This Quantum World/GHZ

The experiment of Greenberger, Horne, and Zeilinger

And yet there is a failsafe strategy.^[1]

Here goes:

Andy, Bob, and Charles prepare three particles (for instance, electrons) in a particular way. As a result, they are able to predict the probabilities of the possible outcomes of any spin measurement to which the three particles may subsequently be subjected. In principle these probabilities do not depend on how far the particles are apart.
Each player takes one particle with him.
Whoever is asked the X question measures the x component of the spin of his particle and answers with his outcome, and whoever is asked the Y question measures the y component of the spin of his particle and answers likewise. (All you need to know at this point about the spin of a particle is that its component with respect to any one axis can be measured, and that for the type of particle used by the players there are two possible outcomes, namely +1 and −1.

Proceeding in this way, the team of players is sure to win every time.

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?

Suppose that the y components of the three spins have been measured. The three equations

X_{A}Y_{B}Y_{C}=1,\quad Y_{A}X_{B}Y_{C}=1,\quad Y_{A}Y_{B}X_{C}=1

of the previous section tell us what we would have found if the x component of any one of the three particles had been measured instead of the y component. If we assume that the x components are in possession of values even though they are not measured, then their values can be inferred from the measured values of the three y components.

Try to fill in the following table in such a way that

each cell contains either +1 or −1,
the product of the three X values equals −1, and
the product of every pair of Y values equals the remaining X value.

Can it be done?

	A	B	C
X
Y

The answer is negative, for the same reason that the four equations

X_{A}Y_{B}Y_{C}=1,\quad Y_{A}X_{B}Y_{C}=1,\quad Y_{A}Y_{B}X_{C}=1,\quad X_{A}X_{B}X_{C}=-1

cannot all be satisfied. Just as there can be no strategy with pre-agreed answers, there can be no pre-existent values. We seem to have no choice but to conclude that these spin components are in possession of values only if (and only when) they are actually measured.

Any two outcomes suffice to predict a third outcome. If two x components are measured, the third x component can be predicted, if two y components are measured, the x component of the third spin can be predicted, and if one x and one y component are measurement, the y component of the third spin can be predicted. How can we understand this given that

the values of the spin components are created as and when they are measured,
the relative times of the measurements are irrelevant,
in principle the three particles can be millions of miles apart.

How does the third spin "know" which components of the other spins are measured and which outcomes are obtained? What mechanism correlates the outcomes?

You understand this as much as anybody else!

↑ D. M. Greenberger, M. A. Horne, and A. Zeilinger, "Going beyond Bell's theorem," in Bell's theorem, Quantum Theory, and Conception of the Universe, edited by M. Kafatos (Dordrecht: Kluwer Academic, 1989), pp. 69-72.

[1] D. M. Greenberger, M. A. Horne, and A. Zeilinger, "Going beyond Bell's theorem," in Bell's theorem, Quantum Theory, and Conception of the Universe, edited by M. Kafatos (Dordrecht: Kluwer Academic, 1989), pp. 69-72.

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