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.
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.
This Quantum World 