> > But to my mind, the biggest take-away is that Einstein's Principle of Relativity absolutely requires that conservation can only be on average. > > And the same request here. Why does the principle of relativity require that?
The way I read this comment was that "the principle of relativity cannot conserve angular momentum on a per-trial basis".
In a Mermin Device a pair of entangled spin particles is set to two Stern-Gerlach experiments. The two particles has net (spin) angular momentum of 0 because that's was the net angular momentum of starting material. But if you measure the angular momentum of the two particles in two non-parallel directions, and if we also require that the only answers you are allowed to get are +hbar/2 or -hbar/2, then the sum of the angular momentum you get by adding +/-hbar/2 times one direction plus +/-hbar/2 times a different direction can never be 0.
If angular momentum cannot be preserved on a per-trial basis, then I suppose it must be preserved on average, because, I suppose if it isn't preserved on average, then I don't think you can say that angular momentum is preserved at all.
> And because of that, when Alice and Bob measure entangled quantum particles, their combined results must violate Bell's inequality.
> > And because of that, when Alice and Bob measure entangled quantum particles, their combined results must violate Bell's inequality. > > Could you be slightly less ridiculously condensed here? Give a one-or-two-paragraph, accessible-to-the-semi-layman explanation of why this means the result must violate Bell's Inequality?
The really short answer is that if we preserve the angular momentum on average then it entails that the correlations we observe from Mermin Device must match the correlations predicted by quantum mechanics, and therefore violate Bell's inequality for the same reason that predictions of quantum mechanics do.
In more detail, if we take the results of a measurement where Alice measures angular momentum in the vertical direction and Bob measures the angular momentum off vertical by theta degrees where Alice gets a result of +hbar/2, then in order for angular momentum to be preserved, Bob's measurement would have to be -cos(theta)hbar/2.
Of course Bob is only allowed to get hbar/2 or -hbar/2, so if we want angular momentum to be preserved on average then when we take an ensemble of trials, and filter out only those trials were Alice measures hbar/2, then the average of all of Bob's measurements for those trials should be -cos(theta)hbar/2. That requires that the probability Bob geting hbar/2 when Alice does is (1-cos(theta))/2 (= sin^2(theta/2)), which I believe is the value predicted by quantum mechanics. Once you have the predictions made by quantum mechanics, a violation of Bell's inequality follows by the usual arguments.
