GHZ (Greenberger—Horne—Zeilinger) Theorem and GHZ States

The GHZ states (Greenberger—Horne—Zeilinger states) are a set of entangled states that can be used to prove the GHZ theorem, which is a significant improvement over ► Bell's Theorem as a way to disprove the concept of “elements of reality”, a concept introduced by ► EPR problem (Einstein—Podolsky—Rosen) in their attempt to prove that quantum theory is incomplete. Conceding that they did not quite know what “reality” is, EPR nonetheless said that it had to contain an “element of reality” as one of its properties. This was that if one could discover a property of a system (i.e., predict it with 100% certainty) by making an experiment elsewhere, that in no way interacted with the system, then this property was an element of reality. The argument was that since one had not in any way interacted with the system, then one could not have affected this property, and so the property must have existed before one performed one's experiment. Thus the property is an intrinsic part of the system, and not an artifact of the measurement one made.

From a common-sense point of view, this proposition seems unassailable, and yet quantum theory denies it. For example, in the Bohm form of the EPR experiment, one has a particle that decays into two, that go off in opposite directions. If the original particle had ► spin 0, while each of the two daughters has spin 1/2, then if the one going to the right has its spin up, the one going to the left will have its spin down, and vice-versa. So the spin of each of the daughters is an element of reality, because if one measures the spin of the particle on the right as up, one can predict with 100% certainty that the other will be spin down, etc. EPR would conclude from this that, since we did not interfere with the particle on the left in any way, then we could not have changed its spin, and so it had to have been spin down from the moment the original particle decayed.