  We study the task of distilling entanglement by a coherent superposition
operation $t\hat{a}+r\hat{a}^\dagger$ applied to a continuous-variable state
under a thermal noise. In particular, we compare the performances of two
different strategies, i.e., the non-Gaussian operation
$t\hat{a}+r\hat{a}^\dagger$ is applied before or after the noisy Gaussian
channel. This is closely related to a fundamental problem of whether Gaussian
or non-Gaussian entanglement can be more robust under a noisy channel and also
provides a useful insight into the practical implementation of entanglement
distribution for a long-distance quantum communication. We specifically look
into two entanglement characteristics, the logarithmic negativity as a measure
of entanglement and the teleportation fidelity as a usefulness of entanglement,
for each distilled state. We find that the non-Gaussian operation after
(before) the thermal noise becomes more effective in the low (high) temperature
regime.
