Remarks on additivity of the Holevo channel capacity and of the entanglement of formation

The purpose of these notes is to discuss the relation between the additivity questions regarding the quantities (Holevo) capacity of a quantum channel T and entanglement of formation of a bipartite state ρ. In particular, using the Stinespring dilation theorem, we give a formula for the channel capacity involving entanglement of formation. This can be used to show that additivity of the latter for some states can be inferred from the additivity of capacity for certain channels. We demonstrate this connection for some families of channels, allowing us to calculate the entanglement cost for many states, including some where a strictly smaller upper bound on the distillable entanglement is known. Group symmetry is used for more sophisticated analysis, giving formulas valid for a class of channels. This is presented in a general framework, extending recent findings of Vidal, Dür and Cirac. In an appendix we speculate on a general relation of superadditivity of the entanglement of formation, which would imply both the general additivity of this function under tensor products and of the Holevo capacity (with or without linear cost constraints).