# Entanglement entropy from entanglement contour: higher dimensions

## TL;DR

This paper explores the entanglement contour and partial entanglement entropy in higher-dimensional quantum field theories, clarifying their relation to geometric and UV regulators, and deriving explicit formulas for spherical regions in conformal field theories.

## Contribution

It provides a detailed classification of geometric regulators, clarifies the ALC proposal for PEE in higher dimensions, and derives explicit entanglement contour functions for various regions in CFTs.

## Key findings

- Derived the exact relation between UV and geometric cutoffs for spherical regions.
- Clarified the proper evaluation of subset entanglement entropies in the ALC proposal.
- Provided explicit formulas for entanglement contour in spherical and shell regions in higher-dimensional CFTs.

## Abstract

We study the \textit{entanglement contour} and \textit{partial entanglement entropy} (PEE) in quantum field theories in 3 and higher dimensions. The entanglement entropy is evaluated from a certain limit of the PEE with a geometric regulator. In the context of the \textit{entanglement contour}, we classify the geometric regulators, study their difference from the UV regulators. Furthermore, for spherical regions in conformal field theories (CFTs) we find the exact relation between the UV and geometric cutoff, which clarifies some subtle points in the previous literature. We clarify a subtle point of the additive linear combination (ALC) proposal for PEE in higher dimensions. The subset entanglement entropies in the \textit{ALC proposal} should all be evaluated as a limit of the PEE while excluding a fixed class of local-short-distance correlation. Unlike the 2-dimensional configurations, naively plugging the entanglement entropy calculated with a UV cutoff will spoil the validity of the \textit{ALC proposal}. We derive the \textit{entanglement contour} function for spherical regions, annuli and spherical shells in the vacuum state of general-dimensional CFTs on a hyperplane.