  Information exchange between two distant parties, where information is shared
without physically transporting it, is a crucial resource in future quantum
networks. Doing so with high-dimensional states offers the promise of higher
information capacity and improved resilience to noise, but progress to date has
been limited. Here we demonstrate how a nonlinear parametric process allows for
arbitrary high-dimensional state projections in the spatial degree of freedom,
where a strong coherent field enhances the probability of the process. This
allows us to experimentally realise quantum transport of high-dimensional
spatial information facilitated by a quantum channel with a single entangled
pair and a nonlinear spatial mode detector. Using sum frequency generation we
upconvert one of the photons from an entangled pair resulting in
high-dimensional spatial information transported to the other. We realise a
d=15 quantum channel for arbitrary photonic spatial modes which we demonstrate
by faithfully transferring information encoded into orbital angular momentum,
Hermite-Gaussian and arbitrary spatial mode superpositions, without requiring
knowledge of the state to be sent. Our demonstration merges the nascent fields
of nonlinear control of structured light with quantum processes, offering a new
approach to harnessing high-dimensional quantum states, and may be extended to
other degrees of freedom too.
