The Bell game: can a plan beat 75%?
The Entanglement lesson describes a game for two players who can't talk to each other. No plan they agree in advance can win more than 75% of rounds on average, but with suitable measuring directions, quantum theory predicts that players sharing entangled pairs can win about 85% on average. Here you can try every possible plan, then share entangled pairs, and see where the difference comes from. (Read the lesson first)
The game
Asha and Ben sit in separate rooms and can't communicate. Each round, a referee hands each of them a card with a random letter, either X or Y. Each must answer with 0 or 1. They win the round if their answers match, except when both were handed Y. In that case they win only if their answers are different.
| Letters | To win, the answers must |
|---|---|
| XX | match |
| XY | match |
| YX | match |
| YY | differ |
Each letter is picked at random, independently, so each of the four pairs turns up a quarter of the time on average.
Before the roundDuring the roundAfter the round
Before the round: they agreed a plan, or they share one entangled pair, one qubit each
Asha's room
Letter: —
Card or measuring direction
Answer: —
Ben's room
Letter: —
Card or measuring direction
Answer: —
After the round: the referee compares the answers. No round played yet.
Step 1 of 5: Can any plan agreed in advance beat 75%?
A plan gives each player a card that says what to answer for each letter. Choose a card for each player.
Make a prediction, then press Check to see the result.
| Letters | Answers | Rule | Result |
|---|
Why no plan can win all four
XX says Asha's X answer equals Ben's X answer. YX says Ben's X answer equals Asha's Y answer. XY says Asha's X answer equals Ben's Y answer. Chain those together and Asha's Y answer must equal Ben's Y answer. But YY demands that they're different. One of the four rules always has to break.
All 16 plans
| Asha's card · Ben's card | Always 0 | Always 1 | Same as the letter | Opposite of the letter |
|---|---|---|---|---|
| Always 0 | ||||
| Always 1 | ||||
| Same as the letter | ||||
| Opposite of the letter |
The best average any plan can reach is 75%; 8 of the 16 plans reach it.
What about random plans?
What about flipping a coin to choose a plan, before or during each round? That only mixes plans, so the score is an average of their scores, and an average of scores of at most 75% is at most 75%.
Play rounds with this plan
Asha's room
Ben's room
| Round | Asha (letter: answer) | Ben (letter: answer) | Rule · result |
|---|
Step 2 of 5: Do matching answers show entanglement?
Before each round, Asha and Ben share one entangled pair: one qubit each, the pair from the Entanglement lesson. Both always ask their qubit the ordinary 0-or-1 question, whatever their letter.
Make a prediction, then press Check to see the result.
| Letters | Answers | Rule | Result |
|---|
Play 100 rounds
| Letters | Rule | Rounds | Wins | Win rate | Expected |
|---|
Compare with the lesson's figure
Pairs prepared to agree in advance match every time too. Matching on its own can't tell entangled pairs from answers agreed in advance. See the three-way comparison in the Entanglement lesson.
See the measuring directions
Each dial is a picture of a measurement choice, drawn on the qubit globe from the Superposition lesson. It isn't a picture of the qubit, and it isn't a direction in the room.
Step 3 of 5: Can different measuring directions beat 75%?
The Superposition lesson pictures a qubit as a point on a globe, with 0 at the North Pole and 1 at the South Pole. The ordinary measurement asks “north or south?”. A player can also turn the question first: tilt the measuring direction by some angle, then ask, which is the same as applying a rotation before the ordinary 0-or-1 measurement. Here, each player picks a direction for each letter before the game. Turning the direction changes how often the two answers agree: the further apart the two players' directions point, the less often their answers match, and opposite directions give opposite answers every time.
- Asha: X → the ordinary question (0°); Y → a quarter turn (90°).
- Ben: X → an eighth turn (45°); Y → an eighth turn the other way (315°, that is −45°).
Make a prediction, then press Check to see the result.
Per letter pair
| Letters | Rule | Rounds | Wins | Win rate | Expected |
|---|
See the measuring directions
Each dial is a picture of a measurement choice, drawn on the qubit globe from the Superposition lesson. It isn't a picture of the qubit, and it isn't a direction in the room.
Last 10 rounds
| Round | Asha (letter: answer) | Ben (letter: answer) | Rule · result |
|---|
What the 75% limit assumes
In this game
On this page, the rules make three things true:
- the referee picks each letter at random, independently of anything the players prepared, so each letter pair turns up a quarter of the time;
- each player answers using only their own letter and whatever they agreed or brought with them; there's no contact during a round;
- every round counts, and each player gives one answer, 0 or 1.
