# Holevo Quantity with Channel

1. Nov 17, 2011

### rmp251

I'm trying to prove that the Holevo quantity does not increase when a channel is applied to the ensemble of states.

So, if

$\Phi$(ε) = { (p(a), $\Phi$(ρa)) : a$\in$$\Gamma$},

then I want to prove that

$\chi$($\Phi$(ε)) ≤ $\chi$(ε)

where $\chi$ refers to the Holevo quantity. I'm trying an approach similar to the proof for Holevo's theorem, but I can's say I totally understand that proof... but I don't think this should be too difficult. Please help!

Thank you!