# Quantum channel Holevo information additivity: proof approach

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.