# The connection between holographic entanglement and complexity of purification

## TL;DR

This paper explores the relationship between entanglement of purification and the complexity of purification in holographic systems, analyzing various black hole solutions and interpreting results through resource theories and bit threads.

## Contribution

It introduces and compares the complexity of purification and interval volume in holographic systems, extending analysis to diverse black hole backgrounds and multipartite states.

## Key findings

- EoP and CoP are related through the $E_P=E_W$ conjecture.
- Complexity measures depend on system parameters like temperature and separation.
- Results are interpreted via resource theories and bit thread models.

## Abstract

In this work we study how entanglement of purification (EoP) and the new quantity of "complexity of purification" are related to each other using the $E_P=E_W$ conjecture. First, we consider two strips in the same side of a boundary and study the relationships between the entanglement of purification of this mixed state and the parameters of the system such as dimension, temperature, length of the strips and the distance between them. Next, using the same setup, we introduce two definitions for the complexity of mixed states, complexity of purification (CoP) and the interval volume (VI). We study their connections to other parameters similar to the EoP case. Then, we extend our study to more general examples of BTZ black holes solution in massive gravity, charged black holes and multipartite systems. Finally, we give various interpretations of our results using resource theories such as LOCC and also bit thread picture.