Under these rules, no plan agreed in advance can average more than 75%. That's a mathematical fact about the game.
In real experiments
A real Bell test can't simply declare these conditions. It has to build them, check them and argue for them:
- No contact: the two measurements are made far enough apart, and fast enough, that no signal travelling at the speed of light could carry one side's letter to the other side's answer.
- Independent letters: the letters come from random-number generators that are assumed to be independent of the source of the pairs. That assumption can be supported, and tested against particular alternatives, but not conclusively proved.
- Every round counts: missed detections must be counted, or be rare enough that leaving them out couldn't explain the result.
- Statistics: the score comes from a limited number of rounds, so the conclusion is a statistical confidence, not a certainty.
Since 2015, experiments have addressed the main loopholes together and found scores above the limit. Under these conditions, that rules out answers agreed in advance. It doesn't rule out explanations that break the conditions, and no finite experiment proves the conditions beyond all question.
Step 4 of 5: Can Ben's choice show up in Asha's results?
Make a prediction, then press Check to see the result.
| Rounds | 0s | 1s | Share of 0s | Expected |
|---|
Fewer than 100 rounds had been played in Step 3, so 1,000 rounds were played when you pressed Check.
Ben's own answers, split by Asha's letter
| Rounds | 0s | 1s | Share of 0s | Expected |
|---|
How the page works out each round
The page works out Asha's answer first and then Ben's, only because it's convenient to calculate that way. The chances are the same whichever is worked out first, and neither player's choice changes the other's results.
Step 5 of 5: How many rounds does it take?
Make a prediction, then press Check to see the result.
| Rounds | A 75% plan | An ideal entangled team | Do they overlap? |
|---|
What the ranges mean
At least 95 runs in 100 of that many rounds would score in this range; it isn't a statistical test.
Real experiments
The two experiments shown here report a CHSH score called S, rather than a win rate in this game's format. The 75% limit corresponds to S = 2 and the quantum ceiling to S of about 2.83. To compare them with this game, the score can be converted into a CHSH-equivalent win probability, ½ + S ÷ 8. This assumes the four letter pairs were used equally often. These are comparison figures calculated from the published scores. They aren't win rates observed in this page's game.
| Experiment | Reported S | Rounds | CHSH-equivalent |
|---|---|---|---|
| 2015: electron spins 1.3 km apart (Hensen and colleagues) | 2.42 ± 0.20 | 245 | about 80% (± 2.5 points) |
| 2023: superconducting circuits 30 m apart (Storz and colleagues) | 2.0747 ± 0.0033 | more than 1 million | about 75.9% (± 0.04 points) |
Both are below the ideal 85.4% average, because real equipment is imperfect. The 2015 test beat the limit with only 245 rounds, so its statistical confidence was modest. The 2023 test was only just above the limit, but with so many rounds that chance is no longer a credible explanation under the experiment's stated assumptions. Three kinds of number appear on this page: theory (75% and 85.4%), this page's simulation (your rounds) and real experiments (the reported S values above).
What the 75% limit assumes
In this game
On this page, the rules make three things true:
- the referee picks each letter at random, independently of anything the players prepared, so each letter pair turns up a quarter of the time;
- each player answers using only their own letter and whatever they agreed or brought with them; there's no contact during a round;
- every round counts, and each player gives one answer, 0 or 1.
Under these rules, no plan agreed in advance can average more than 75%. That's a mathematical fact about the game.
In real experiments
A real Bell test can't simply declare these conditions. It has to build them, check them and argue for them:
- No contact: the two measurements are made far enough apart, and fast enough, that no signal travelling at the speed of light could carry one side's letter to the other side's answer.
- Independent letters: the letters come from random-number generators that are assumed to be independent of the source of the pairs. That assumption can be supported, and tested against particular alternatives, but not conclusively proved.
- Every round counts: missed detections must be counted, or be rare enough that leaving them out couldn't explain the result.
- Statistics: the score comes from a limited number of rounds, so the conclusion is a statistical confidence, not a certainty.
Since 2015, experiments have addressed the main loopholes together and found scores above the limit. Under these conditions, that rules out answers agreed in advance. It doesn't rule out explanations that break the conditions, and no finite experiment proves the conditions beyond all question.
