Maximization of capacity and lp norms for some product channels

It is conjectured that the Holevo capacity of a product channel Ω⊗Φ is achieved when product states are used as input. Amosov, Holevo, and Werner have also conjectured that the maximal lp norm of a product channel is achieved with product input states. In this article we establish both of these conjectures in the case that Ω is arbitrary and Φ is a CQ or QC channel (as defined by Holevo). We also establish the Amosov, Holevo and Werner conjecture when Ω is arbitrary and either Φ is a qubit channel and p=2, or Φ is 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⊗Φ, when Φ is a qubit channel. We show that this conjecture in some cases also implies additivity of the Holevo capacity.