  We give two strengthenings of an inequality for the quantum conditional
mutual information of a tripartite quantum state recently proved by Fawzi and
Renner, connecting it with the ability to reconstruct the state from its
bipartite reductions. Namely we show that the conditional mutual information is
an upper bound on the regularised relative entropy distance between the quantum
state and its reconstructed version. It is also an upper bound for the measured
relative entropy distance of the state to its reconstructed version. The main
ingredient of the proof is the fact that the conditional mutual information is
the optimal quantum communication rate in the task of state redistribution.
