Entropy of a subalgebra of observables and the geometric entanglement entropy

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.