  We propose and demonstrate an effective mode-filtering technique of
non-Gaussian states generated by photon-subtraction. More robust non-Gaussian
states have been obtained by removing noisy low frequencies from the original
mode spectrum. We show that non-Gaussian states preserve their non-classicality
after quantum teleportation to a higher degree, when they have been
mode-filtered. This is indicated by a stronger negativity $-0.033 \pm 0.005$ of
the Wigner function at the origin, compared to $-0.018 \pm 0.007$ for states
that have not been mode-filtered. This technique can be straightforwardly
applied to various kinds of photon-subtraction protocols, and can be a key
ingredient in a variety of applications of non-Gaussian states, especially
teleportation-based protocols towards universal quantum information processing.
