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 lp-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⩾2p−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.

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