Conditions for multiplicativity of maximal ℓp-norms of channels for fixed integer p

We introduce a condition for memoryless quantum channels which, when satisfied guarantees the multiplicativity of the maximal ${\ell }_p$ -norm with $p$ a fixed integer. By applying the condition to qubit channels, it can be shown that it is not a necessary condition, although some known results for qubits can be recovered. When applied to the Werner-Holevo channel, which is known to violate multiplicativity when $p$ is large relative to the dimension $d$ , the condition suggests that multiplicativity holds when $d⩾2^{p-1}$ . This conjecture is proved explicitly for $p=2,3,4$ . Finally, a new class of channels is considered which generalizes the depolarizing channel to maps which are combinations of the identity channel and a noisy one whose image is an arbitrary density matrix. It is shown that these channels are multiplicative for $p=2$ .