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Transmission of coherent information at the onset of interactions

In this work, we investigate the parameters governing the rate at which a quantum channel arises at the onset of an interaction between two systems, $A$ and $B$. In particular, when system $A$ is pre-entangled with an ancilla, $\tilde{A}$, we quantify the early-time transmission of pre-existing entanglement by calculating the leading order change in coherent information of the complementary channel ($A\rightarrow B'$). We show that, when $A$ and $B$ are initially unentangled and $B$ is pure, there is no change in coherent information to first order, while the leading (second) order change is divergent. However, this divergence may be regulated by embedding the conventional notion of coherent information into what we call the family of $n$-coherent informations, defined using $n$-Rényi entropies. We find that the rate of change of the $n$-coherent information at the onset of the interaction is governed by a quantity, which we call the $n$-exposure, which captures the extent to which the initial coherent information of $A$ with $\tilde{A}$ is exposed to or `seen by' the interaction Hamiltonian between $A$ and $B$. We give examples in qubit systems and in the light-matter interaction.