# Towards a Bit Threads Derivation of Holographic Entanglement of Purification

## TL;DR

This paper uses the bit thread approach to relate the entanglement of purification in holography to geometric minimal surfaces, providing a new derivation and insights into holographic entanglement structures.

## Contribution

It introduces a novel bit thread configuration that constructs a purification of reduced states, supporting the $E_P = E_W$ conjecture in holography.

## Key findings

- Established conditions under which $E_P = E_W$ holds.
- Constructed a specific bit thread configuration for purification.
- Discussed implications for black hole geometries and minimal purifications.

## Abstract

We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured $E_P = E_W$ relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.