Counterexamples to the Maximal p-Norm Multiplicativity Conjecture for all p > 1

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 Rényi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Rényi 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.