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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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Bell's theorem says 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 says 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). $\endgroup$
    – user107153
    Commented Dec 25, 2018 at 12:35
  • $\begingroup$ @tfb: That's actually not right. We cannot observe non-local hidden variables in full, but that by no means implies that we cannot observe them at all. It's not conceptually any different from a wave-function, which we can never measure in full but can measure approximately at a chosen point. $\endgroup$
    – user21820
    Commented Mar 16, 2021 at 15:37
  • $\begingroup$ Non-local hidden variables could work like global variables in computers, where a single location in memory gets referred to from multiple different local scopes that don't share local variables. The existence of non-local hidden variables could weakly support the idea that the universe is a simulation. $\endgroup$
    – CommaToast
    Commented Jun 20, 2021 at 19:06
  • $\begingroup$ @CommaToast philosophy.stackexchange.com/questions/56796/… $\endgroup$
    – alanf
    Commented Jun 21, 2021 at 20:09
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    $\begingroup$ @alanf I'm not convinced by Deutsch's arguments or your summary of them. The Halting problem and Incompleteness Theorem are limits on the ability to make statements about things, however they don't block you from creating a full simulation because youtu.be/xP5-iIeKXE8 the Game of Life can run in the Game of Life without having to know if a given pattern ever halts. A simulation can be a full simulation without needing to be able to predict what will happen next in inside itself. $\endgroup$
    – CommaToast
    Commented Jul 14, 2021 at 0:55
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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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what does a non-local hidden variable theory mean

In physics, the speed of light is also known the maximum speed with which information or causality can travel. If Alice is 1 LY away from Bob, then there is nothing Bob can do to send a message to Alice or affect her reality in a meaningful way faster than 1 year.

An effect is considered "local" if it has come to pass as a result of some force or particle that travelled at light speed, or slower.

One might be tempted to think this is an illusion of the macroscopic world. To a thing travelling at light speed, where is "here" vs. "there"? Locality is, to us, "where I am now." However, to an object moving at the speed of light, there is no sense in which time passes, from its perspective, except as bookends between two events in different points in spacetime where its wavepacket merges or splits into some other kind of wavepacket.

Now it stands to reason that whatever you do to such an object would affect its entire existence across time, and indeed it does—but not in a predictable enough way to allow for information to be sent FTL.

Yet some people don't like the idea that random things could affect each other in a manner that, given enough time has passed to collect all the information, seem less than random at a certain level. Like, I have good luck whenever you have bad luck, and vice versa, but we would never know our lucks were correlated unless we talked about it (and otherwise they would seem random.)

Perhaps if that were the case you might think God knows what both our lucks will be, since how can they be linked without a third, "non-local, hidden variable" that is the source of truth for both?

However, there's no evidence for non-local, hidden things, and by definition, there can't be. Think about it... if it's non-local and hidden, that means by definition you can neither interact with it, nor know about it.

Someone might argue that from God's point of view, non-local and hidden things are local and visible. That he/she/it knows in advance which way every photon will be polarized, etc. Maybe so, but it can never be proven nor can there be evidence of it without breaking the whole universe.

So most scientists publicly prefer Occam's razor since it's usually simpler to deal only with things we can measure and test in a repeatable way.

And since we can only make observations by smashing things into each other, once you get to the smallest level there is a hard limit to how much useful info you can get because everything you touch flies away instantly.

But in short, non-local hidden variables do not have any relationship whatsoever with Bell Inequality, which is just a test for whether the outcomes of two random measurements are correlated in some way beyond the fact that, by themselves, they're just purely random.

Are they correlated with each other, or with some hidden, third thing? Doesn't matter; a violation of Bell's Inequality is still a violation no matter how many layers of hidden realities you imagine there might be.

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