Correlations without Correlata?
While Bell’s theorems aim at the derivation of a statistical inequality that violates quantum predictions, the physicsts Greenberger, Zeilinger and Horne provided in 1989 a new theorem “without inequalities”, therefore yielding a “stronger” result than Bell’s, that again heated up the ongoing debate about the philosophical implications of quantum theory. We will focus on a simplified version of the proof given by Mermin [1].
Imagine a gedanken setup (figure above) that consists of a particle source that creates triples of spin ½ particles. After the emission, they are sent into areas of spacetime that are far apart from each other - so far that the velocity of light would not suffice to signal information between them while experimenters do nearly simultaneous spin measurements on each of the particles (spacelike separation). Each measurement yields binary results, one can only get +1 or -1, and the measurements are modeled in the quantum mechanical formalism using Pauli spin matrices σ. Their upper case index {1, 2, 3} denotes the number of the particle, whereas the lower case index {i,j,k} denotes the spatial direction in which the spin is measured. One can choose i,j,k=x,y,z independently for each of the particles. In particular the following four “observables” are of interest, meaning that we either measure all combinations of one x-setting and two y-settings or the triple x-setting. Those combinations can be simultaneously mesured because all four operators do indeed commute (though the single particle Pauli matrices do not commute of course).
At this point it is useful to become aware of the fact that the dimension of the whole system’s Hilbert space is 2^3=8, since each particle’s Hilbert space is 2-dimensional and the dimension of the whole Hilbert space is the product of the constituent’s dimensions due to the tensor product. Bell’s setup involves only two particles and therefore the Hilbert space dimension is only 2*2=4. Having noted this, the following maximally entangled wavefunction describes the joint state of those three particles:
Where the kets denote the basis of the σ_z Pauli matrix. Some computation yields the following eigenvalues:
Note that the triple x-measurment (last line) does have a negative eigenvalue - this will be the source of the derived contradiction. So far so good, we’re done with the part that relies on the purely quantum mechanical formalism. The interesting thing is, that with this particular setup one can, after the measurement of two particles, determine with certainty what the result of the third must be. E.g. we measure the x-direction on particle 1, the y-direction on particle 2 and the y-direction on particle 3 (first line of the equations above). Then the product of all outcomes must be +1. In the case that the outcomes of particle 2 and 3 are -1, we can directly determine that the first particle’s outcome must be +1. More generally, if both y-components yield the same result, the x-outcome must be +1. If both y-measurements yield a different result, the x-outcome must be -1. This is in accordance with the quantum mechanical predictions above. The possibility of determining values with certainty makes one suspect that it might be reasonable to assume that there are some predetermined values prior to the measurement. Let’s elaborate this train of thought: We denote the predetermined measurement value with the variable m. Its upper case index stands for the particle’s number and the lower case index for the spin component. Then, according to our findings, the products of any combination of two y-measurements and one x-measurements yield +1.
If these three equations hold, also the product of these equations must be:
Since all of these are scalars, we should be able to simply rearrange them. Any y-value appears twice, which is why they contribute each a +1 [simply because (-1)^2=(1)^2=1].
But then only the x-values remain and their product must be +1. But this is a clean contradiction to the quantum mechanical prediction! The eigenvalue to the operator corresponding to a triple x-measurement is -1. Since the mathematics is clear, the assumptions leading to the formulation of predetermined values must be flawed somehow - and there’s by far no consensus which assumption is responsible for this mess. Either some notion of “locality” must fail, or there are no “elements of reality” - the possibilites are various. Just to mention one conclusion that could be drawn from this: “Correlations have physical reality; that which they correlate does not.” [3] So one might give up the idea that we have to deal with “real” correlata - but this is not mandatory. One can save the idea of “reality” with certain restrictions, it’s a matter of the own philosophical worldview, so no one can say for sure that the GHZ proof or Bell-type proofs in general are “no go theorems” for hidden variable accounts. Having noted this, I recommend to really understand what’s going on in the GHZ proof above, before jumping to conclusions by adopting a certain interpretation.
Further reading:
[1] Mermin: “What’s wrong with these elements of reality?” doi: 10.1063/1.2810588
[2] Greenberger, Zeilinger, Horne: “Bell’s theorem without inequalities” doi: 10.1119/1.16243
[3] Mermin: “What is Quantum Mechanics trying to tell us?” arXiv:quant-ph/9801057v2
