# Necessary criterion for approximate recoverability

## TL;DR

This paper investigates the necessary conditions for approximate quantum state recovery, establishing a lower bound involving conditional mutual information and a disturbance measure, thus deepening understanding of quantum Markov chains.

## Contribution

It provides a lower bound on the relative entropy for approximate recovery, highlighting the necessary conditions involving conditional mutual information and disturbance.

## Key findings

- Lower bound on relative entropy for approximate recovery
- Conditional mutual information as a key factor
- Quantification of disturbance in the recovery process

## Abstract

A tripartite state $\rho_{ABC}$ forms a Markov chain if there exists a recovery map $\mathcal{R}_{B \to BC}$ acting only on the $B$-part that perfectly reconstructs $\rho_{ABC}$ from $\rho_{AB}$. To achieve an approximate reconstruction, it suffices that the conditional mutual information $I(A:C|B)_{\rho}$ is small, as shown recently. Here we ask what conditions are necessary for approximate state reconstruction. This is answered by a lower bound on the relative entropy between $\rho_{ABC}$ and the recovered state $\mathcal{R}_{B\to BC}(\rho_{AB})$. The bound consists of the conditional mutual information and an entropic correction term that quantifies the disturbance of the $B$-part by the recovery map.