What is the correct mathematical notation to describe the following setup? I have classical state in register $A$ which I can think of as $\sum_i p_i \vert i\rangle\langle i \vert_A$. I measure this to obtain outcome $i$ and then perform operation $N_i$ on a different register $A'$.
How does one denote this type of conditional channel? The channel has to have input system $AA'$ and output $B$ (which is the output of $N_i$ for any $i$). I currently say
$$\sum_i N_i \otimes \langle i\vert \cdot \vert i\rangle $$
but this looks awkward.