Explore
Explore unlocks when you check Step 5, or press Skip to Explore.
Set up
Play
Limit of 10,000 rounds reached. Clear results to start again.
Result
Expected values come from theory; your rounds come from this page's simulation.
Latest round
Asha's room
Ben's room
Per letter pair
| Letters | Rule | Rounds | Wins | Win rate | Expected |
|---|
Last 10 rounds
| Round | Asha (letter: answer) | Ben (letter: answer) | Rule · result |
|---|
Each player's own answers
| Rounds | 0s | 1s | Share of 0s | Expected |
|---|
| Rounds | 0s | 1s | Share of 0s | Expected |
|---|
A plan's answers depend only on the player's own letter, so these tables never depend on the other player's letter either.
Typical ranges
| Rounds | A 75% plan | An ideal entangled team | Do they overlap? |
|---|
Measuring directions
Each dial is a picture of a measurement choice, drawn on the qubit globe from the Superposition lesson. It isn't a picture of the qubit, and it isn't a direction in the room.
Show the maths unlocks when you check Step 3, or press Skip to Explore.
Show the maths
The plans. A plan fixes each player's answer to X and to Y. Winning all four pairs would need a₀ ⊕ b₀ = 0, a₀ ⊕ b₁ = 0, a₁ ⊕ b₀ = 0 and a₁ ⊕ b₁ = 1. Adding all four, each answer appears twice on the left, giving 0, but the right-hand sides add to 1. So every plan fails at least one pair: 8 of the 16 plans win 3 pairs (75%) and 8 win 1 pair (25%). Mixing plans at random averages their scores, so nothing agreed in advance averages more than 75%.
| Asha's card · Ben's card | Always 0 | Always 1 | Same as the letter | Opposite of the letter |
|---|---|---|---|---|
| Always 0 | 75% | 25% | 75% | 25% |
| Always 1 | 25% | 75% | 25% | 75% |
| Same as the letter | 75% | 25% | 25% | 75% |
| Opposite of the letter | 25% | 75% | 75% | 25% |
The entangled pair. The shared state is (|00⟩ + |11⟩)/√2. A measuring direction t means: apply the rotation Ry(−t) to your own qubit, then measure 0 or 1; the result 0 corresponds to cos(t/2)|0⟩ + sin(t/2)|1⟩. With Δ = tAsha − tBen, the chances are P(0,0) = P(1,1) = ½cos²(Δ/2) and P(0,1) = P(1,0) = ½sin²(Δ/2). So the answers agree with chance cos²(Δ/2), and each player's own answers are 0 half the time whatever the other does.
The best-known directions. Asha 0° and 90°, Ben 45° and 315°. XX: Δ = −45°, agree cos²(22.5°) ≈ 0.854, win. XY: Δ = −315°, agree cos²(157.5°) ≈ 0.854, win. YX: Δ = 45°, agree ≈ 0.854, win. YY: Δ = −225°, agree cos²(112.5°) ≈ 0.146, so they differ ≈ 0.854, win. Every pair is won with probability cos²(π/8) = (2 + √2)/4 ≈ 85.4%.
The CHSH score. Coding answers as +1 and −1, the correlation is E = cos Δ, and S = E(XX) + E(XY) + E(YX) − E(YY). The win chance is exactly ½ + S/8; this is also the conversion used for the CHSH-equivalent figures from real experiments. Plans give S ≤ 2 (75%). Quantum theory gives S ≤ 2√2 for any state and any measurements (Tsirelson's bound), so no quantum strategy averages more than about 85.4%. These are limits on the average over many rounds; a short run can score above either one by chance. The best-known directions reach S = 2√2.
Angles and noise. These are globe angles; photon experiments usually quote polariser angles, which are half these numbers (0°, 45°, 22.5° and −22.5°). With imperfect pairs of “visibility” V between 0 and 1, the best average becomes ½ + V√2/4, which exceeds 75% only if V is above 1/√2, about 0.71.
A link to Interference. The Interference experiment's result P(0) = cos²(φ/2) has the same shape. There the turn is about the vertical axis, followed by recombining. Here the measuring direction tilts in a plane through the poles.
What this shows
No plan agreed in advance can beat 75% on average, and 8 of the 16 possible plans reach it. Entangled pairs measured in suitable directions are predicted to win about 85.4% on average, and no quantum strategy does better on average. Each player's own results stay 50/50 whatever the other does, so nothing is signalled. A short run can land above or below either average by chance, so telling 75% from 85% takes many rounds.
