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 \textit{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 $d^2$. 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 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.