  We show that, for a class of planar determinantal point processes (DPP) X,
the entanglement entropy of X on a compact region grows exactly at the rate of
the variance fluctuation in that region.
  Therefore, such DPPs satisfy an area law if they are of Class I
hyperuniformity, while the area law is violated if they are of Class II
hyperuniformity. As a result, the entanglement entropy of Weyl-Heisenberg
ensembles (a family of DPPs containing the Ginibre ensemble and Ginibre-type
ensembles in higher Landau levels), satisfies an area law, as a consequence of
its hyperuniformity.
