  For all 1 < p < 2, we demonstrate the existence of quantum channels with
non-multiplicative maximal p-norms. Equivalently, the minimum output Renyi
entropy of order p of a quantum channel is not additive for all 1 < p < 2. The
violations found are large. As p approaches 1, the minimum output Renyi entropy
of order p for a product channel need not be significantly greater than the
minimum output entropy of its individual factors. Since p=1 corresponds to the
von Neumann entropy, these counterexamples demonstrate that if the additivity
conjecture of quantum information theory is true, it cannot be proved as a
consequence of maximal p-norm multiplicativity.
