  It is well known that photons can carry a spatial structure akin to a
"twisted" or "rippled" wavefront. Such structured light fields have sparked
significant interest in both classical and quantum physics, with applications
ranging from dense communications to light-matter interaction. Harnessing the
full advantage of transverse spatial photonic encoding using the
Laguerre-Gaussian (LG) basis in the quantum domain requires control over both
the azimuthal (twisted) and radial (rippled) components of photons. However,
precise measurement of the radial photonic degree-of-freedom has proven to be
experimentally challenging primarily due to its transverse amplitude structure.
Here we demonstrate the generation and certification of full-field
Laguerre-Gaussian entanglement between photons pairs generated by spontaneous
parametric down conversion in the telecom regime. By precisely tuning the
optical system parameters for state generation and collection, and adopting
recently developed techniques for precise spatial mode measurement, we are able
to certify fidelities up to 85\% and entanglement dimensionalities up to 26 in
a 43-dimensional radial and azimuthal LG mode space. Furthermore, we study
two-photon quantum correlations between 9 LG mode groups, demonstrating a
correlation structure related to mode group order and inter-modal cross-talk.
In addition, we show how the noise-robustness of high-dimensional entanglement
certification can be significantly increased by using measurements in multiple
LG mutually unbiased bases. Our work demonstrates the potential offered by the
full spatial structure of the two-photon field for enhancing technologies for
quantum information processing and communication.
