  High-dimensional encoding of quantum information provides a promising method
of transcending current limitations in quantum communication. One of the
central challenges in the pursuit of such an approach is the certification of
high-dimensional entanglement. In particular, it is desirable to do so without
resorting to inefficient full state tomography. Here, we show how carefully
constructed measurements in two bases (one of which is not orthonormal) can be
used to faithfully and efficiently certify bipartite high-dimensional states
and their entanglement for any physical platform. To showcase the practicality
of this approach under realistic conditions, we put it to the test for photons
entangled in their orbital angular momentum. In our experimental setup, we are
able to verify 9-dimensional entanglement for a pair of photons on a
11-dimensional subspace each, at present the highest amount certified without
any assumptions on the state.
