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It is said that Bell's Inequality basically denies all possible local hidden variables theories as solutions to entanglement but what does a non-local hidden variable theory mean and how does it get around Bell's Inequality?

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If you know what are local hidden variables, then any variables outside that is non-local variable.

Local variables (hidden or otherwise) is the information/plan stored inside the entangled particles at the time they depart. Whether hidden or not is a different question. I think they are called hidden because they would be stored in the entangled particles and not visible to outside observers.

Any other mechanism/plan/influence would be non-local.

Not necessarily true, but an example can be - Suppose the measurement of previous pairs somehow are remembered by the environment and that memory influences outcome of measurement of subsequent pairs in such a way that quantum predictions are matched. By environment, I mean one or more of - creation equipment, measuring equipment, space in the vicinity of the experiment.

This would be considered a non-local influence because it is not stored inside entangled particle at the time of creation. It would rather accumulate in the environment as we measure more and more entangled pairs and the accumulation would steer the overall outcome towards quantum predictions. This kind of influence does not need to act at FTL. Simple sub luminal speeds would be sufficient in such a mechanism as it has plenty of time to act over duration of experiment.

This phenomena is named as memory loophole. There can be other possibilities which can be given some other name. All non-local possibilities are called loopholes by QM community.

Allmost all entanglement experiments geared towards proving two things -

  1. Bell's inequality is violated
  2. All loopholes (non-local influences) are closed.

Any data sets that do not prove these two things, are discarded as erroneous data.

I am ready for the down votes:)

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Bell's theorem sez the following. Suppose that each measurable quantity for a system is described by a stochastic variable - a single number picked out of a hat. The stochastic variable's value might depend in some way on other values you don't know about or can't measure - hidden variables. In order to match the predictions of quantum mechanics, the variables of spatially separated systems would have to influence one another non-locally - without any signal passing between them.

So Bell's theorem means that any other theory that reproduces the predictions of quantum mechanics either works by some means other than hidden variables or it is non-local. A non-local hidden variable theory would just say that there are hidden variables but they are non-local. Such a theory wouldn't get around Bell's inequality - it would claim that the inequality is correct and sez that the laws of physics are non-local.

I would also say it seems strange to talk about getting past Bell's inequality. The inequality is either right or wrong. You should be clear about either accepting it or refuting it - getting past is a vague description that leaves your position unclear.

There are other responses to Bell's inequality that don't involve accepting that the world is non-local, such as trying to explain the outcomes of the relevant experiments by applying quantum mechanics instead of trying to find another theory that reproduces its predictions. Quantum mechanics doesn't have hidden variables - rather each system is described in terms of observables represented by Hermitian operators:

https://arxiv.org/abs/quant-ph/9906007

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  • $\begingroup$ I think it would be worth pointing out the consequences of non-local hidden variables ever becoming non-hidden: causality violation. So because hidden variable theories must be non-local by Bell's theorem, such hidden variables must (assuming we don't want causality violation) not be observable even in principle. That's a bit like the aether or something: this thing in the theory which can never be detected, and that makes many people pretty uncomfortable with such theories I think (certainly me). $\endgroup$ – tfb Dec 25 '18 at 12:35

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