  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.
