  When can noiseless quantum information be sent across noisy quantum devices?
And at what maximum rate? These questions lie at the heart of quantum
technology, but remain unanswered because of non-additivity -- a fundamental
synergy which allows quantum devices (aka quantum channels) to send more
information than expected. Previously, non-additivity was known to occur in
very noisy channels with coherent information much smaller than that of a
perfect channel; but, our work shows non-additivity in a simple low-noise
channel. Our results extend even further. We prove a general theorem concerning
positivity of a channel's coherent information. A corollary of this theorem
gives a simple dimensional test for a channel's capacity. Applying this
corollary solves an open problem by characterizing all qubit channels whose
complement has non-zero capacity. Another application shows a wide class of
zero quantum capacity qubit channels can assist an incomplete erasure channel
in sending quantum information. These results arise from introducing and
linking logarithmic singularities in the von-Neumann entropy with quantum
transmission: changes in entropy caused by this singularity are a mechanism
responsible for both positivity and non-additivity of the coherent information.
Analysis of such singularities may be useful in other physics problems.
