Maximization of capacity and p-norms for some product channels

It is conjectured that the Holevo capacity of a product channel Ω\otimes Φis achieved when product states are used as input. Amosov, Holevo and Werner have also conjectured that the maximal p-norm of a product channel is achieved with product input states. In this paper we establish both of these conjectures in the case that Ωis arbitrary and Φis a CQ or QC channel (as defined by Holevo). We also establish the Amosov, Holevo and Werner conjecture when Ωis arbitrary and either Φis a qubit channel and p=2, or Φis a unital qubit channel and p is integer. Our proofs involve a new conjecture for the norm of an output state of the half-noisy channel I \otimes Φ, when Φis a qubit channel. We show that this conjecture in some cases also implies additivity of the Holevo capacity.

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