Entanglement-assisted capacities of constrained quantum channels

In this paper we fill a gap in previous works by proving the conjectured formula for the classical entanglement-assisted capacity of a quantum channel with additive constraint (such as the Bosonic Gaussian channel). Our main tools are the coding theorem for classical-quantum constrained channels and a finite-dimensional approximation of the input density operators for entanglement-assisted capacity. We also give sufficient conditions under which suprema in the capacity formulas are achieved.