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A Quantum Game
By koantum, Section Reviews Posted on Wed Apr 26, 2006 at 11:34:58 PM PST
Three-particle entanglement was first discussed by Greenberger, Horne, and Zeilinger. Whereas the two-particle entanglement discussed by Bell exhibits inconsistencies with common sense expectations at the statistical level, the GHZ experiment demonstrates the inconsistency of the quantum world with common sense by a single measurement outcome.

Andy, Bob, and Charles play as a team. Everyone is asked one of two questions: "What is the value of X" or "What is the value of Y?" Only two answers are allowed: +1 or -1. Either everyone is asked the first question, or one player is asked the first question and two players are asked the second question. Andy, Bob, and Charles win if the product of their answers is -1 in case only the first question is asked, and if the product of their answers is +1 in case the second question is asked. If we call the answers XA, XB, XC and YA, YB, YC, then the four winning combinations satisfy the following equations: XA x XB x XC = -1 XA x YB x YC = 1 YA x XB x YC = 1 YA x YB x XC = 1 The players are not allowed to communicate which each other once the questions are asked. Before that, they are allowed to work out a strategy. Is there a failsafe strategy? Can they make sure that they will win? With pre-agreed answers XA, XB, XC, YA, YB, YC there is no failsafe strategy. Find out why. However, there is a failsafe strategy without pre-agreed answers. It goes like this: Andy, Bob, and Charles prepare three particles (say, electrons) in a particular quantum state. This means that the particles are described by joint probability distributions over the possible outcomes of measurements to which they may be subjected. These probabilities are independent of how far the particles are apart. Each player takes with him one of these particles. Whoever is asked the X question measures the x component of the spin of his particle (which has two possible values, +1/2 or -1/2) and answers with twice his outcome. Whoever is asked the Y question measures the y component of the spin of his particle and answers likewise. In this way our team is sure to win every time. But this brings up a new question: Is it possible for the x and y components of the spins of the three particles to be in possession of values before their values are actually measured? Find our why the answer is no. Credits: L. Vaidman ("Variations on the theme of the Greenberger-Horne-Zeilinger proof," Foundations of Physics 29, 615-30, 1999) D. M. Greenberger, M.A. Horne, and A. Zeilinger ("Going beyond Bell's theorem" in Bell's theorem, Quantum Theory, and Conceptions of the Universe, edited by M. Kafatos, Kluwer, 1989, pp. 69-72)

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