# Bit threads and holographic entanglement of purification

## TL;DR

This paper introduces a 'bit thread' approach to holographic entanglement of purification, providing new proofs, flow interpretations, and bounds related to quantum correlations and information transmission.

## Contribution

It proposes a novel 'bit thread' formulation of EoP, proves properties and monogamy relations, and derives tighter bounds involving quantum advantage of dense coding.

## Key findings

- Bit thread formulation of EoP established.
- Flow interpretation of quantum advantage of dense code.
- New tighter lower bound for $S(AB)$ in terms of QAoDC.

## Abstract

The entanglement of purification (EoP), which measures the classical correlations and entanglement of a given mixed state, has been conjectured to be dual to the area of the minimal cross section of the entanglement wedge in holography. Using the surface-state correspondence, we propose a `bit thread' formulation of the EoP. With this formulation, proofs of some known properties of the EoP are performed. Moreover, we show that the quantum advantage of dense code (QAoDC), which reflects the increase in the rate of classical information transmission through quantum channel due to entanglement, also admits a flow interpretation. In this picture, we can prove the monogamy relation of QAoDC with the EoP for tripartite states. We also derive a new lower bound for $S(AB)$ in terms of QAoDC, which is tighter than the one given by the Araki-Lieb inequality.