  We provide a rigorous treatment of the entanglement properties of two-mode
Gaussian states in atmospheric channels by deriving and analyzing the
input-output relations for the corresponding entanglement test. A key feature
of such turbulent channels is a non-trivial dependence of the transmitted
continuous-variable entanglement on coherent displacements of the quantum state
of the input field. Remarkably, this allows one to optimize the entanglement
certification by modifying local coherent amplitudes using a finite, but
optimal amount of squeezing. In addition, we propose a protocol which, in
principle, renders it possible to transfer the Gaussian entanglement through
any turbulent channel over arbitrary distances. Therefore, our approach
provides the theoretical foundation for advanced applications of Gaussian
entanglement in free-space quantum communication.
