  High-dimensional entanglement is a promising resource for quantum
technologies. Being able to certify it for any quantum state is essential.
However, to date, experimental entanglement certification methods are imperfect
and leave some loopholes open. Using a single-photon sensitive time-stamping
camera, we quantify high-dimensional spatial entanglement by collecting all
output modes and without background subtraction, two critical steps on the
route towards assumptions-free entanglement certification. We show
position-momentum Einstein-Podolsky-Rosen (EPR) correlations and quantify the
entanglement of formation of our source to be larger than 2.8 along both
transverse spatial axes, indicating a dimension higher than 14. Our work
overcomes important challenges in photonic entanglement quantification and
paves the way towards the development of practical quantum information
processing protocols based on high-dimensional entanglement.
