A criterion for coincidence of the entanglement-assisted classical   capacity and the Holevo capacity of a quantum channel
  It is easy to show coincidence of the entanglement-assisted classical capacity and the Holevo capacity for any c-q channel between finite dimensional quantum systems. In this paper we prove the converse assertion: coincidence of the above-mentioned capacities of a quantum channel implies that the $\chi$-essential part of this channel is a c-q channel (the $\chi$-essential part is a restriction of a channel obtained by discarding all input states useless for transmission of classical information).   The above observations are generalized to infinite dimensional quantum channels with linear constraints by using the obtained conditions for coincidence of the quantum mutual information and the constrained Holevo capacity. 
M.E. Shirokov
quant-ph