We derive a Lipschitz continuity bound for quantum-classical conditional
entropies with respect to angular distance, with a Lipschitz constant that is
independent of the dimension of the conditioning system. This bound is sharper
in some situations than previous continuity bounds, which were either based on
trace distance (where Lipschitz continuity is not possible), or based on
angular distance but did not include a conditioning system. However, we find
that the bound does not directly generalize to fully quantum conditional
entropies. To investigate possible counterexamples in that setting, we study
the characterization of states which saturate the Fuchs--van de Graaf
inequality and thus have angular distance approximately equal to trace
distance. We give an exact characterization of such states in the invertible
case. For the noninvertible case, we show that the situation appears to be
significantly more elaborate, and seems to be strongly connected to the
question of characterizing the set of fidelity-preserving measurements.Comment: 21 pages, 3 figure