7 The GHZ experiment

As Green­berger, Horne, and Zeilinger have shown,[1] quantum mechanics allows us to pre­pare three par­ti­cles A, B, C and to sub­ject each to either of two mea­sure­ments X, Y in such a way that

  • the out­come of each mea­sure­ment is either +1 or −1,
  • the product of the three out­comes is −1 if each par­ticle is sub­jected to a mea­sure­ment of X,
  • the product of the three out­comes is +1 if one par­ticle is sub­jected to a mea­sure­ment of X and the two other par­ti­cles are sub­jected to a mea­sure­ment of Y.

To imple­ment the fail-​​safe strategy men­tioned in the pre­vious sec­tion, Andy, Bob, and Charles pre­pare three par­ti­cles in this par­tic­ular manner. Each player keeps one par­ticle with him. When asked for the value of X, he will mea­sure the x com­po­nent of his particle’s spin, and when asked for the value of Y, he will mea­sure the y com­po­nent. His answer will be +1 or −1 according as his out­come is pos­i­tive or neg­a­tive. Pro­ceeding in this way, the players are sure to win.

Sup­pose now that the quan­ti­ties being mea­sured have values irre­spec­tive of whether they are actu­ally mea­sured. Let us call these pur­port­edly pre-​​existent values XA, XB, XC and YA, YB, YC. If YA, YB, and YC have actu­ally been mea­sured, we can then argue that a mea­sure­ment of XA would have yielded the product YB YC since the product XA YB YC equals 1 (so that XA = 1 if YB YC = 1 and XA = −1 if YB YC = −1). In the same way we can argue that a mea­sure­ment of XB would have yielded the product YA YC, and that a mea­sure­ment of XC would have yielded the product YA YB. It fol­lows that if we had mea­sured XA, XB, and XC instead of YA, YB, and YC, the product of the out­comes would have been

XA, XB, XC = YB YC YA YC YA YB = (YA)2 (YB)2 (YC)2 = +1.

But we know that if we had mea­sured XA, XB, and XC, the product of the out­comes would have been −1!

What went wrong? Which assump­tion has just been reduced to absurdity?

Most physi­cists would agree that it is the assump­tion that phys­ical quan­ti­ties are in pos­ses­sion of values irre­spec­tive of whether they are actu­ally mea­sured, though there appears to be a narrow escape route for the pro­po­nents of pre-​​existent values. Those who are stead­fast in their belief in the reality of unmea­sured values argue that at least some phys­ical quan­ti­ties are con­tex­tual. By this they mean that a quan­tity such as YB has more than one pre-​​existent value, and that the value that will show up in a mea­sure­ment depends on the mea­sure­ment con­text (that is, on the other quan­ti­ties together with which it is measured).

To illus­trate this notion, sup­pose that XA = −XB = XC = +1, and that YA = −YC = +1. What, in this case, would be the value of YB? If YB is mea­sured together with XA and YC, its value has to be −1 because XA YB YC = +1, and if it is mea­sured together with YA and XC, its value has to be +1 because YA YB XC = +1.

The inherent absur­dity of con­tex­tu­ality is that it is intended to allow phys­ical quan­ti­ties to exist inde­pen­dently of mea­sure­ments even though their values can only be defined as mem­bers of sets of mea­sure­ments that are per­formed together.

By the time Green­berger, Horne, and Zeilinger pub­lished their paper, it was all but taken for granted that the con­tra­dic­tions between quantum mechanics and the common-​​sense view that phys­ical quan­ti­ties have values regard­less of whether or not they are mea­sured, are essen­tially sta­tis­tical. Bell-​​type inequal­i­ties, for instance, which codify a common-​​sense expec­ta­tion, are vio­lated by sta­tis­tical dis­tri­b­u­tions of mea­sure­ments. Hence when Green­berger et al. showed that one can dis­pose of (non-​​contextual) pre-​​existent values through a pre­dic­tion that can be refuted by a single mea­sure­ment, it caused quite a stir.

In their sem­inal paper of 1935, Ein­stein, Podolsky, and Rosen had argued that

If, without in any way dis­turbing a system, we can pre­dict with cer­tainty (i.e., with prob­a­bility equal to unity) the value of a phys­ical quan­tity, then there exists an ele­ment of phys­ical reality cor­re­sponding to this phys­ical quantity.

When GHZ pub­lished their find­ings, Mermin quipped: “So farewell ele­ments of reality! And farewell in a hurry”.[2]

The first GHZ-​​type exper­i­ment was per­formed by Bouwmeester et al.[3] Need­less to say, it was in agree­ment with the pre­dic­tions of quantum mechanics.

Next


1. [↑] Green­berger, D.M., Horne, M.A., and Zeilinger, A. (1989). Going beyond Bell’s the­orem, in M. Kafatos (ed.), Bell’s The­orem, Quantum Theory, and Con­cep­tion of the Uni­verse, Kluwer Aca­d­emic, 69–72.

2. [↑] Mermin, N.D. (1990). What’s wrong with these ele­ments of reality?, Physics Today 43 (6), 9–11.

3. [↑] Bouwmeester, D., Pan, J–W., Daniell, M., Wein­furter, H. and Zeilinger, A. (1999). Obser­va­tion of three-​​photon Greenberger-​​Horne-​​Zeilinger entan­gle­ment, Phys­ical Review Let­ters 82, 1345–1349.