Necessary Criterion for Approximate Recoverability

Abstract

A tripartite state ρ_ABC forms a Markov chain if there exists a recovery map R_B→_BC acting only on the B-part that perfectly reconstructs ρ_ABC from ρ_AB. To achieve an approximate reconstruction, it suffices that the conditional mutual information I(A:C|B)_ρ 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 ρ_ABC and the recovered state R_B→_BC(ρ_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.