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Conditions for multiplicativity of maximal \(l_ p\)-norms of channels for fixed integer \(p\). (English) Zbl 1067.81011

Summary: 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 \(d \geq 2^{p-1}\). 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 \(p=2\).

MSC:

81P68 Quantum computation
94A40 Channel models (including quantum) in information and communication theory
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