  Recently, there have been a number of works investigating the entanglement
properties of distinct noncomplementary parts of discrete and continuous
Bosonic systems in ground and thermal states. The Fermionic case, however, has
yet to be expressly addressed. In this paper we investigate the entanglement
between a pair of far-apart regions of the 3+1 dimensional massless Dirac
vacuum via a previously introduced distillation protocol [B. Reznik et al.,
Phys. Rev. A 71, 042104 (2005)]. We show that entanglement persists over
arbitrary distances, and that as a function of L/R, where L is the distance
between the regions and R is their typical scale, it decays no faster than
exp(-(L/R)^2). We discuss the similarities and differences with analogous
results obtained for the massless Klein-Gordon vacuum.
