A significant strengthening of strong subadditivity was proved in 2014, which was subsequently improved in and. In 2017, it was shown that the recovery channel can be taken to be the original Petz recovery map. These improvements of strong subadditivity have physical interpretations in terms of recoverability, meaning that if the conditional mutual information I ( A ; B | E ) = S ( A E ) + S ( B E ) − − S ( E ) − − S ( A B E ) of a tripartite quantum state ρ ρ A B E is nearly equal to zero, then it is possible to perform a recovery channel R E → → A E (from system E to AE) such that ρ ρ A B E ≈ ≈ R E → → A E ( ρ ρ B E ) . These results thus generalize the exact equality conditions mentioned above.