  Complementing recent progress on the additivity conjecture of quantum
information theory, showing that the minimum output p-Renyi entropies of
channels are not generally additive for p>1, we demonstrate here by a careful
random selection argument that also at p=0, and consequently for sufficiently
small p, there exist counterexamples.
  An explicit construction of two channels from 4 to 3 dimensions is given,
which have non-multiplicative minimum output rank; for this pair of channels,
numerics strongly suggest that the p-Renyi entropy is non-additive for all p <
0.11. We conjecture however that violations of additivity exist for all p<1.
