017 Bell test: the CHSH game
SimulatorCan two players who cannot talk beat 75% of a referee's questions? Ordinary plans cannot. Linked coins win about 85.4% of rounds.
Simulator running in your browser. Real hardware: coming soon (planned provider: IBM Quantum).
What's happening?
Play the CHSH Bell test game: no ordinary plan wins more than 75% of rounds, but measuring linked coins wins about 85.4%. Check all 16 plans. Simulator only.
You will learn: Why ordinary plans top out at 75%, and how linked coins reach about 85.4% (the CHSH inequality).
How the game works
A referee gives Alice a random question bit x and gives Bob a random question bit y. Each question bit is 0 or 1, fair and independent. Alice answers a bit a and Bob answers a bit b, and they cannot talk to each other once the questions are asked. The answers are bits too: 0 means Heads and 1 means Tails. They win the round when (a XOR b) equals (x AND y). In plain words, their answers should match, unless both questions are 1, in which case they should differ.
Why 75% is the best ordinary result
An ordinary plan fixes Alice's answer for x=0 and for x=1, and Bob's answer for y=0 and y=1. That makes 2 to the power 4, which is 16 plans. Each plan is checked on the four equally likely question pairs. Every plan wins either 3 of the 4 pairs or only 1, so the best one wins 75%, and 8 of the 16 reach it. Mixing plans cannot help either: the average of plans never beats the best single plan. Answering at random only wins 50%. Press the check on this page to see all 16 rows.
This is the CHSH game, named after Clauser, Horne, Shimony and Holt. Its sum is the usual test: for ordinary plans the sum can reach 2, and the win rate is one half plus the sum divided by 8. That gives exactly 75%.
What the linked coins do
Alice and Bob start with a linked pair [entangled pair], two coins in the state (|00> + |11>) divided by the square root of 2. Each coin on its own looks like a fair coin, but the two agree when they are both looked at in the same direction. Alice measures her coin along an angle in the X-Z plane: 0 degrees for x=0 and 90 degrees for x=1. Bob uses +45 degrees for y=0 and -45 degrees for y=1. When the two angles are alpha and beta, the chance that their answers match is the square of the cosine of (alpha minus beta) divided by 2.
For the pairs (0,0), (0,1) and (1,0) the angle difference is 45 degrees, so the answers should match, with chance cos squared of 22.5 degrees, about 85.4%. For (1,1) the difference is 135 degrees, so the answers should differ, with chance 1 minus cos squared of 67.5 degrees, which is also about 85.4%. Every round is therefore won with chance cos squared of 22.5 degrees, about 85.4%, above the 75% limit. This is Tsirelson's bound: the CHSH sum can reach 2 times the square root of 2, about 2.83, where ordinary plans stop at 2. The simulator draws each linked round from these chances, so the win rate wobbles around 85.4% just as real coin flips do.
Honest notes
- This is a simulator. It uses ordinary random numbers from your browser. Each linked round is drawn from the exact chances above, not from a full quantum calculation.
- Nothing travels faster than light. Alice's answers alone look like fair random bits, and so do Bob's. The win only appears when the two sets of answers are compared, and that needs a message at normal speed.
- Real experiments did this. Loophole-free Bell tests were done in 2015 by several teams using entangled particles, closing the gaps that earlier experiments left open.
- Nobel Prize. The 2022 Nobel Prize in Physics went to Alain Aspect, John F. Clauser and Anton Zeilinger for experiments with entangled photons.
Run on real hardware: coming soon. The planned provider is IBM Quantum. It is not connected yet. Every result on this page comes from a simulator running in your browser, not from a quantum computer.
How it works: the simulator keeps the exact quantum state of the qubits as a list of complex numbers, applies each gate to it, and draws random samples from the resulting probabilities. The random numbers are ordinary pseudo random numbers from your browser, and a real device would add noise. No QNT, wallet or account is needed.
Sources
This is an educational simulator, not financial advice. QUANTUM (QNT) is an independent community token on Solana and does not perform quantum computation.