Determination of any pure spatial qudits from a minimum number of measurements by phase stepping interferometry

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 4 d measurement outcomes are needed, implying a significant reduction with respect to the standard schemes for quantum state tomography which require of the order of d². 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 in 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.