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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.