The key is that a measurement "projects" a particle into a definite state -- just as the measurement of a photon collapses it into either vertical or horizontal polarization. So even though photons 2 and 3 start out unentangled, physicists can set up a "projective measurement" that asks, are the two in one of two distinct entangled states or the other? That measurement entangles the photons, even as it absorbs and destroys them. If the researchers select only the events in which photons 2 and 3 end up in, say, the first entangled state, then the measurement also entangles photons 1 and 4. (See diagram, top.) The effect is a bit like joining two pairs of gears to form a four-gear chain: Enmeshing to inner two gears establishes a link between the outer two.
In recent years, physicists have played with the timing in the scheme. For example, last year a team showed that entanglement swapping still works even if they make the projective measurement after they've already measured the polarizations of photons 1 and 4. Now, Eisenberg and colleagues have shown that photons 1 and 4 don't even have to exist at the same time, as they report in a paper in press at Physical Review Letters.
To do that, they first create entangled pair 1 and 2 and measure the polarization of 1 right away. Only after that do they create entangled pair 3 and 4 and perform the key projective measurement. Finally, they measure the polarization of photon 4. And even though photons 1 and 4 never coexist, the measurements show that their polarizations still end up entangled. Eisenberg emphasizes that even though in relativity, time measured differently by observers traveling at different speeds, no observer would ever see the two photons as coexisting.