EPR paradox shows that quantum physics is not complete. And Bell's inequality demonstrates by assuming that there is a hidden variable due to incompleteness, and it shows that does not match with the result of quantum effects.

My question is that, if I look at Bell's inequality using hidden variable it seems to be a classical approach. Meaning that the formulas it took to derive Bell's inequality doesn't seems to have any indication of quantum effects, so it seems obvious that it would turn out not matching quantum effective results.

I know that this inequality was to prove that there is no hidden variable in order to reply to incompleteness. But how was hidden variable (which seems classical) supposed to have been a solution to incompleteness in the first place?

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    Bells inequality try’s to put a bound on correlations you would expect from a local hidden variable theory. You need to derive it in a classical setting because the bound is broken in quantum mechanics. – Shane P Kelly Aug 6 at 12:12
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    It is hard to answer this question since there are so many confused assumptions in it. I can only recommend the very good article about Bell's inequality on scholarpedia: scholarpedia.org/article/Bell's_theorem – Luke Aug 6 at 15:14
up vote 3 down vote accepted

There seem to be two questions here.

Meaning that the formulas it took to derive Bell's inequality doesn't seems to have any indication of quantum effects, so it seems obvious that it would turn out not matching quantum effective results.

The setting used to derive Bell's inequalities is neither classical nor quantum. Rather, it only assumes that measurement results are described by probability distributions. One then imposes some "natural" assumptions over these probability distributions and investigates the correlations that can arise from them.

The fact that measurement outcomes are to be described, in the most general case, by a probability distribution, is general enough to encompass any kind of theory, classical, quantum, or other. Indeed, it is hard to imagine a physical theory in which this would not be the case.

But how was hidden variable (which seems classical) supposed to have been a solution to incompleteness in the first place?

Indeed, hidden variable theories are "classical". The reason why one wonders whether a kind of hidden variable theory can explain the predictions of quantum mechanics is that many don't feel very comfortable with the idea that nature really works like quantum mechanics seems to be telling us.

For example, the intrinsic indeterminism associated with measurements is completely at odds with the classical (deterministic) way of describing the world. The fact that quantum mechanics predicts nonlocal correlations that however do not allow for superluminal communication can also be seen as quite odd.

It is therefore only natural that people wondered whether it is possible to go back to using a more "natural" and intuitive description of the world. However, Bell's inequalities tell us that this is indeed not possible.

Suppose you have a bunch of radioactive nuclei... You cannot determine which nucleus is going to disintegrate next. And why is it that you cant determine that? Its because you dont have any parameter which differentiates between the various nuclei. If, in the future we were to find out some parameter which sets these nuclei apart, and hence provides us with a probability of their disintegrating, say within the next t seconds, the situation wont be so random anymore. This parameter is what is referred to as a hidden variable.

  • You have a pool of water, can you say which water molecule is going to evaporate next? I do not think so. So, is evaporation of water a quantum phenomena? Nop. As we go smaller and smaller (in other words quantum), we become limited in figuring out those parameters because the tools we use to measure, are either bigger or same size as what is being measured. Environment as a whole become a parameter and it is not practical to isolate the cause of individual event. That is what is randomness. It is always present at quantum level, but does not mean is missing at classical level. – kpv Aug 7 at 9:59
  • In addition, the wave/particle duality at quantum level also makes things look more weird than they are. – kpv Aug 7 at 10:01

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