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I've seen the Veritassium video about Bell's theorem which made me think about what the hidden variables could be.

I'd like somebody to explain to me what's wrong with this extremely simple hidden variable approach:

Upon entanglement, the hidden variables consist only of a vector $v$ with a random direction. Upon the two measurements in direction $w$, we define the outcome of the measurements to be spin up if and only if $\langle v,w\rangle > 0$ for the one particle and the opposite for the other.

Where does that not agree with observations?

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    $\begingroup$ There is no distribution for the $v$s of repeated samples that will result in a violation of Bell's inequalities but experimentally they are violated, in agreement with the predictions of quantum mechanics do. $\endgroup$ Sep 11 at 16:37
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In your scheme, the measurement results will be equal if the plane normal to $v$ is between the measurement vectors, and unequal otherwise, so the chance that they'll be equal is linearly proportional to the angle between the measurement vectors. That's the red curve labeled "classical" in this diagram:

The red curve makes Bell's inequality into an equality, i.e., Bell's theorem says that you can't do better, but it isn't good enough to match the quantum prediction.

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