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I am working on a problem related to average secret key rate calculation in satellite quantum key distribution (QKD). The problem is similar to a paper on arXiv:1712.09722 (P. No. 21).

In a single mode transfer version of entanglement based QKD, we send one of the modes to the satellite via atmospheric fading channel, while the other mode is retained at the ground. We assume that the atmospheric fading channel transmissivity $\eta$ is distributed according to probability distribution $P(\eta)$ with maximum transmissivity $\eta_0$. The resultant mixed state can be written as $$ \rho_t^{\prime} = \int_0^{\eta_0} P(\eta) \rho^{\prime}(\eta) d \eta. $$

The average secret key rate can be calculated as $$ K = \int_0^{\eta_0} P(\eta) K[\rho^{\prime}(\eta) ]d \eta. $$

However, I am unable to follow the above equation. I first tried to see how the mutual information can be evaluated in such a scenario. I came to know that ``the mutual information is neither convex nor concave in the joint distribution or density matrix" Ref: arXiv:1404.6496

However, I don't know how to proceed further in this special case also. Any help will be appreciated.

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