What the 75% limit assumes
In this game
On this page, the rules make three things true:
- the referee picks each letter at random, independently of anything the players prepared, so each letter pair turns up a quarter of the time;
- each player answers using only their own letter and whatever they agreed or brought with them; there's no contact during a round;
- every round counts, and each player gives one answer, 0 or 1.
Under these rules, no plan agreed in advance can average more than 75%. That's a mathematical fact about the game.
In real experiments
A real Bell test can't simply declare these conditions. It has to build them, check them and argue for them:
- No contact: the two measurements are made far enough apart, and fast enough, that no signal travelling at the speed of light could carry one side's letter to the other side's answer.
- Independent letters: the letters come from random-number generators that are assumed to be independent of the source of the pairs. That assumption can be supported, and tested against particular alternatives, but not conclusively proved.
- Every round counts: missed detections must be counted, or be rare enough that leaving them out couldn't explain the result.
- Statistics: the score comes from a limited number of rounds, so the conclusion is a statistical confidence, not a certainty.
Since 2015, experiments have addressed the main loopholes together and found scores above the limit. Under these conditions, that rules out answers agreed in advance. It doesn't rule out explanations that break the conditions, and no finite experiment proves the conditions beyond all question.
What it leaves out
This page is an ordinary computer program, not an experiment. It works out both players' answers in one place, from quantum theory's predicted chances, which separated players couldn't do. So it shows what quantum theory predicts. The evidence that nature behaves this way comes from real experiments. The simulation is ideal:
- no noise;
- every round gives two answers;
- no timing or distance;
- the letters come from the browser's random number generator.
Real tests have to deal with imperfect equipment, missed detections, keeping the two measurements too far apart and too fast for any signal, and choosing the letters independently. Entanglement on its own doesn't make a computer faster.
Check yourself
- Why must every plan lose at least one letter pair?
- Asked the ordinary question every time, entangled pairs always match. Why is their expected average only 75%?
- Ben changes his measuring direction. What changes in Asha's own results?
- A team wins 9 of 10 rounds. Have they beaten 75%?
Answers
- Every plan has to break one of the four rules (the chain in Step 1 shows why), so it always loses at least one letter pair. The best is 75% on average.
- Their answers do always match, but matching loses every Y-and-Y round, where the rule says the answers must differ. Always matching gives an expected average of 75%.
- Nothing. Asha's own answers are 0 half the time whatever letter Ben gets and whatever direction he uses. The link appears only when their two lists are compared, and comparing needs an ordinary message.
- Not from 10 rounds. A 75% plan wins 9 or more of 10 rounds about one time in four. Short runs vary too much to tell 75% from 85%.
The experiment runs entirely in your browser. Your choices, your predictions and your results aren't stored, and this page's own code doesn't send them anywhere. Cloudflare Web Analytics runs on every Plain Quantum page and measures how each page loads and responds, including to clicks and key presses, and the typefaces load from Google; see Privacy. The model is tested automatically: every plan, every measuring direction and the sampling are checked against an independent calculation.
Sources and further reading
Tags show what kind of source each one is (how this site is made). Spotted an error? Email hello@plainquantum.com.
- Peer-reviewed paperProposed experiment to test local hidden-variable theories (the CHSH inequality)Clauser, Horne, Shimony and Holt, Physical Review Letters, 1969
- Peer-reviewed paperConsequences and limits of nonlocal strategies (the CHSH game: 3/4 for plans, cos²(π/8) with entanglement)Cleve, Høyer, Toner and Watrous, IEEE Conference on Computational Complexity, 2004
- Peer-reviewed paperQuantum generalizations of Bell's inequality (the quantum limit, S at most 2√2)B. S. Cirel'son, Letters in Mathematical Physics, 1980
- ExperimentLoophole-free Bell inequality violation using electron spins separated by 1.3 kilometresHensen et al., Nature, 2015
- ExperimentLoophole-free Bell inequality violation with superconducting circuitsStorz et al., Nature, 2023
- ReferenceThe Nobel Prize in Physics 2022Nobel Prize Outreach, 2022
- BookQuantum Computation and Quantum Information (10th anniversary edition)Michael Nielsen and Isaac Chuang, Cambridge University Press, 2010