  A condition for reversibility (sufficiency) of a channel with respect to a
given countable family of states with bounded rank is obtained.
  This condition shows that a quantum channel preserving the Holevo quantity of
at least one (discrete or continuous) ensemble of states with rank $\leq r$ has
the r-partially entanglement-breaking complementary channel. Several
applications of this result are considered. In particular, it is shown that
coincidence of the constrained Holevo capacity and the quantum mutual
information of a quantum channel at least at one full rank state implies that
this channel is entanglement-breaking.
