  We investigate the coherence of quantum channels using the
Choi-Jamio\l{}kowski isomorphism. The relation between the coherence and the
purity of the channel respects a duality relation. It characterizes the allowed
values of coherence when the channel has certain purity. This duality has been
depicted via the Coherence-Purity (Co-Pu) diagrams. In particular, we study the
quantum coherence of the unital and non-unital qubit channels and find out the
allowed region of coherence for a fixed purity. We also study coherence of
different incoherent channels, namely, incoherent operation (IO), strictly
incoherent operation (SIO), physical incoherent operation (PIO) etc.
Interestingly, we find that the allowed region for different incoherent
operations maintain the relation $PIO\subset SIO \subset IO$. In fact, we find
that if PIOs are coherence preserving operations (CPO), its coherence is zero
otherwise it has unit coherence and unit purity. Interestingly, different kinds
of qubit channels can be distinguished using the Co-Pu diagram. The unital
channels generally do not create coherence whereas some nonunital can. All
coherence breaking channels are shown to have zero coherence, whereas, this is
not usually true for entanglement breaking channels. It turns out that the
coherence preserving qubit channels have unit coherence. Although the coherence
of the Choi matrix of the incoherent channels might have finite values, its
subsystem contains no coherence. This indicates that the incoherent channels
can either be unital or nonunital under some conditions.
