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Bell's Inequalities are a set of inequalities that establish that theories with "counterfactual definiteness" and "locality" require a set of inequalities (describing the probabilities of making any measurement) to hold. Since quantum mechanics violates these inequalities, it therefore rules out theories with these two properties.

Is Bell's Inequality still considered to rule out a significant amount of meaningful alternative "hidden-variable theories"?

Specifically, are there meaningful theories with both locality and "counterfactual definiteness" that are ruled out by Bell's Inequality?

The writers of this paper (which I wasn't able to completely follow) seem to think that the requirement of counter-factual definiteness "reduces the generality of the physics of Bell-type theories so significantly that no meaningful comparison of these theories with actual Einstein-Podolsky-Rosen experiments can be made."

Starting things off: what exactly is counterfactual definiteness? According to this source:

Let us define “counterfactual-definite” [14, 15] a theory whose experiments uncover properties that are pre-existing. In other words, in a counterfactual-definite theory it is meaningful to assign a property to a system (e.g. the position of an electron) independently of whether the measurement of such property is carried out.

Maybe I'm misunderstanding, but doesn't this rule out even "'classical' Heisenberg Uncertainty"? What I mean is that early students before they learn quantum mechanics are taught a sort of "classical intution" for Heisenberg's Uncertainty (People who understand QM know this is inaccurate, but it didn't even stop Feynman from using this analogy in his lectures for first year undergrads.)

Intution is: If you ever need to observe something sufficiently small, you must physically "hit it" to see it. This interaction imparts some energy and you end up changing the state.

Therefore isn't even a simple "billiard ball" model of the universe contained within "counter-factual definite" theories because measurements of "billiard balls" require collisions with other billiard balls (measurements therefore change the properties of the states that are being observed)?

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  • $\begingroup$ I'd like to mention that Bell's Inequality doesn't take dynamic hidden variables into account. This paper [ researchgate.net/publication/… ] suggests that if such hidden variables are taken into consideration, one can have both locality and determinism in a quantum theory. $\endgroup$
    – Laff70
    Aug 17 '20 at 21:56
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    $\begingroup$ Actually in the first paper by Bell (if I correctly remember), he briefly discussed this point declaring that the parameter $\lambda$, usually denoting hidden variables, may indicate initial conditions of some dynamical process, so that hidden variables in Bell's discussion include dynamical ones. $\endgroup$ Aug 18 '20 at 10:47
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Possibly I shouldn't write an answer about a paper (the one by Hess et al) that I haven't managed to understand, but my impression is that it's just gibberish. Parts of it read like a postmodernist critique of the scientific method, especially the use of the word "trespass".

One mistake that I've seen in an anti-Bell paper before, and which may also be the core mistake here, is that they aren't talking about the same experiment. To show that a broad class of hidden-variable models can't match the predictions of QM, Bell only had to show that there exists one experiment, and one way of analyzing the data from it, for which they can't match the QM prediction. There are certainly many other experiments, and probably other ways of analyzing the data from Bell's experiment, that don't yield any conflict with QM. This paper complains on page 2 that Bell didn't consider all of the possible correlations between different measurement results. I suppose that's true, but it doesn't matter—the correlations that he did consider are enough for a contradiction.

All that Bell's argument requires of hidden-variable models is that they be able to answer questions in the form of an angle $θ$ with either "up" or "down". The locality assumption is that the answers can't be influenced by questions asked at spacelike separated locations. A hidden-variable model that treats each particle as an independent (nonentangled) quantum spin system, and chooses an answer based on the Born rule followed by wavefunction collapse, would probably not be counterfactually definite in the view of the authors of that paper (since measurements on the same particle don't commute), but it's still vulnerable to Bell's argument.

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