  We study the continuous-variable (CV) quantum teleportation protocol in the
case that one of the two modes of the shared entangled resource is sent to the
receiver through a Gaussian Quantum Brownian Motion noisy channel. We show that
if the channel is engineered in a non-Markovian regime, the information
backflow from the environment induces an extra dependance of the phase of the
two-mode squeezing of the shared Gaussian entangled resource on the transit
time along the channel of the shared mode sent to the receiver. Optimizing over
the non-Markovianity dependent phase of the squeezing yields a significant
enhancement of the teleportation fidelity. For short enough channel transit
times, essentially unit fidelity is achieved at realistic, finite values of the
squeezing amplitude for a sufficiently large degree of the channel
non-Markovianity.
