  In this paper we fill the gap in previous works by proving the formula for
entanglement-assisted capacity of quantum channel with additive constraint
(such as bosonic Gaussian channel). The 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. The new
version contains improved formulation of sufficient conditions under which
suprema in the capacity formulas are attained.
