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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?

Thank you for your input.

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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.

The Holevo information is defined for a probabilistic mixture of density matrices, or for a cq-state (cq = classical quantum state).

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