[magicalhippo]

> I suppose all that is left in the intuition busting, is how the probabilities don't add up as expected?

Lets imagine electrons are objects in a program, then the "electron class" has a private field containing a seed value to a pseudo-random number generator (ie deterministic), and the two electrons are initialized with the same seed value.

Further imagine that performing a measurement of an electron amounts to taking the seed, generating a random number between 0 and 360 degrees (sample value), and then comparing that random number to the measurement angle. If the sample value and the measurement angle is closer than +/- 90 degrees we say the measurement result is up, otherwise down.

Alright, so, if we imagine that when Charlie prepared the electrons, he creates two "electron objects", and passes one to Alice and the other to Bob.

If Charlie prepares entangled electrons, he'll ensure both instances have the same seed value. If he wants to create regular non-entangled electrons, he'll make each have a random seed value.

For non-entangled electrons, Alice and Bob will not see any correlation if they later compare notes.

For entangled electrons, if Alice and Bob uses the same angle they must get the same result per definition[1]. And indeed one can find the correlation as a function of the difference in angle, and it's a linear function from perfect correlation if the angles are the same (zero difference) to zero correlation (perfect anti-correlation) when the angles are 180 degrees apart.

However on real, entangled electrons in the lab things are different. There you'll find that the correlation is higher than the linear function when the difference is smaller than 90 degrees, and less than the linear function when the difference is greater than 90 degrees[2].

Thus if we measure the entanglement at not just 0 and 90 degrees difference but also 45 degrees difference, we'll find that our lab measurements do not agree with our simulated measurements.

Hence we conclude that entangled electrons do not behave like small objects that were created with the same "hidden value", ie the seed value in my example.

That's the essence of Bell's theorem and the tests of it (at least according to my memory).

[1]: Note that measuring real entangled electrons Alice and Bob will get exactly the opposite result, they're perfectly anti-correlated, but this matters not for this explanation and not worrying about it will make the explanation easier.

[2]: IIRC it goes like cos(a/2)^2 or something along those lines, ie https://www.wolframalpha.com/input?i=plot+%7B1-abs%28x%2Fpi%...

[taeric]

I thought the hidden variable idea was proven not to hold, though? Like, I thought that was the point? That the behavior observed can only be explained using the state of the remote measure as part of the explanation?

I have not looked back at the book I read. I definitely remember it had examples that were not paired off. I'm assumingy memory is simply off.

[magicalhippo]

Yes, that's what I was trying to say.

Hidden variables, like my "electron objects", would give a linear relationship.

However what quantum mechanics predicts and what we measure in the lab is a non-linear relationship that for some angles yield stronger correlations. Hence the phrase "violating Bell's inequality".

After Bell people devised other inequalities for other experimental setups which also can be used to rule out local hidden variables. A popular example are the CHSH inequalities[1], which is easier to realize experimentally, and give a stronger disagreement with hidden variables.

I've never seen the particles not paired up, and I don't see how that would work.

The whole point is that according to quantum mechanics, the pair of entangled particles aren't two separate systems, but that they must be treated as having a single state.

[1]: https://en.wikipedia.org/wiki/CHSH_inequality