  The geometric entanglement entropy of a quantum field in the vacuum state is
known to be divergent and, when regularized, to scale as the area of the
boundary of the region. Here we introduce an operational definition of the
entropy of the vacuum restricted to a region: we consider a subalgebra of
observables that has support in the region and a finite resolution. We then
define the entropy of a state restricted to this subalgebra. For Gaussian
states, such as the vacuum of a free scalar field, we discuss how this entropy
can be computed. In particular we show that for a spherical region we recover
an area law under a suitable refinement of the subalgebra.
