Conditions for the multiplicativity of maximal l_p-norms of channels for fixed integer p

Abstract

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 $d⩾2^{p-1}$. 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.

20 pages: new material added including additional details supporting the conjecture that the Werner-Holevo channel is multiplicative for all p when d > 2^{p-1}; typos corrected. To appear on JMP

References in corpus (5)

• Multiplicativity of maximal output purities of Gaussian channels under Gaussian inputs
• Additivity properties of a Gaussian Channel
• Purity and State Fidelity of Quantum Channels via Hamiltonians
• Minimum Renyi and Wehrl entropies at the output of bosonic channels
• Classical phase-space descriptions of continuous-variable teleportation

Cited by in corpus (9)

• Quantum channels and their entropic characteristics
• Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1
• Classical information capacity of a class of quantum channels
• Nonadditivity effects in classical capacities of quantum multiple-access channels
• Maximal output purity and capacity for asymmetric unital qudit channels
• Extending additivity from symmetric to asymmetric channels
• Quantum Channels and Representation Theory
• Capacities of a two-parameter family of noisy Werner-Holevo channels
• An application of decomposable maps in proving multiplicativity of low dimensional maps