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We often see the statements that a violation of Bell inequality means that the system shows quantum behavior or that a Local hidden variable theoretic description is not possible. One also finds that, typically the inequality gets violated in certain regions. Does it mean that the system is admitting a classical description for such regions ?

What do these regions of no violation mean?

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The point of Bell's inequality is to distinguish between two assumptions about reality. All we care about is whether mother nature violates the bounds derived by John Bell for $\textit{some angle}$. The fact that it does, is really a statement about $ \textit{all angles}$ : namely that the underlying system is quantum mechanical despite what may happen for some experimental set ups i.e angles where the inequality is not violated. The fact that the bound is not violated for some angles does not mean that for those angles the system deviates from what quantum mechanics predicts, it just means that I can set up an experiment where I could not distinguish a realist view of the world and a quantum mechanical view of the world. That's ok, nothing mysterious is happening because ultimately there is an angle where the two views of the world differ.

To see my point, think of a different situation. Consider a qubit in superposition $$ \Psi= \frac{1}{\sqrt{2}} (\psi_0 + \psi_1), $$ where $ \psi_0$ is the 0 state and $ \psi_1 $ is the 1 state. One can think of experiments for which it would act differently from a classical bit. But there are experiments where this state $ \Psi$ would act the same as a classical bit. For example, if the classical bit changed randomly between 0 and 1 state with probability $\frac{1}{2}$, then observing and re-preparing the state $\Psi$ would show exactly the same behavior. Does this mean that something strange and mysterious is happening? No, it just means that although the underlying system is quantum mechanical there are experiments where it's behavior can't be distinguished from a classical bit.

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A violation of a Bell inequality means that, for the apparatus being considered, no local hidden variable can reproduce the observed correlations (so, roughly speaking, that you cannot reproduce the results with only classical physics).

This is not always true: many predictions of quantum mechanics are easily reproduced with classical theories. Nor you should expect otherwise, given that most systems we interact with in everyday life behave "classically", even though they are really "quantum mechanical" underneath.

The point of Bell's theorem is that quantum mechanics, in some circumstances, gives outcomes that cannot be reproduced via a local hidden variable theory. There is therefore nothing strange in a graph like the one you show, and you should expect that kind of behaviour to be the norm rather than the exception.

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