# Entanglement of Purification and Projective Measurement in CFT

## TL;DR

This paper explores the entanglement of purification in conformal field theory, establishing a holographic formula and examining the effects of projective measurements, revealing a constant difference related to boundary contributions.

## Contribution

It provides a holographic proof of the EoP formula using the surface/state correspondence and analyzes the role of projective measurements in the purification process.

## Key findings

- Holographic EoP formula is proven using the Reeh-Schlieder theorem.
- The difference between holographic EoP and post-measurement entanglement entropy is a constant.
- Projective measurements can approximate purification states with minimal EoP.

## Abstract

We investigate entanglement of purification in conformal field theory. By using Reeh-Schlieder theorem, we construct a set of the purification states for $\rho_{AB}$, where $\rho_{AB}$ is reduced density matrix for subregion $AB$ of a global state $\rho$. The set can be approximated by acting all the unitary observables,located in the complement of subregion $AB$, on the global state $\rho$, as long as the global state $\rho$ is \text{cyclic} for every local algebra, e.g., the vacuum state. Combining with the gravity explanation of unitary operations in the context of the so-called surface/state correspondence, we prove the holographic EoP formula. We also explore the projective measurement with the conformal basis in conformal field theory and its relation to the minimization procedure of EoP. Interestingly, though the projective measurement is not a unitary operator, the difference in some limits between holographic EoP and the entanglement entropy after a suitable projective measurement is a constant $\frac{c}{3}\log 2$ up to some contributions from boundary. This suggests the states after projective measurements may approximately be taken as the purification state corresponding to the minimal value of the procedure.