The history and physical implications of this derivation is discussed on EPR page.

Briefly: based on certain assumptions about the the microscopic world, which include

and may be other, less obvious assumptions,a mathematical relation (namely an inequality) is derived concerning outcome of some measurements of microsocopic particles. Experiments violate that relation. Conclusion is often that those assumptions, and in particular realism and locality, are not compatible, they cannot both be true in any consistent theory.

The following is a simplified description of the EPR scenario, developed by Bohm and Wigner.

We follow the approach in Sakurai (1994).

Alice and Bob are two spatially separated observers. Between them is an apparatus that continuously produces pairs of electrons. One electron in each pair is sent towards Alice, and the other towards Bob. The setup is shown below:

The electron pairs are specially prepared so that if both observers measure the spin of their electron along the same axis, then they will always get opposite results. For example, suppose Alice and Bob both measure the z-component of the spins that they receive. According to quantum mechanics, each of Alice's measurements will give either the value +1/2 or -1/2, with equal probability. For each result of +1/2 obtained by Alice, Bob's result

will inevitably be -1/2, and vice versa.

Mathematically, the state of each two-electron composite system can be described by the state vector

- .

b c a b c freq + + + - - - NEach row describes one type of electron pair, with their respective hidden variable values and their probabilites N. Suppose Alice measures the spin in the_{1}+ + - - - + N_{2}+ - + - + - N_{3}+ - - - + + N_{4}- + + + - - N_{5}- + - + - + N_{6}- - + + + - N_{7}- - - + + + N_{8}

- P(a+,b+) = N
_{3}+ N_{4}

- P(a+,c+) = N
_{2}+ N_{4}

- P(c+,b+) = N
_{3}+ N_{7}

- N
_{3}+ N_{4}< N_{3}+ N_{4}+ N_{2}+ N_{7}

- P(a+,b+) < P(a+,c+) + P(c+,b+)

σ_{y} is the second Pauli matrix, which generates the rotation operator
D(**y**,θ). The other two probabilities can be obtained with similar calculations. Bell's inequality then
becomes:

- 1/2 sin
^{2}2θ < 1/2 sin^{2}θ + 1/2 sin^{2}θ

- 0.25 < 0.1464...

There are several popular responses to this situation:

The second is to abandon the notion of *hidden variables* and to argue that the wave function does not contain any
information about the outcome of the measurement of the values in the particles. This corresponds to the Copenhagen interpretation of quantum mechanics.

One may also give up locality: the violation of Bell's inequality can be explained by a *non-local* hidden variable
theory, in which the particles exchange information about their states. This is the basis of the Bohm interpretation
of quantum mechanics. However, this type of interpretation is regarded as inelegant, since it requires all particles in
the universe to be able to instantaneously exchange information with all other particles in the universe.

Finally, one subtle assumption of the Bell's inequality is counterfactual definiteness. In reality, one can only measure the particles once without collapsing the wave function, and yet Bell's inequality involves talking about alternative measurements that cannot be performed and assuming that these would result in well defined outcomes. But relaxing this assumption one can also resolve Bell's inequality. In the Everett many-worlds interpretation, the assumption of counterfactual definiteness is abandoned because this interpretation assumes that the universe branches into many different observers each which measures a different observation.

One active area of theoretical research is to attempt to find other hidden assumptions in Bell's inequality.

The CHSH inequality, developed in 1969 by Clauser, Horne, Shimony, and Holt, generalizes Bell's inequality to arbitrary observables. It is expressed in a form more suitable for performing actual experimental tests.

Bell's thought experiment is statistical: Alice and Bob must carry out several measurements to obtain P(a+,b+), and the other probabilities. In 1989, Greenberger, Horne, and Zeilinger produced an alternative to the Bell setup, known as the GHZ experiment. It uses three observers and three electrons, and is able to distinguish hidden variables from quantum mechanics in a single set of observations.

In 1993, Hardy proposed a situation where nonlocality can be inferred without using inequalities.

Beginning with the Kocher and Commins experiment in 1967, several experiments have been carried out to test the above results, and Bell's inequality was found to be violated, in one case by tens of [[standard deviation|standard deviations]].

Experiments generally test the CHSH generalization of Bell's inequality, and use observables other than spin (which is in practice not easy to measure.) Most use the polarization of photon pairs produced during radioactive decay. However, the basic approach is very similar to the simple model presented above.

In 1998, Weihs, Jennewein, *et al.* at the University of Innsbruck first demonstrated the violation for
space-like separated observations (that is to say, there is no time for even a light signal to propagate from one
observation event to the other.)

- Hardy, L.:
*Nonlocality for 2 particles without inequalities for almost all entangled states*. Physical Review Letters 71: (11) pp. 1665-1668 (1993) - Sakurai, J.J.:
*Modern Quantum Mechanics*. Addison-Wesley, USA 1994, pp. 174-187, 223-232 - A. Aspect:
*Bell's inequality test: more ideal than ever*. Nature, vol 398, 18 March 1999. http://www-ece.rice.edu/~kono/ELEC565/Aspect_Nature.pdf