# Entanglement of purification and disentanglement in CFTs

## TL;DR

This paper investigates the entanglement of purification in conformal field theories, exploring its relation to disentanglement processes, and derives constraints on the states involved using holographic principles and entropy inequalities.

## Contribution

It introduces a new constraint on the state producing EoP in CFTs and analyzes the dissimilarity of candidate disentangled states with the actual EoP state.

## Key findings

- Derived a commutation constraint on the EoP state.
- Identified limitations of candidate disentangled states.
- Connected EoP to holographic and entropy inequality frameworks.

## Abstract

We study the entanglement of purification (EoP) of subsystem $A$ and B in conformal field theories (CFTs) stressing on its relation to unitary operations of disentanglement, if the auxiliary subsystem $\tilde{A}$ adjoins $A$ and $\tilde{A}\tilde{B}$ is the complement of $AB$. We estimate the amount of the disentanglement by using the holographic EoP conjecture as well as the inequality of Von Neumann entropy. Denote the state that produces the EoP by $|\psi\rangle_M$. We calculate the variance of entanglement entropy of $A\tilde{A}$ in the state $|\psi(\delta)\rangle:=e^{i\delta H_{\tilde{A}\tilde{B}}}|\psi\rangle_M$. We find a constraint on the state $|\psi\rangle_M$, $[K_{A\tilde{A},M},O_{\tilde{A}}]=0$, where $K_{A\tilde{A},M}$ is the modular Hamiltonian of $A\tilde{A}$ in the state $|\psi\rangle_M$, $O_{\tilde{A}}\in \mathscr{R}(\tilde{A})$ is an arbitrary operator. We also study three different states that can be seen as disentangled states. Two of them can produce the holographic EoP result in some limit. But we show that none of they could be a candidate of the state $|\psi\rangle_M$, since the distance between these three states and $|\psi\rangle_M$ is very large.