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The notion of the Holevo capacity for arbitrarily constrained infinitedimensional quantum channels is introduced. It is shown that despitenonexistence of an optimal ensemble in this case it is possible to define thenotion of the output optimal average state for such a channel. Thecharacterization of the output optimal average state and a "minimax" expressionfor the Holevo capacity are obtained. This makes it possible to proveequivalence of several additivity properties for infinite dimensional quantumchannels.  The notion of the $\chi$-function for an infinite dimensional channel isconsidered, its strong concavity and lower semicontinuity are shown.  The problem of continuity of the Holevo capacity is also discussed. It isshown that the Holevo capacity is continuous function of a channel in thefinite dimensional case while in general it is only lower semicontinuous. Thisconclusion is confirmed by the example.  The main result of this note is the statement that additivity of the Holevocapacity for all finite dimensional channels implies additivity of the Holevocapacity for all infinite dimensional channels with arbitrary constraints. Thesubadditivity of the $\chi$-function for two infinite dimensional channels withone of them noiseless or entanglement breaking is also proved.Comment: 32 page

The Holevo Capacity of Infinite Dimensional Channels and the Additivity Problem