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Q: How is it that Bell’s Theorem proves that there are no “hidden variables” in quantum mechanics? How do we know that God really does play dice with the universe?

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Just for the hell of it, take a look at: L = A 0 B 22.5 + A 45 B 22.5 + A 45 B 67.5 – A 0 B 67.5 = (A 0 + A 45 )B 22.5 + (A 45 – A 0 )B 67.5L = (A 0 + A 45 )B 22.5 + (A 45 – A 0 )B 67.5 = ±2, since either (A 0 + A 45 ) = ±2 and (A 45 – A 0 ) = 0, or (A 0 + A 45 ) = 0 and (A 45 – A 0 ) = ±2. So if you could fill out each of these values (A 0 , A 45 , B 22.5 , B 67.5 ), then L = A 0 B 22.5 + A 45 B 22.5 + A 45 B 67.5 – A 0 B 67.5 = ±2 which, notably, is less than or equal to 2. E[A 0 B 22.5 ] + E[A 45 B 22.5 ] + E[A 45 B 67.5 ] – E[A 0 B 67.5 ] = E[A 0 B 22.5 + A 45 B 22.5 + A 45 B 67.5 – A 0 B 67.5 ] ≤E[|A 0 B 22.5 + A 45 B 22.5 + A 45 B 67.5 – A 0 B 67.5 |] = E[2] = 2. So E[A 0 B 22.5 ] + E[A 45 B 22.5 ] + E[A 45 B 67.5 ] – E[A 0 B 67.5 ] ≤ 2. This is one version of “Bell’s Inequality”, and it holds if each term (A 0 , A 45 , B 22.5 , B 67.5 ) has a value.