, the test used the CH–Eberhard inequality and reports a p-value rather than an S, so this page quotes no score for it.⁴⁰

The BIG Bell Test used people. On 30 November 2016 about 100,000 volunteers playing an online game for 12 hours produced 97,347,490 binary choices, which set the questions for 13 experiments in 12 laboratories on five continents. Human choices are too slow to be space-like separated from the measurements, so in all 13 the locality loophole stays open as far as the choices are concerned. Only experiment 13, at NIST with stations 187 m apart, had both space-like separated measurements and detection efficient enough to need no fair sampling. Most experiments, in the paper's words, "observed statistically strong violations of their respective inequalities". The paper says laboratory methods can tighten the freedom-of-choice loophole but never close it, and that human choices, taken as free, do close it; because that assumes the volunteers' choices owed nothing to the hidden variables, this page counts the test as a different approach, not a closure.⁴¹

Figure 4's "How old the questions" view lines up the three tests whose settings came from starlight or quasars by the age of that light. Chapter 13 returns to the minority view that the assumption fails.

Part IIIWhat gives way

Chapter 12What the experiments rule out

The mainstream summary is that the experiments rule out local realism: theories in which results are fixed by properties carried with the particles (realism) and no influence travels faster than light (locality), with settings chosen independently of those properties. The Nobel committee put it this way: Bell showed that "all attempts to construct a local realist model of quantum phenomena are doomed to fail".²⁹ Every such statement assumes that each measurement has a single outcome and that the questions were chosen freely (Chapter 11).

The committee's background defines the words like this: "Bell used the words local and realist here in a technical sense: the former indicates the impossibility of instantaneous signalling, limited by the finite speed of light, and the latter means that the outcome of any experiment is fully determined by properties of the system, often referred to as hidden, that exist independently of any actual or potential measurement."²⁹ This page does not adopt that definition of "local". Quantum mechanics itself never signals (Chapter 5), so by that definition quantum mechanics would count as local, and the only thing left to give up would be realism. This page uses Bell's own notion instead, local causality: no causal influence of any kind, signal or not, travels faster than light.²⁴

Give up predetermined answers

Keeps: No signal crosses the gap: the odds at each end ignore the distant setting
Gives up: Answers fixed before the questions
The correlation: Has no explanation that stays on each side
Typical of: Operational readings of quantum mechanics

Give up locality

Keeps: Outcomes fixed by real properties
Gives up: Local causality: something far away matters
The correlation: Explained by a nonlocal influence that cannot be used to signal
Typical of: Realist readings, Bohm's theory among them

The first column is not the easy escape it looks like. Giving up predetermined answers does not save local causality, because answers tossed up on the spot at each end are just another crib sheet, capped at 75% (Chapter 3).¹⁰ It keeps only the weaker kind of locality, no signalling, and the correlation itself has no local explanation. Tim Maudlin and Travis Norsen argue that this weaker sense does not deserve the name locality, and that "realism" is not a separate assumption: the violation of Bell's inequality shows that nature itself is nonlocal, whatever else one believes.^42,43 That is one camp's reading, contested by others, and this page does not adopt it.

The second column has a working theory in it. Bohm's theory of 1952 keeps definite positions and gives up locality openly, and it reproduces every prediction of this page, 85.4% included.¹³ So Bell did not show that hidden variables are impossible, and the experiments did not refute realism. What they showed is narrower and firmer: no crib sheet carried with the particles, read only by nearby events, with freely chosen questions, reproduces what the detectors record.

Where does that leave Einstein? EPR's argument assumed that nothing done at one station could affect the other. The local picture Einstein hoped for cannot be kept, if each measurement has one outcome and the questions are chosen freely. His other claim, that quantum mechanics is incomplete, is a separate question that the experiments do not settle.

Chapter 13Escape routes

The minority routes keep locality in Bell's sense by denying one of the theorem's other assumptions. None is ruled out by experiment, and each has serious critics.

