# This Quantum World/GHZ

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**\< PREVIOUS**

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## 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

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X_AY_BY_C=1,\\quad Y_AX_BY_C=1,\\quad Y_AY_BX_C=1`</math>`{=html}

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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

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X_AY_BY_C=1,\\quad Y_AX_BY_C=1,\\quad Y_AY_BX_C=1,\\quad
X_AX_BX_C=-1`</math>`{=html}

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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!*

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[^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.
