They asked the question "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?" in the title to the famous EPR paper. Although there may be disagreement as to what, exactly, a complete theory of nature must be, they said that any complete theory must meet the following requirement:

every element of the physical reality must have a counter-part in the physical theory... If, without in any was disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity. (EPR, 777)

But, what does this mean? The first part of the statement is merely setting up a one-to-one correspondence between reality and theory. Basically, " The correctness of the theory is judged by the degree of agreement between the conclusions of the theory and human experience... [which] in physics, takes the form of experiment and measurement" (EPR, 777). The second part of the statement attempts to define what quantities of a physical theory we should consider real. Another way of stating his second assertion is to say that if it is a logical necessity that a quantity have a certian value (i.e., that quantity has probability one for that particular value and probability zero for all other values) then that quantity must in fact have that value, independent of everything else. Since this value exists absolutely, one can therefore say that it is real. This appears acceptable if one thinks of unitary probabilities as arising from a need to satisfy some real physical constraint which would be violated if the quantity were not real or if it had a different value.

Now, lets examine the predictions of Quantum Mechanics. According to the Heisenberg Uncertainty Principle, there are certain sets of variables which, if one of them is known exactly, the other is completely unknown. More specifically, if the operators for two observables do not commute, then only one of the two quantities can be known precisely at a given moment. Take, for example, position and momentum. The Heisenberg Uncertainty principle tells us that, if we know the exact momentum of a system, say a free particle, then its position is completely unknown. So, if we know, with a probabilitiy of one, the momentum of a particle, we cannot know with probabilitiy one the position of that particle. Now, in general, we consider the position of a particle a real quantity and its momentum a real quantity. So, "either (1) the quantum-mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality. " (EPR 778).

To begin with, lets assume that assertion 1 above is false and that the wavefunction can provide one with a complete description of reality. Lets assume for a moment that we have two systems, A and B. Now, lets allow them to interact and then move apart so that they are spacelike separated (meaning, no signals can pass from one system to the other).

Now, what happens if I try to make a measurement on system A? Lets say, for example, that I choose to measure the position of system A. The wavefunction which describes this situation (both system A and system B) will be "reduced" into a form which is an eigenfunction of the position operator. Now, in the instant after I measure the position of system A, I choose to measure the position of system B. Since the two systems (A and B) are spacelike separated, then nothing I can do to system A should have any effect on system B. The wavefunction describing this situation is in the form of an eigenfunction of the position operator due to my measurement on A. So, if I measure the position of system B, the wavefunction should be able to predict a definite value.

What would happen if I had the same situation, but measured the momentum of system A first and then, as before, measured the position of system B? In this case, the wavefunction describing the situation is an eigenfunction of the momentum operator. If I choose to measure the position of system B, it will be completely undetermined! So, depending on what type of measurement I make on system A, I get two different results in my measurement on system B. But, how can this happen? System A and system B are so far apart that nothing I can do to system A can have any effect on system B.

The problem here is that I can assign two different wavefunctions to deccribe system B which have very different properties. But, in both cases, system B is in the SAME state so both wavefunctions must describe the same reality. With the first wavefunction, system B's position is 100% determined, so it must be a real quantity. With the second wavefunction, system B's momentum is 100% determined, so it must be a real quantity also. This same argument will work with any two non-commuting observables.