Urbasi Sinha recently gave a seminar talk at IQIS about her test of the Born rule using a triple-slit setup, as well as an implementation of a single-qutrit system using the same setup. The bulk of the talk was based on an arXiv posting, and while it contains no earth-shattering results (Born was right, probably!), the motivation and experiment are quite interesting.
Born's rule is so ubiquitous in quantum mechanics that I had actually forgotten it had a special name and had to look it up before the talk; it's the rule which tells us that the probability distribution for a quantum system is given by |Ψ|². We know of course that this rule holds very well in a wide variety of situations, but then we know the same about Newton's laws, and look what happened there.
The fundamental principle being examined in the experiment is a rule for describing interference terms when combining probabilities, and was originally discussed by Rafael Sorkin in 1994. The original double-slit experiment demonstrates the wave nature of light (and other particles) by showing that the probability distribution resulting from a photon's passing through two slits is not equal to the sum of the probability distributions of having passed through one of the slits but not the other. The difference between these two possibilities is the interference term,
I(A,B) = P(A,B) - P(A) - P(B).
That is, the interference arising when slits A and B are both open is given by adding up the distributions when the slits are each open on their own, P(A) and P(B), and then subtracting the distribution arising when they are both open. If there is no interference, then I(A,B) vanishes. This is what we naïvely expect with our classical intuition for a single particle, and it is the fact that this is not the case (among other things) that gave rise to quantum mechanics.
What if there are three slits though? The corresponding rule is
I(A,B,C) = P(A,B,C) - P(A,B) - P(A,C) - P(B,C) + P(A) + P(B) + P(C).
If we assume that Born's rule is correct, then I(A,B,C) = 0. Running an experiment in which single quantum particles are sent through a three-slit apparatus can tell us what nature yields for I(A,B,C), and if it doesn't equal zero then something new and different is going on!
As alluded to earlier of course, no such violation was found. As far as they were able to determine, Born's rule does indeed hold. An interesting side note is that after the result appeared in Science last year (arXiv listing here), Max Born's son (in his nineties) wrote to the experimenters to thank them for their effort to prove his father's decades-old theoretical prediction.