# Entropy of a subalgebra of observables and the geometric entanglement entropy

## TL;DR

This paper introduces an operational approach to define the entropy of a quantum field's vacuum state restricted to a region, using a subalgebra of observables with finite resolution, and demonstrates an area law for spherical regions.

## Contribution

It proposes a new operational definition of vacuum entropy based on subalgebras with finite resolution, recovering the area law for spherical regions.

## Key findings

- Operational entropy aligns with the geometric area law.
- Finite resolution subalgebras yield finite entropy.
- Area law is recovered for spherical regions.

## Abstract

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.