Many-region vacuum entanglement: Distilling a W state

We investigate the correlations between any number of arbitrarily far-apart regions of the vacuum of the free Klein-Gordon field by means of its finite duration coupling to an equal number of localized detectors. We show that the correlations between any N such regions enable us to distill an N-partite W state, and therefore exhibit true $N$-fold entanglement. Furthermore, we show that for N=3, the correlations cannot be reproduced by a hybrid local-nonlocal hidden-variable model. For N >= 4 the issue remains open.