  We extend the definition of the conditional min-entropy from bipartite
quantum states to bipartite quantum channels. We show that many of the
properties of the conditional min-entropy carry over to the extended version,
including an operational interpretation as a guessing probability when one of
the subsystems is classical. We then show that the extended conditional
min-entropy can be used to fully characterize when two bipartite quantum
channels are related to each other via a superchannel (also known as supermap
or a comb) that is acting on one of the subsystems. This relation is a
pre-order that extends the definition of "quantum majorization" from bipartite
states to bipartite channels, and can also be characterized with semidefinite
programming. As a special case, our characterization provides necessary and
sufficient conditions for when a set of quantum channels is related to another
set of channels via a single superchannel. We discuss the applications of our
results to channel discrimination, and to resource theories of quantum
processes. Along the way we study channel divergences, entropy functions of
quantum channels, and noise models of superchannels, including random unitary
superchannels, and doubly-stochastic superchannels. For the latter we give a
physical meaning as being completely-uniformity preserving.
