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To check the correlation between Hidden Variable Theory and Quantum Mechanics, Bell calculated the expectation value

$<\sigma_{e}(\vec a,\vec V) \sigma_{p}(\vec b,\vec V)> = \int d^n V \rho(\vec V) \sigma_{e}(\vec a,\vec V) \sigma_{p}(\vec b,\vec V)$

Here I am assuming that "Alice" is measuring the spin of an electron e along $\vec a$ and "Bob" is measuring the spin of the positron p along $\vec b$. Then $\sigma_{e}(\vec a,\vec V)$ and $\sigma_{p}(\vec b,\vec V)$ are the resulting spin values ($\pm \frac{1}{2}$) of the electron and positron, respectively. The vector $\vec V$ is an n-dimensional vector containing the hidden variables and $\rho(\vec V)$ is a probability distribution for the hidden variables.

But does this not assume QM is probabilistic? I thought Einstein disagreed with the probabilistc nature of Quantum Mechanics, as he said: I am convinced that He (God) does not play dice.

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I'm no historian (so won't post an answer) but I believe Einstein accepted well enough that quantum mechanics makes probabilistic predictions. What he objected to was the view that these probabilities are "fundamental". In statistical mechanics you also have probabilities, but these don't reflect any underlying stochasticity in the microscopic dynamics. I think Einstein wanted something similar for quantum mechanics. – Nathaniel Aug 8 '12 at 14:58
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I would also point out that it doesn't matter what Einstein thought about a particular issue. What matters is what the experimental result actually is. – Colin McFaul Aug 8 '12 at 15:01
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Einstein was a coauthor on the EPR paradox paper, so I'm sure he was well aware of these sorts of issues. – Jerry Schirmer Aug 8 '12 at 15:02

You are correct that Einstein believed that their should be an underlying, deterministic explanation for the probabilistic results of quantum theory, just as their is an underlying, if unknowable, deterministic explanation for statistical mechanics and observed phenomena like Brownian motion, both fields in which he made contributions.

Bell simply expresses Einstein's ideas in the form of a very general and clever experiment, satisfying the conditions of local realism. He then goes on to show that their exist particular states, what we call maximally entangled, where the predictions of quantum theory violate these mathematical rules.

This demonstration is quite simple (as quantum theory goes!), and the upshot is that if an underlying, realistic, deterministic theory does exist, then it must be non-local; e.g., the de Broglie-Bohm pilot wave model can be made to satisfy the Bell inequalities, but is grossly non-local -- the pilot wave violates special relativity.

Einstein may have sought alternative theories, but Bell's theorem (and later improvements) all go towards showing that their is no underlying theory that satisfies Einstein's criteria.

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