Lower estimates for distances from a given quantum channel to certain classes of quantum channels

By using estimates for the variation of quantum mutual information and the relative entropy of entanglement, we obtain $\varepsilon$-exact lower estimates for distances from a given quantum channels to sets of degraded, anti-degraded, and entanglement-breaking channels. As an auxiliary result, we obtain $\varepsilon$-exact lower estimates for the distance from a given two-particle state to the set of all separable states.