  Recently there has been considerable activity on the subject of additivity of
various quantum channel capacities. Here, we construct a family of channels
with sharply bounded classical, hence private capacity. On the other hand,
their quantum capacity when combined with a zero private (and zero quantum)
capacity erasure channel, becomes larger than the previous classical capacity.
  As a consequence, we can conclude for the first time that the classical
private capacity is non-additive. In fact, in our construction even the quantum
capacity of the tensor product of two channels can be greater than the sum of
their individual classical private capacities.
  We show that this violation occurs quite generically: every channel can be
embedded into our construction, and a violation occurs whenever the given
channel has larger entanglement assisted quantum capacity than (unassisted)
classical capacity.
