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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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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 $\endgroup$ – Frédéric Grosshans Jan 7 at 13:55
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    $\begingroup$ Furthermore, answering your own question is encouraged: cf physics.stackexchange.com/help/self-answer $\endgroup$ – Frédéric Grosshans Jan 7 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 at 11:33

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