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

Coherent information of a quantum channel or its complement is generically positive