Is the Holevo Quantity Preserved under Channel Applications?

[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), $\Phi$(ρ_a)) : 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!

 

[Nate810]

First, let us recall the definition of the Holevo quantity. The Holevo quantity of a given ensemble \Phi is defined as\chi(\Phi) = S(\sum_{a \in \Gamma} p_a \Phi(ρ_a)) - \sum_{a \in \Gamma} p_a S(\Phi(ρ_a))where S is the von Neumann entropy. Now, suppose that a channel \mathcal{N} is applied to the ensemble \Phi. By linearity of the channel, we have \Phi'(ε) = { (p(a), \mathcal{N}(\Phi(ρa)) : a\in\Gamma}, where \Phi' refers to the new ensemble after the channel has been applied. For this, the Holevo quantity of the new ensemble \Phi' is then \chi(\Phi') = S(\sum_{a \in \Gamma} p_a \mathcal{N}(\Phi(ρ_a))) - \sum_{a \in \Gamma} p_a S(\mathcal{N}(\Phi(ρ_a)))Since the von Neumann entropy is non-increasing under quantum operations, it follows that \chi(\Phi') ≤ S(\sum_{a \in \Gamma} p_a \Phi(ρ_a)) - \sum_{a \in \Gamma} p_a S(\Phi(ρ_a)) = \chi(\Phi)Therefore, the Holevo quantity does not increase when a channel is applied to an ensemble of states.