  Compressive sensing is used to perform high-dimensional quantum channel
estimation with classical light. As an example, we perform a numerical
simulation for the case of a three-dimensional classically non-separable state
that is propagated through atmospheric turbulence. Using singular value
thresholding algorithm based compressive sensing, we determine the channel
matrix, which we subsequently use to correct for the atmospheric turbulence
induced distortions. As a measure of the success of the procedure, we calculate
the fidelity and the trace distance of the corrected density matrix with
respect to the input state, and compare the results with those of the density
matrix for the uncorrected state. Furthermore, we quantify the amount of
classical non-separability in the density matrix of the corrected state by
calculating its negativity. The results show that compressive sensing could
contribute in the development and implementation of free-space quantum and
optical communication systems.
