Conditions for the multiplicativity of maximal l_p-norms of channels for   fixed integer p
  We introduce a condition for memoryless quantum channels which, when satisfied guarantees the multiplicativity of the maximal l_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 $d \geqslant 2^{p-1}$. 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. 
Vittorio Giovannetti, Seth Lloyd, and Mary Beth Ruskai