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Let $\mathcal N^{A_1\rightarrow B_1}_1,..,\mathcal N^{A_1\rightarrow B_1}_k$ be a set of valid quantum evolutions with equal input and output dimensions. And let the effect of a channel on a system $\rho_{A_1A_2}$ be:

$$\mathcal N^{A_1A_2\rightarrow B_0B_1}(\rho_{A_1A_2})=\int dB_0|b_0\rangle\langle b_0|\otimes\sum_ip_i\mathcal N^{A_1\rightarrow B_1}_i(\rho_{A_2})$$

where $B_0$ is a random basis which is given as output and $p_i=tr(B_0^i\rho_{A_1})$ is the probability of obtaining outcome $i$ after measuring the system $A_1$ in the $B_0$ basis.

If the maps $\mathcal N^{A_1\rightarrow B_1}_1,..,\mathcal N^{A_1\rightarrow B_1}_k$ are random unitaries, I have seen without proof that the optimal input for the coherent information, is a product state between $A_1$ and $A_2$, does anyone know how to prove it? Is that also true for general channels, i.e. non random unitaries?

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