We present a proof-of-principle demonstration of a method to characterize any pure spatial qudit of arbitrary dimension d, which is based on the classic phase shift interferometry technique. In the proposed scheme a total of only 4d measurement outcomes are needed, implying a significant reduction with respect to the standard schemes for quantum state tomography which require of the order of d 2 . By using this technique, we have experimentally reconstructed a large number of states ranging from d = 2 up to 14 with mean fidelity values higher than 0.97. For that purpose the qudits were codified in the discretized transverse momentum-position of single photons, once they are sent through an aperture with d slits. We provide an experimental implementation of the method based on a Mach-Zehnder interferometer, which allows to reduce the number of measurement settings to 4 since the d slits can be measured simultaneously. Furthermore, it can be adapted to consider the reconstruction of the unknown state from the outcome frequencies of 4d − 3 fixed projectors independently of the encoding or the nature of the quantum system, allowing to implement the reconstruction method in a general experiment.

In this Letter, we propose a simple optical architecture based on phase-only programmable spatial light modulators, in order to characterize general processes on photonic spatial quantum systems in a d>2 Hilbert space. We demonstrate the full reconstruction of typical noises affecting quantum computing, such as amplitude shifts, phase shifts, and depolarizing channels in dimension d=5. We have also reconstructed simulated atmospheric turbulences affecting a free-space transmission of qudits in dimension d=4. In each case, quantum process tomography was performed in order to obtain the matrix χ that fully describes the corresponding quantum channel, E. Fidelities between the states are experimentally obtained after going through the channel, and the expected ones are above 97%.