  The large number of spatial modes supported by standard multimode fibers is a
promising platform for boosting the channel capacity of quantum and classical
communications by orders of magnitude. However, the practical use of long
multimode fibers is severely hampered by modal crosstalk and polarization
mixing. To overcome these challenges, we develop and experimentally demonstrate
a vectorial time reversal technique, which is accomplished by digitally
pre-shaping the wavefront and polarization of the forward-propagating signal
beam to be the phase conjugate of an auxiliary, backward-propagating probe
beam. Here, we report an average modal fidelity above 80% for 210
Laguerre-Gauss and Hermite-Gauss modes by using vectorial time reversal over an
unstabilized 1-km-long fiber. We also propose a practical and scalable
spatial-mode-multiplexed quantum communication protocol over long multimode
fibers to illustrate potential applications that can be enabled by our
technique.
