# Notes on entanglement wedge cross sections

## TL;DR

This paper investigates the properties of entanglement wedge cross sections in holography, comparing them to the entanglement of purification, across various geometries and theories, revealing non-monotonic behavior and scale-dependent purification.

## Contribution

It provides the first detailed analysis of entanglement wedge cross sections in diverse holographic settings, including finite temperature, confining theories, and complex RG flows.

## Key findings

- $E_W$ is non-monotonic with scale, similar to mutual information.
- $E_W$ exhibits non-trivial behavior in complex RG flow geometries.
- Supports the idea that purification can occur through scale expansion or reduction.

## Abstract

We consider the holographic candidate for the entanglement of purification $E_P$, given by the minimal cross sectional area of an entanglement wedge $E_W$. The $E_P$ is generally very complicated quantity to obtain in field theories, thus to establish the conjectured relationship one needs to test if $E_W$ and $E_P$ share common features. In this paper the entangling regions we consider are slabs, concentric spheres, and creases in field theories in Minkowski space. The latter two can be mapped to regions in field theories defined on spheres, thus corresponding to entangled caps and orange slices, respectively. We work in general dimensions and for slabs we also consider field theories at finite temperature and confining theories. We find that $E_W$, similarly to holographic mutual information, is not a monotonic function of a scale. We also study a full ten-dimensional string theory geometry dual to a non-trivial RG flow of a three-dimensional Chern-Simons matter theory coupled to fundamentals. We show that also in this case $E_W$ behaves non-trivially, which if connected to $E_P$, lends further support that the system can undergo purification simply by expansion or reduction in scale.