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Following https://en.wikipedia.org/wiki/Bell%27s_theorem:

enter image description here

The best possible local realist imitation (red) for the quantum correlation of two spins in the singlet state (blue), insisting on perfect anti-correlation at 0°, perfect correlation at 180°.

Just for my own understanding and learning of math, let's pretend that indeed the experimental data was the red curve rather than the blue curve. Then following the same article, we would be able to model this using a local realist model:

$$ {\displaystyle C_{h}(a,b)=E(A(\mathbf {a} ,\lambda )B(\mathbf {b} ,\lambda ))=\int _{\Lambda }A(\mathbf {a} ,\lambda )B(\mathbf {b} ,\lambda )p(\lambda )d\lambda .} $$

What choices of $A(\mathbf {a} ,\lambda )$, $B(\mathbf {b} ,\lambda )$, and $p(\lambda )$ would yield the red curve?

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  • $\begingroup$ There are examples of hidden variables that produce results matching the blue line. For example Coherent photons could have an oscillating variable added to them and you would produce the blue line. Mathematically matching cos2theta $\endgroup$ Jan 12, 2022 at 16:37

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Assuming $\mathbf {a}, \mathbf{b} \in \mathbb{R}^2$, with $\alpha, \beta$ being the corresponding angles, the following model yields the red graph: \begin{array}{l} \lambda \sim{\rm{Uniform}}\left[ {0,2\pi } \right] \to \mathop P\left( \lambda \right) = \frac{1}{{2\pi }} \\ A(\mathbf {a} ,\lambda ) = {\mathop{\rm sgn}} \cos (\alpha - \lambda) \\ B(\mathbf {b} ,\lambda ) = -{\mathop{\rm sgn}} \cos (\beta - \lambda) \end{array}

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    $\begingroup$ Why the negative votes ? This model is precisely the one given in example in the origanl paper by Bell $\endgroup$ Jan 7, 2019 at 13:55
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    $\begingroup$ Furthermore, answering your own question is encouraged: cf physics.stackexchange.com/help/self-answer $\endgroup$ Jan 7, 2019 at 13:57
  • $\begingroup$ @FrédéricGrosshans Thank you for the friendly comments, I really appreciate it! Yeah it was my first time answering a question and I'm like "why does everyone hate me here?" :) $\endgroup$
    – Gabi
    Jan 8, 2019 at 11:33
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For a while this question has haunted me more than the whole theorem and implications of its violation. @Gabi I found what I'm seeing now is exactly the same solution by doing a simulation, understanding it as a sort of "non-probabilistic limit" (taking it from a different angle so to speak - pun intended). It's not completely self-evident or very intuitive per sé. @FrédéricGrosshans thank you for pointing out that the same was given by Bell. For anyone still reading this, I've attached a screenshot of the sim, which should be quite self-explanatory. It's ultra low-res, so you can literally count the green matching dots at the left, which are then printed as a function of the angle at the right. Don't mind it looking a bit messy, it has other uses too. enter image description here

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