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Start with two particles in the state $UD+DU$, where $U$ and $D$ are the "up" and "down" eigenstates of the measurement Alice plans to make. Put $u=U+D$ and $d=U-D$, and suppose Bob makes a measurement with these eigenstates. Note that $UD+DU=ud+du$. (The factor of 2 doesn't matter). Alice makes her measurement and finds her electron is in state, ...

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Under your assumption of simultaneously well-defined x and z value, you reach predictions which are inconsistent with quantum theory. This is exactly what leads to Bell's inequality which is (experimentally!) violated by quantum theory, see https://en.wikipedia.org/wiki/Bell%27s_theorem or the explanation in Preskill's lecture notes ...

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My issue here is it seems as though if his experiment were to be performed, it would violate known laws of physics. The picture and words in the section of the video you cited seem a little too vague to uniquely specify a specific experiment. But in the whole first 40 minutes of the video you see that all he is claiming is that we don't see departures ...

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