Consider a $n\times n$ density matrix $\rho$ and decompose it as $$ \rho=\left[\begin{array}{c|c} \rho_A & \rho_B \\ \hline \rho_B^\dagger & \rho_C\end{array}\right], $$ with $\rho_A$, $\rho_b$ and $\rho_c$ of dimension $p\times p$, $p\times q$ and $q\times q$, respectively ($p+q=n$).
My question is the following one: there exists a Completely Positive Trace Preserving (CPTP) map that maps every density matrix $\rho$ into the $p\times p$ upper diagonal block without annealing the off-diagonal terms $\rho_B$, $\rho_B^\dagger$?
Probably the answer is a rather obvious fact but I can't figure it out.
Thanks in advance for your help.
N.B. A CPTP map mapping every density matrix $\rho$ into the $p\times p$ upper diagonal block without the additional constraint on the off-diagonal terms does indeed exist. As a matter of fact, $$ \mathcal{E}(\rho)=\Pi_A \rho\Pi_A + \frac{\Pi_A}{p}\mathrm{Tr}(\Pi_A^\perp\rho), $$ where $\Pi_A$ is the orthogonal projection onto $\mathrm{supp}(\rho_A)$ and $\Pi_A^\perp=I_n-\Pi_A$, does the job.