# Maximum entropy of a pure state passed through a quantum channel

What is $\underset{\rho\ \text{pure}}{\text{max}}S(\mathcal{N}^{\otimes N}(\rho))$?

Denote $S$ as the Von Neumann entropy defined $S=-\text{tr}(\rho\ln\rho)$, $\mathcal{N}$ a quantum channel (i.e., a linear, trace preserving, completely positive map) and $\rho$ is a $n$ qubit pure density matrix. I.e., there exists some $2^n$-dimensional vector $\psi$ such that $\rho=\psi^{\dagger}\psi$. Some important properties of such a density matrix: $\rho^2=\rho$ (projector), $\rho^{\dagger}=\rho$ (hermitian), $\text{Tr}(\rho)=1$ (normalization), $\rho\geq 0$ (positive), and $\text{tr}(\rho^2)=1$ (purity).

It might not be possible to give an answer to this for general $\mathcal{N}$. If not, how about the specific case where $\mathcal{N}$ is the so called depolarizing channel?

$\mathcal{N}(\rho)=\lambda\rho+\frac{(1-\lambda)}{2}I$ with $-\frac{1}{3}\leq\lambda\leq 1$

or, equivalently,

$\mathcal{N}(\rho)=(1-p)\rho +\frac{p}{3}X\rho X+\frac{p}{3}Y\rho Y+\frac{p}{3}Z\rho Z$

I haven't been able to solve this problem. I thought possibly that for an even $N$ , $N$ bell pairs might work, but it doesn't seem to be so.

I have proven that $\underset{\rho\ \text{pure}}{\text{max}}S(\mathcal{N}_{BF}^{\otimes N}(\rho))=Nh(p)$ where $\mathcal{N}_{BF}$ is the bitflip channel and $h(p)$ is the entropy of a single qubit passing through a bit flip channel.