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Bell assumes that there exists a function $f$ which takes the hidden variable $\lambda$ and the setting of the measurement device : $\vec{a}$ or $\vec{b}$, as the input and gives the result of the measurement as output. That is, the function $f(\lambda , \vec{a})$ where $\vec{a}$ is the setting of the measurement device.

However, more realistically, a hidden variable theory would attribute hidden variables $\lambda$ to the electrons and hidden variables $\mu$ to the measurement device (as we don't know the exact state of the measurement device. We only know the detector setting). Then there is a function $f(\lambda, \mu, \vec{a})$ which gives you the result of the measurement . All this is because the final outcome is arrived at after time evolving the exact state of the system, which here is the exact hidden state of the particles + measurement device.

The correlation formula changes to :

$$P(\vec{a} , \vec{b})=\int d\lambda d\mu _i \rho (\lambda ,\mu _i) f(\lambda, \mu _1, \vec{a}) g(\lambda, \mu _2 , \vec{b})$$

One might think that this is equivalent to the standard Bell setup because we can just group $\lambda$ and $\mu _i$ into a new hidden variable $h$ by defining $h=(\lambda, \mu _1, \mu _2)$, and the resulting formula becomes identical to the one in Bell's result. The problem is that $\mu _i$ can be statistically correlated to the setting of the measurement device $\vec{a}$ or $\vec{b}$. So the assumption of statistical independence between the hidden variable and the device can be violated now.

Is this correct, and if yes, how to address this loophole?

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    $\begingroup$ The assumption of statistical independence is just that, an assumption, so if you decide to break it to begin with of course it is violated, but so what? Your are not in the framework of Bell's analysis anymore, that's all. But also, if a hidden variable relative to the measuring device has no effect on the detector setting, as you posit, then it is irrelevant to the analysis, since by definition it is not involved in the outcome of the measurement. So I would say no, there is no loophole here. $\endgroup$ Commented 16 hours ago
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    $\begingroup$ @StéphaneRollandin But this is a loophole because it allows for statistical dependence without having superdeterminism. This loophole has to be ruled out to reasonably rule out hidden variable theories. Also, the outcome of the measurement depends on these additional hidden variables, by hypothesis. The function $f(\lambda, \mu_1, \vec {a})$ is hypothesized to depend on all three arguments. To address this loophole, one would have to show that this hypothesis can't reproduce the predictions of quantum mechanics $\endgroup$ Commented 16 hours ago
  • $\begingroup$ In Bell's work, $\lambda$ refers to all relevant physical quantities (be-ables) that the probability can depend on. This includes quantities attached spacetime-wise to the system and the surroundings, and thus also the measurement devices. So yes, all hidden variables are covered formally by Bell's argument. The statistical independence was considered by Bell to be implicitly true, but of course, given the experimental results, it is suspect. $\endgroup$ Commented 3 hours ago

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So the assumption of statistical independence between the hidden variable and the device can be violated now.

Actually, there is no need to assume that which can be demonstrated experimentally. The measurement devices do not contribute to the observed outcome in any significant manner. If they did, you could not have perfect correlations.

Perfect correlations (or anti-correlations) occur when both entangled particles are measured on the same basis.

If the devices themselves were a factor, the measurement devices would need to perfectly sync to each other remotely, so as to have an exactly offsetting effect. Otherwise, the expected statistics would not be seen.

No assumption, and no loophole to address. Note that experiments such as Weihs et al (1998) already demonstrated that the measurement devices themselves cannot be in communication with each other if such communication respects the speed of light.

Violation of Bell's inequality under strict Einstein locality conditions

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