I have an interesting idea for a proof approach that someone might find useful. Here it is.
Suppose we are given a quantum qubit channel $N$ (for example the amplitude damping channel) whose Holevo information we are trying to prove is additive for two uses, i.e, we are trying to show that $\chi_{N \otimes N} = 2 \chi_N$.
Given the channel $N$ and another channel $M$ that we know has strongly additive Holevo information, suppose we construct a channel $N^\prime$ that simulates the channel $N$ with probability $(\chi_{N \otimes N} - 2 \chi_N)$, and simulates the channel $M$ with probability $1 - (\chi_{N \otimes N} - 2 \chi_N)$, i.e,
$$N^\prime = (\chi_{N \otimes N} - 2 \chi_N) N + (1 - (\chi_{N \otimes N} - 2 \chi_N))M.$$
Note that for qubit channels $\chi_{N \otimes N} \leq 2$ and $\chi_N \leq 1$, so the assumption that $(\chi_{N \otimes N} - 2 \chi_N) \leq 1$ -- and so can be used as a probability -- is reasonable. From the construction of the channel, we can see that if $N$ is additive, then $(\chi_{N \otimes N} - 2 \chi_N) = 0$, and $N^\prime$ is strongly additive.
That was the idea; to use $(\chi_{N \otimes N} - 2 \chi_N)$ in the construction of a new channel. May be this idea, or a variant of it, can be used to prove something.
I would be interested to know about similar ideas that have been used in proofs before.
