  A necessary condition for reversibility (sufficiency) of a quantum channel
with respect to complete families of states with bounded rank is obtained. A
full description (up to isometrical equivalence) of all quantum channels
reversible with respect to orthogonal and nonorthogonal complete families of
pure states is given. Some applications in quantum information theory are
considered.
  The main results can be formulated in terms of the operator algebras theory
(as conditions for reversibility of channels between algebras of all bounded
operators).
