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?
[Physics] What are non-local hidden variables
bells-inequalityquantum mechanicsquantum-entanglementquantum-interpretations
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Realism refers to a philosophical position that says that certain attributes of the world of experience are independent of our observations. Let's take a physics example. In classical physics we used to say that a particle has a definite position and a definite momentum at a certain instant of time. These are represented by real numbers and they have those definite numbers independent of any observation. This seemed to be the only sane position one can take about the objective world. However the uncertainty principle of quantum mechanics tells us that a particle can not have both a well defined value of position and well defined value of momentum at the same time along the same direction independent of measurement. The more accurately one tries to measure one the less accurately one can have the knowledge of the other. Philosophically it means that position and its conjugate momentum can not have simultaneous reality. This realization had led the founding fathers of quantum theory to reformulate mechanics into a new theory called quantum mechanics. In QM a system is represented by a state vector in an abstract space. The length (norm) of this vector remain unchanged but with time its direction changes (for simplicity I am discussing Schrödinger picture). The various components of this state vector along the axes are various eigenstates with definite value of certain observables. Obviously the state vector is the linear combination of these eigenstates. Whenever a measurement is performed the state vector collapses to one of the eigenstates with certain probability determined by the Schrödinger's equation.
The so-called realists claim that the system was already in a definite state characterized by some additional hidden parameters before the measurement and since we are not aware of those hidden parameters we have an incomplete knowledge of the system. The random outcome reflects our incomplete knowledge of the system. There are number of hidden variable theories developed which reproduced the results of ordinary quantum mechanics.
Then surprisingly Bell discovered the famous Bell's inequality and showed that not all results are identical for both qm and local hidden variable theories. Experiment carried out and the verdict was clear. QM won. Nature supported QM. Therefore local hidden variable theories were ruled out. However there are nonlocal hidden variable theories which still survived like Bohmian mechanics. (I would also like to emphasize that MWI is an interpretation which is to some extent realist in spirit and it is by no means ruled out)
But what is locality? Locality is the assumption that an object can be influenced only by its immediate surroundings by the events which took place in its immediate past. All classical and quantum field theories depends on this assumption in an essential way. Non locality implies that two events which are separated from each other by space-like separation can affect each other. Some people demand (imho) falsely that EPR type entanglement violates locality. In reality in never does. All one need to abandon is realism. Entanglement just shows that there exists quantum correlations between particles which were in past had some common origin. It also shows that if it were a classical world then the EPR entanglements effects were nonlocal. But we live in a quantum world and there is no non locality.
Therefore in a nutshell, locality is certainly not ruled out. Realism is ruled out to a large extent.
The statement "The observed violations of the Bell inequalities disprove local hidden variable theories" is a profound one, and certainly does not qualify as a didactically over-simplification.
In essence, what these violations tell us is that you can't build quantum physics from some hidden variable theory, unless you are happy to include into this theory some magic that 1) makes the hidden variable theory unphysical (non-locality) or 2) makes it at least as strange as quantum mechanics (negative probabilities).
Best Answer
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