I’ve got only time for a brief answer, but I’ll elaborate if any questions are left open…
Basically, what they’re doing is nothing new, in the sense that it’s not surprising – essentially, they’ve realized a thought experiment from Asher Peres from 1999, I believe, and nobody was having much doubt that things would work out as predicted. (By the way, here’s the article in free preprint form, and here’s Peres’ original work.)
What they’re testing, however, is a somewhat counterintuitive effect (well, a lot of quantum effects are), and is unfortunately liable to over-interpretation. The key point in their setup is the delayed-choice component, which basically says that in the quantum world, you can determine (in some sense) a measurement result after the measurement has taken place – for instance, in the familiar double slit experiment, you can decide whether or not to observe a photon after it has passed the barrier, thus leading either to the observation of wave- or particle-like behaviour (roughly). This is Wheeler’s original delayed-choice experiment.
However, big caveats need to be put into place here. Most importantly, this doesn’t really lend itself to ‘influencing the past’-type interpretations; in particular, as always with entanglement, you couldn’t send any message with it, nor do anything else that’s causally efficacious. Indeed, from the point of view of entanglement the effect isn’t all that surprising: measuring one of a pair of entangled particles leads to instantaneous ‘collapse’ of the other into some definite state (if you like to talk that way, which I usually don’t), where instantaneous here means without delay across arbitrary distances – but according to special relativity, of course, this is in some sense equal to action ‘backwards into the past’ (if two parties, A and B, share an entangled pair of qubits, and both measure at the same time in their shared frame of reference, there are inertial observers seeing both A measure first, causing the collapse of B’s particle, and seeing B measure first).
So in that sense, it’s not really so weird after all – at least not weirder than however weird you consider entanglement to be.
The additional twist they put onto the story is that of entanglement swapping, which is a different, counterintuitive effect in quantum mechanics. Basically, you have three parties, A, B, and C, and create two pairs of entangled photons, AC1, and BC2. A gets one of the AC1-entangled pair, while C gets the other; similarly, B gets one of the BC2 entangled pair, while C gets the other. So the situation is somewhat like this:
o~~~~~~o o~~~~~~o
A C1 C2 B
Now, C can make an operation on his two photons, entangling them – and as a result, A and B end up entangled with one another, as well, even though they have never interacted with one another! In the end, the situation is thus this one:
o~~~~~~~~~~~~~~~o
o~~o
A C1 C2 B
Now, of course, A and B can, through measurement and comparing their results afterwards, determine whether their photons were entangled.
But, and here’s the twist, C can decide whether or not to entangle the photons after both B and A have already carried out their measurements, thus entangling them a posteriori!
However, the whole shebang only becomes noticeable once all three observers meet and compare notes – in particular, C needs to tell A and B in which cases she entangled their photons. Otherwise, there are many possible ways to select subsets of ‘entangled’ states from A’s and B’s combined data, even if they have in fact only undertaken random measurements on uncorrelated systems. So the only ‘active’ role C plays, ultimately, is just that of a selector of a certain subset of A’s and B’s measurements – like in the case above, not quite the ‘influence on the past’ kind of thing that is often suggested (but still, really neat).
Seems this got a little longer than anticipated…