  A system-ancilla bipartite state capable of containing the complete
information of an unknown quantum channel acting on the system is called
faithful. The equivalence between faithfulness of state and invertibility of
the corresponding Jamiolkowski map proved by D'Ariano and Presti has been a
useful characterization for ancilla-assisted process tomography albeit the
proof was incomplete as they assumed trace nonincreasing quantum operations,
not quantum channels. We complete the proof of the equivalence and introduce
the generalization of faithfulness to various classes of quantum channels. We
also explore a more general notion we call sensitivity, the property of quantum
state being altered by any nontrivial action of quantum channel. We study their
relationship by characterizing both properties for important classes of quantum
channels such as unital channels, random unitary operations and unitary
operations. Unexpected (non-)equivalence results among them shed light on the
structure of quantum channels by showing that we need only two classes of
quantum states for characterizing quantum states faithful or sensitive to
various subclasses of quantum channels. For example, it reveals the relation
between quantum process tomography and quantum correlation as it turns out that
only bipartite states that has no local classical observable at all can be used
to sense the effect of unital channels.
