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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
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
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
The presumptions that lead to the Bell inequality are the Kolmogorov axioms for probability (or much weaker variants thereof). If you're asking whether Einstein ever questioned classical probability theory, then the answer at least is that he certainly never did so in print, or in any other format that's become part of the historical record. – WillO May 22 at 18:17

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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Everything is fundamentally deterministic. However, randomly complex and unknown factors make certain phenomena appear probabilistic. For example tossing a coin has 50/50 probability. But we know that complex/unknowable factors and their random nature make the toss a probabilistic phenomenon. The factors are so random at an immeasurable level that the overall process behaves probabilistic. Another example is - throwing a bowling ball appears probabilistic for me as far as number of pins knocked down is concerned. But for an expert bowler, it is relatively more deterministic. So, it all depends upon knowing and understanding, and being able to exploit the random factors. In case of QM, the factors are so immeasurably and minutely random, that we have to accept it as a probabilistic phenomena based upon the experimental results. Another factor that makes Quantum Entanglement even more complex is that it is so sensitive that measuring process impacts what is being measured, making it impossible to repeat the measurement with same entangled pair.

So, I think Einstein was rightly convinced that things are deterministic. But he was trying to explain/prove it, which is not yet done.

QM is at such a minute level, that nature itself appears to not know well enough and we have no option other than considering it probabilistic.

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