Necessary criterion for approximate recoverability

Abstract

A tripartite state $\ddot{I}{}_{ABC}$ forms a Markov chain if there exists a recovery map ${\mathcal{R}}_{B\to BC}$ acting only on the $B$-part that perfectly reconstructs $\ddot{I}{}_{ABC}$ from $\ddot{I}{}_{AB}$. To achieve an approximate reconstruction, it suffices that the conditional mutual information $I(A:C|B{)}_{\ddot{I}}$ 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 $\ddot{I}{}_{ABC}$ and the recovered state ${\mathcal{R}}_{B\to BC}(\ddot{I}{}_{AB})$. 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.

v2: 18 pages, final version published in Annales Henri Poincaré