Given two channels ${A_1,B_1,P_1}$ and ${A_2, B_2, P_2}$ with capacities $c_1, c_2$ respectively, where $A_1,A_2$ are disjoint sets of input symbols, $B_1,B_2$ disjoint sets of output symbols and $P_1, P_2$ are transition probability matrices, what is the capacity $c$ of ${A_1\cup A_2, B_1\cup B_2, P}$ where $P=\begin{pmatrix}P_1&0 \0&P_2\end{pmatrix}$.
I thought I should use the formula $c=\displaystyle\max_{P_X}I(X;Y)$ but I don't get any clue to find/bound the mutual information $I(X;Y)$ interms of $c_1, c_2$.
Some hints would be appreciated.
