# Classification of Non-local Actions: Area versus Volume Entanglement Entropy

## TL;DR

This paper investigates how non-local actions with fractional Laplacians can be transformed into local theories, revealing conditions under which entanglement entropy follows an area law or scales with volume, thus providing a test for locality.

## Contribution

It establishes a criterion based on Hilbert-space transforms to determine when non-local actions are equivalent to local theories, affecting entanglement entropy behavior.

## Key findings

- Locality corresponds to an area law for entanglement entropy.
- Failure of reduction leads to volume-scaling entanglement entropy.
- Proposes a new test for the locality of Gaussian theories.

## Abstract

From the partition function for two classes of classically non-local actions containing the fractional Laplacian, we show that as long as there exists a suitable (non-local) Hilbert-space transform the underlying action can be mapped onto a purely local theory. In all such cases the partition function is equivalent to that of a local theory and an area law for the entanglement entropy obtains. When such a reduction fails, the entanglement entropy deviates strongly from an area law and can in some cases scale as the volume. As these two criteria are coincident, we conjecture that they are equivalent and provide the ultimate test for locality of Gaussian theories rather than a simple inspection of the explicit operator content.