Holevo Information and Quantum Mutual Information

This question is about the difference between Quantum Mutual Information and Holevo Information of quantum channels. From http://arxiv.org/pdf/1004.2495.pdf equation 7 we know that the sum of quantum mutual informtions of a channel and that of its complementary channel is equal to two times the von Neumannn entropy of the input state. Can we say the same about the Holevo informations of a channel and its complementary channel?:

$$\chi(\rho,\Phi)+\chi(\rho,\bar{\Phi})=? 2H(\rho)$$

So from what I understand the Holevo information is equal to the mutual information when mutual information is maximized over the channel's input:

$$\chi(\rho,\Phi)=\max_{\rho}I(\rho,\Phi)$$

If this is true what is the resulting corrlation between the Holevo informations of the channel and that of the complementary channel?

Your definition of Holevo information is wrong. It corresponds to $C_{ea}$, the entanglement assisted capacity of the channel. See equation (5) of the paper.