Necessary criterion for approximate recoverability

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 $I(A:C\mathrm{\mid }B{)}_{\rho }$ 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.