@ACuriousMind From time to time it becomes easier to consider in the way I mentioned.
For each block $\rho_n$ I know that the entropy is invariant under unitary transformation $U_n$ i.e. $S(\rho_n) = S(U_n \rho_n U_n^{\dagger})$. Further if we define $U_n \otimes I$ then $S(U_n \rho_n U_n^{\dagger} \otimes |n \rangle \langle n|) = S(\rho_n) + S(|n \rangle \langle n |) = S(\rho_n)$.
But, do you see though how for $V = \sum_n p_n \rho_n \otimes |n \rangle \langle n |$ we have that $S(V) = S(
\sum_n p_n U_n\rho_n U_n^{\dagger} \otimes |n \rangle \langle n |)$. Since $S$ is not a linear operat…