## GREENBERGER-HORNE-ZEILINGER (GHZ) THEOREM

 The GHZ Game Alice, Bob, and Charles decide to participate in the televised game called “Guess My Number”. In accordance with the rules of the game each contestant is stationed at their isolation booths without the ability to communicate with each other. The host divides four apples between them so that the cut pieces are no smaller than half an apple. Thus, each contestant could receive either nothing at all, or half an apple, or one apple, or even one-and-a-half apples, although, Alice, Bob, and Charles will not know how many apples were distributed among them. But the number shared is always a whole number, or none at all! Alice, as the team leader, will have to correctly guess whether the members possess between them an even number (0, 2, or 4) or an odd number (3 or 4) of apples. She is entitled to get a little help from her fellow contestants who are each given a small flag by the host. After looking at their share of apples, Bob and Charles will be allowed to hold their flags up, or down. But Alice will not be able to see Bob and Charles’s actions—the host will have to relay the information to her about the orientations of each flag (either up or down) and with this information, and with her own apple, Alice will then have to make her choice. The contestants are allowed to “conspire” before they enter their respective booths (for example, they could agree on simple schemes such as flag up means zero, flag down means one), but after they do so, they are not allowed to communicate with each other anymore. Now, the million-dollar question is whether it is possible for Alice, Bob, and Charles to devise a strategy to always win this game? It turns out that within the boundaries of our classical world there is no possible way for Alice, Bob, and Charles to always win this game. But interestingly, there is a way for them to always win this game if, when they get together to conspire a strategy, they distribute an entangled quantum system between them, namely the GHZ state, and use this entangled quantum system in the game.

The Greenberger-Horne-Zeilinger (GHZ) theorem (also known as Bell’s theorem without inequalities) represents a significant theoretical advancement in the foundations of quantum mechanics following Bell’s work. In 1989, the American physicists Daniel Greenberger and Michael Horne collaborated with Austrian physicist Anton Zeilinger to introduce an innovative way of testing entanglement, thus extending Bell’s theorem in an interesting way (Greenberger, Horne, Shimony, & Zeilinger, 1990).

Recall, Bell’s 1964 theorem states that certain statistical correlations predicted by quantum mechanics involving measurements on two-particle systems cannot be explained by a realistic picture based on the local properties of each individual particle. This is true even if the two particles are separated by large distances. It was Einstein, Podolsky and Rosen who first recognized the vital significance of these two-particle quantum correlations (termed “entanglement” by Schrödinger). Bell’s theorem can be stated in terms of an inequality, which in GHZ’s words, “is violated by certain quantum mechanical statistical predictions. Statistical predictions concern imperfect or statistical correlations, in which the outcome of a measurement on one system determines not the outcome of a measurement on the other system but rather the probabilities of various outcomes.”

GHZ considered a system consisting of three or more correlated sin-½ particles but unlike Bell’s original argument, “GHZ’s demonstration of the incompatibility of quantum mechanics with EPR’s propositions concerned only perfect correlations rather than statistical correlations, and it completely dispenses with inequalities. Since EPR’s argument for the incompleteness of quantum mechanics was based upon perfect correlations, GHZ’s analysis lies close to the heart of EPR’s ideas, but with the surprising turnabout of exhibiting a contradiction.” Perfect correlation, according to EPR, is where “without in any way disturbing a system, we can predict with certainty (i.e., a probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.” GHZ came up with a scenario where one could predict with certainty, and without any disturbance, the remote measurement outcome of the third local observable, once the other two local results of measurements were known. This always-versus-never refutation of Einstein, Podolsky and Rosen is a remarkable feature of the GHZ entangled states involving three or four correlated spin-½ particles, first noted by Greenberger, Horne, and Zeilinger and according to N. David Mermin, GHZ “found a clever and powerful extension of the two-particle EPR experiment to gedanken decays that produce more than two particles.”

 Figure 12: The experimental set-up for the Greenberger-Horne-Zeilinger (GHZ) tests of quantum nonlocality.

However, experimental verification of the GHZ theorem proved difficult, as it involved entanglement between at least three particles. In 2000, Zeilinger and his team (Pan, Bouwmeester, Daniell, Weinfurter, & Zeilinger, 2000) eventually reported the observation of three photon entanglement, or GHZ states using a new method. In the words of the experimenters “The results of three specific experiments, involving measurements of polarization correlations between three photons, lead to predictions for a fourth experiment; quantum physical predictions are mutually contradictory with expectations based on local realism. We find the results of the fourth experiment to be in agreement with the quantum prediction and in striking conflict with local realism.”