Every outcome happens. Bell's theorem assumes that each measurement has a single outcome. In Everett's many-worlds theory both outcomes occur, each in its own branch, and some of its supporters argue that the theory is then local. In 2000 David Deutsch and Patrick Hayden gave an account of local information flow in entangled systems, and Harvey Brown and Christopher Timpson made a similar case in 2016.^44,45 A 2026 volume edited by Alyssa Ney collects both sides: Lev Vaidman argues that many-worlds is the only way to avoid action at a distance, and Mordecai Waegell and Kelvin McQueen find nonlocal action within it.⁴⁶ Many-worlds framing appears on this site in Nº 3, and Nº 10 explains why decoherence, the process that splits the branches, does not by itself say why we see one outcome: that question is open.

The questions were never free. Superdeterminism holds that the hidden variables and the settings are correlated, so measurement independence fails. Sabine Hossenfelder and Tim Palmer argue for taking it seriously, and Jonte Hance and Hossenfelder argue that Bell's theorem "allows local theories of quantum mechanics" once that assumption is dropped.^37,47 It is a minority view and logically open. Critics since Shimony, Horne and Clauser in 1976 have argued that it would undermine randomised experiments of every kind.⁴⁸ The starlight and quasar tests squeeze the simplest versions but cannot exclude the idea in principle.

Influences run backwards. Retrocausal accounts let a measurement setting influence the particles' past, inside the experiment, so that a local, time-symmetric story can produce the correlation. Kenneth Wharton and Nathan Argaman reviewed such "locally mediated" reformulations of quantum mechanics in 2020. It is another minority proposal.⁴⁹

One more assumption hides in the word "measurement": when does it end? If the collapse of the quantum state happened late, for example on a gravitational timescale, the outcomes might not be space-like separated after all. Adrian Kent named this the collapse-locality loophole in a paper posted in 2002 and published in 2005.⁵⁰ In 2008 Daniel Salart, Nicolas Gisin, Hugo Zbinden and colleagues in Geneva built a test aimed at it: each detection triggered a piezoelectric actuator that moved a 2 mg mirror, with the stations 18 km apart, 60 µs for light. Under the Diósi–Penrose criterion for when a superposition of a mass collapses, each measurement finished within 7.1 µs, the paper estimates, "almost one order of magnitude shorter" than the light time. The authors say the experiment closes the loophole on that criterion; it inferred the violation from two-photon interference and did not address detection.⁵¹ For other ideas about when collapse happens, the loophole was, in Kent's words of 2020, "largely untested".⁵² Since then a preprint from CEILAP, a laser research centre in Villa Martelli, Argentina, posted in March 2026 and not refereed as of October 2026, reports a test with stations 24 m apart. It says the test closes, whatever the theory of collapse, a weaker "essential" form of the loophole, in which a late collapse could depend only on the other station's result. Its settings were known well in advance, so the full form, in which the collapse could depend on the other station's setting, stays open, as far as we can find.⁵³ Nº 10 describes the Diósi–Penrose idea in its chapters on collapse models and Penrose's clock.

Most of the readings in this chapter and the last leave the 85.4% untouched: many-worlds, Bohm's theory and the operational readings all predict it, and the superdeterministic and retrocausal proposals are built to reproduce it, so Bell tests of this kind are not designed to decide between them. Some versions do predict differences elsewhere. Hossenfelder and Palmer suggest how superdeterminism might be tested,³⁷ and Kent's causal quantum theory, from which the collapse-locality loophole comes, predicts no violation at all once the collapses are space-like separated.⁵⁰

Chapter 14From puzzle to tool

On 4 October 2022 the Royal Swedish Academy of Sciences announced that the Nobel Prize in Physics would be awarded jointly, in equal thirds, to Alain Aspect, John F. Clauser and Anton Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science".⁵⁴ Bell had died in 1990, and the prize is not awarded posthumously.

