  In this paper we give several equivalent formulations of the additivity
conjecture for constrained channels, which formally is substantially stronger
than the unconstrained additivity. To this end a characteristic property of the
optimal ensemble for such a channel is derived, generalizing the maximal
distance property. It is shown that the additivity conjecture for constrained
channels holds true for certain nontrivial classes of channels.
  Recently P. Shor showed that conjectured additivity properties for several
quantum information quantities are in fact equivalent. After giving an
algebraic formulation for the Shor's channel extension, its main asymptotic
property is proved. It is then used to show that additivity for two constrained
channels can be reduced to the same problem for unconstrained channels, and
hence, "global" additivity for channels with arbitrary constraints is
equivalent to additivity without constraints.
