  The task of determining whether a given quantum channel has positive capacity
to transmit quantum information is a fundamental open problem in quantum
information theory. In general, the coherent information needs to be computed
for an unbounded number of copies of a channel in order to detect a positive
value of its quantum capacity. However, in this Letter, we show that the
coherent information of a single copy of a randomly selected channel is
positive almost surely if the channel's output space is larger than its
environment. Hence, in this case, a single copy of the channel typically
suffices to determine positivity of its quantum capacity. Put differently,
channels with zero coherent information have measure zero in the subset of
channels for which the output space is larger than the environment. On the
other hand, if the environment is larger than the channel's output space,
identical results hold for the channel's complement.