A Bell violation certifies something no crib sheet can fake, and that makes it useful. If two devices beat 75% with the loopholes closed, their answers cannot have been written down in advance, by the manufacturer or anyone else. Given freely chosen settings, the answers must contain genuine randomness, and that guarantee rests on the observed statistics, not on trusting how the devices work. That is the idea of device-independent protocols.

Random numbers. In 2018 two experiments, one at NIST and one at USTC, generated random numbers certified by loophole-free Bell tests.^55,56 USTC's test won 75.03% of its 6.895 × 10¹⁰ trials, with p ≤ 10^−204792 (the paper's bound, which assumes the settings were chosen uniformly at random in every trial; the score is this page's arithmetic from the paper's CHSH game value).⁵⁶ NIST and the University of Colorado then built CURBy, a public randomness beacon (a service that posts random numbers for anyone to use) whose numbers were certified the same way, with stations about 110 m apart (367 ns for light, this page's arithmetic). The guarantee holds against an adversary who has only ordinary, non-quantum information about the devices. The Bell experiment ran intermittently, and its public record of certified output spans October 2023 to 22 August 2025 (checked 25 October 2026). The paper, published in Nature in June 2025, reports that the protocol ran to completion in 7,434 of 7,454 attempts in its first 40 days.⁵⁷

Secret keys. In 2022 an Oxford experiment with two trapped-ion nodes about 2 m apart, linked by optical fibre, produced the first key certified by a Bell test without trusting the hardware, as far as we can find; its devices were shielded from each other rather than separated in space.⁵⁸ Munich showed the approach could work with atoms between buildings 400 m apart, reaching a positive key rate of 0.07 per entanglement event on estimates that assume unlimited rounds; security for the finite number of rounds actually run was analysed but not achieved.⁵⁹ A USTC photon experiment the same year was, in its own title, a step "toward" a photonic demonstration.⁶⁰ In February 2026 a USTC team reported device-independent key distribution between two single atoms in one laboratory, joined through spooled fibre. It found positive key rates for up to 100 km of fibre on the same unlimited-rounds estimates as Munich's and, over 11 km, an estimated key rate that allows for the finite number of rounds actually run.⁶¹

A recent test. A preprint from a group in Padova, posted in August 2026 and not refereed as of October 2026, reports a Bell test with time-bin photons, entangled in their arrival times, over fibre between stations 49 m apart, 163 ns for light, measuring S = 2.583 ± 0.002, more than 265 standard deviations: 82.3% among detected pairs (the light time and the score are this page's arithmetic). It closes the locality loophole and the postselection loophole, which arises when a time-bin test keeps only the rounds whose arrival times line up, and, like the 2010 Canary Islands test, addresses freedom of choice by choosing each setting space-like separated from the pair's emission; as it states, it does not close the detection loophole.⁶²

Chapter 15What is settled, and what is open

Settled: no plan agreed in advance wins more than 75% of rounds in the long run; quantum mechanics allows up to 85.4% and no more; since 2015, tests that closed the locality and detection loopholes together have beaten 75%, scoring from just above 75% to about 80% where a score can be worked out; and none has shown the odds of one player's answers depending on the other's question. Open: which assumption gives way, and why nature stops at 85.4% rather than at 75% or 100%. The freedom-of-choice assumption can be pushed back but never removed, and the collapse-locality loophole, in its full form, remains open for collapse criteria other than the one tested in Geneva, as far as we can find.

John Bell found the limit in 1964; Clauser, Horne, Shimony and Holt turned it into a test in 1969; Boris Tsirelson found the quantum ceiling in 1980; Clauser (1972), Aspect (1982) and Zeilinger's group (from 1998) led the experiments, and the 2022 Nobel Prize went to Clauser, Aspect and Zeilinger.

