Sending quantum information over a distance is a challenging task and the challenge increases, if the task is to be performed several times, to send a message over a large distance. In general, this can only be done with a certain probability. On several quantum networks, a recently introduced procedure, which uses special quantum measurements, enhances this probability, as follows from the properties of related classical percolation models. ALL of the above terms will be explained in the talk, which is aimed at a general mathematical audience. While mathematically rigorous, the presented results were obtained in collaboration with a physics group and are of current physical interest, leading to a number of open problems, which will be mentioned.

In this talk, I will present recent results on wave localization
in nonlinear random media in the frame work of the stochastic
Gross-Pitaevskii equation (describing Bose-Einstein condensation). In particular, it is shown numerically that the disorder average spatial extension of the stationary density profile decreases with an increasing strength of the disordered potential both for repulsive and attractive interactions.