Free quantum tool · Level: First-year university
Entanglement and the Bell test
Two particles, two detectors, and a dial for how entangled they are. The gap between the two lines is the whole argument.
Try this on the tool above
- Set both detectors to 30°. The correlation reads 1,00 and the two results always agree, which on its own is not strange at all.
- Now put detector A on 0° and B on 45°, and turn on the classical comparison. The dashed line allows 0,50 where the curve gives 0,71. Nothing with pre-decided answers reaches up there.
- Turn on CHSH and press set the best angles: S reads 2,828. Now drag the entanglement down to 0 and S drops under 2 and stays there.
An entangled pair has one state that cannot be split into a state for each particle, so neither half has a state of its own. Measure both along axes an angle apart and the correlation traces a cosine. Any theory where both answers were decided in advance is stuck under a straight line, and the CHSH total of four such measurements cannot pass 2 for such a theory, while quantum mechanics reaches 2,83.
The usual mistake
Measuring one particle sends a signal to the other one.
Turn on the outcomes layer and read Alice on her own. It sits at 50/50 at every angle, whatever Bob does with his dial, so there is nothing in her results to decode. The correlation only shows up once the two lists are put side by side, and carrying a list across a room is as slow as everything else.
Set both detectors the same and nothing is odd
With the pair fully entangled and both dials on one angle, the two results always agree. Put a red sock in one envelope and a blue one in another, post them to opposite ends of the country, and opening one tells you the other instantly. Nobody calls the post office spooky.
The sock story makes a promise, though: the colours were fixed before posting. Everything below is a test of that promise, and the only place to test it is at angles where the two dials disagree.
Where the two lines come apart
The dashed line is the best a theory with pre-decided answers can manage while still agreeing perfectly at 0°. The curve is what quantum mechanics predicts and what laboratories measure. Between 0° and 90° the curve rides above the line, and no amount of cleverness about socks closes that gap.
That picture is Bell's theorem. It does not say the maths is strange, it says a whole family of reasonable explanations is ruled out by numbers you can go and measure. Aspect, Clauser and Zeilinger took the 2022 Nobel Prize for closing the loopholes in exactly these experiments.
Entanglement is a quantity, not a switch
Drag the entanglement to 0 and the state becomes an ordinary product of two independent particles. Turn on CHSH and S will not pass 2 at any settings you try. Drag it back to 1, press the best angles, and S climbs to 2,83.
Do that once before accepting any sentence with the word spooky in it. The strange behaviour does not come from two particles having once shared a room. It belongs to a particular shared state, it arrives by degrees, and it has a number.
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Frequently asked questions
- What is quantum entanglement?
- A state of two or more particles that cannot be split into a separate state for each one. Neither particle has a state of its own, only the pair does. It is not a hidden wire between them, and it moves nothing faster than light.
- What does the CHSH inequality say?
- Combine four correlation measurements at two settings each and any local hidden-variable theory gives at most 2. Quantum mechanics reaches 2√2, about 2,83, and stops there. Measuring above 2 rules out that entire family of theories rather than one of them.
- Does entanglement send information faster than light?
- No. Each side alone sees pure randomness whatever the other does, so there is nothing in the local data to decode. The correlation only appears once the two lists of results are compared, and comparing them means sending the lists by ordinary means.
- Are all entangled states equally strange?
- No, and the entanglement slider shows it. Partially entangled states give a weaker correlation and a smaller CHSH value, and below a certain amount the inequality is not violated at all. Entanglement is a quantity, not a yes or no.
More interactive quantum tools
- Quantum teleportation, two ways →The same protocol as a circuit and as a bent wire pulled straight.
- Quantum gates and interference →Stack gates on a qubit and watch +0,5 and −0,5 cancel to nothing.
- Measurement and the Born rule →Turn the detector, run a thousand shots, and watch probability come out of amplitude.