  High-dimensional entanglement promises to increase the information capacity
of photons and is now routinely generated exploiting spatio-temporal degrees of
freedom of single photons. A curious feature of these systems is the
possibility to certify entanglement despite strong noise in the data. We show
that it is also possible to exploit this noisy high-dimensional entanglement
for quantum key distribution by introducing a protocol that uses mutliple
subspaces of the high-dimensional system simultaneously. Our protocol can be
used to establish a secret key even in extremely noisy experimental conditions,
where qubit protocols fail. To show that, we analyze the performance of our
protocol for noise models that apply to the two most commonly used sources of
high-dimensional entanglement: time bins and spatial modes.
