  For all p > 1, we demonstrate the existence of quantum channels with
non-multiplicative maximal output p-norms. Equivalently, for all p >1, the
minimum output Renyi entropy of order p of a quantum channel is not additive.
The violations found are large; in all cases, 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 any channel-independent guarantee of maximal p-norm
multiplicativity. We also show that a class of channels previously studied in
the context of approximate encryption lead to counterexamples for all p > 2.
