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Complementing recent progress on the additivity conjecture of quantuminformation theory, showing that the minimum output p-Renyi entropies ofchannels are not generally additive for p>1, we demonstrate here by a carefulrandom selection argument that also at p=0, and consequently for sufficientlysmall 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.

Counterexamples to additivity of minimum output p-Renyi entropy for p close to 0