  High-dimensional entangled states are of significant interest in quantum
science as they increase the information content per photon and can remain
entangled in the presence of significant noise. We develop the analytical
theory and show experimentally that the noise tolerance of high-dimensional
entanglement can be significantly increased by modest increases to the size of
the Hilbert space. For example, doubling the size of a Hilbert space with local
dimension d=300 leads to a reduction of the threshold detector efficiencies
required for entanglement certification by two orders of magnitude. This work
is developed in the context of spatial entanglement, but it can easily be
translated to photonic states entangled in different degrees of freedom. We
also demonstrate that knowledge of a single parameter, the signal-to-noise
ratio, precisely links measures of entanglement to a range of experimental
parameters quantifying noise in a quantum communication system, enabling
accurate predictions of its performance. This work serves to answer a simple
question: "Is high-dimensional photonic entaglement robust to noise?". Here we
show that the answer is more nuanced than a simple "yes" or "no" and involves a
complex interplay between the noise characteristics of the state, channel, and
detection system