Five things the result does not mean:

A message faster than light: The odds of each player's answers ignore the other's question, so no message can be sent (Chapter 5).
An instant change in the far particle: That is one interpretation's picture, the collapse picture, not a measured fact (Chapters 12 and 13).
The end of realism: Bohm's theory keeps definite positions and gives up locality instead (Chapter 12).
A measured 85%: 85.4% is the quantum ceiling. Loophole-free tests scored from just above 75% to about 80%; 85.3% was reached only among detected pairs, with loopholes open (Chapters 6 and 9).
Every loophole closed: Locality and detection were closed together; freedom of choice can only be pushed back (Chapter 11).

Two stations in the dark, a photon at each, and neither knows the other's question. Each notebook, on its own, looks like a list of coin tosses. Side by side, they agree more often than any crib sheet allows, and no message travelled.

Glossary

CHSH game: Two players get random questions, 0 or 1, and win if their answers agree, unless both were asked 1.
Bell inequality, Bell test: A limit every local hidden-variable theory obeys, and an experiment testing it.
Crib sheet: This page's name for a local hidden-variable strategy: answers, or their odds, fixed before the questions.
Hidden variables: Properties beyond quantum mechanics that would fix measurement results.
CHSH value, S: Four correlations combined (Chapter 2): at most 2 for a crib sheet, 2√2 for quantum mechanics.
Equivalent game score: 1/2 + S/8 (for USTC's 2018 randomness test, which reports the CHSH game itself, 3/4 plus the paper's game value), counted over every trial ("per valid trial" or "per heralded trial") where detection was closed, and among detected pairs where it was open.
Tsirelson's bound: S ≤ 2√2, a win rate of 85.36%: the most any quantum strategy achieves (1980).
Superposition: A quantum state that holds two or more possibilities together without being either.
Entanglement: A joint quantum state of two systems that cannot be described as separate states of each.
Polarisation: The direction in which light's electric field oscillates.
Singlet: An entangled pair of spin-½ particles (a spin measured along any direction gives one of two results) that always come out opposite when both are measured along the same direction.
Visibility, V: A source's correlations relative to ideal pairs; S = 2√2 V in the simple model.
No-signalling: The odds of one party's results do not depend on the other party's choice of measurement.
PR box: A hypothetical device that wins every round of the CHSH game without signalling (Popescu and Rohrlich, 1994).
Loophole (locality, detection), loophole-free: A way a crib sheet could fake a violation: learning the far setting in time, or not clicking. A loophole-free test closes both together.
Fair sampling: The assumption that the detected pairs are a representative sample of all pairs.
Freedom of choice (measurement independence): The assumption that the settings are independent of whatever fixes the results.
CH–Eberhard inequality: A form of Bell's limit, equivalent to CHSH with each non-detection counted as an answer, that tolerates efficiencies above two-thirds; the papers that use it report p-values, not S.
Heralded (event-ready) trial: A round that counts only after a separate signal confirms that the distant particles are entangled.
Space-like separated: Two events so far apart, and so close together in time, that light could not travel from one to the other.
p-value: How often a world obeying the hypothesis tested would give a result at least this strong.
Standard deviation (σ): The typical size of chance fluctuations.
Local realism: Results fixed by properties carried with the particles, with no influence faster than light.
Local causality: Bell's condition that no causal influence of any kind travels faster than light.
Operational reading: Quantum mechanics read as rules for predicting measurement results, with no picture of what happens between source and detector.
Bohm's theory: David Bohm's 1952 completion of quantum mechanics: every particle has a definite position, and its motion can depend on what is done far away.
Collapse: In some readings of quantum mechanics, the abrupt change of the quantum state to match the single outcome a measurement gives.
Many-worlds: Everett's interpretation: every outcome occurs, in different branches.
Superdeterminism: The minority view that the settings were never independent of the hidden variables.
Retrocausality: Accounts in which a measurement setting can influence the particles' past.
Device-independent: Security or randomness certified by an observed Bell violation, without trusting how the devices work.

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