Parallelized projective measurements for spatial photonic qudits estimation

We present a quantum state tomography method that enables the reconstruction of <i>arbitrary</i><i>d</i>-dimensional quantum states encoded in the discretized transverse momentum of photons, by using <i>only</i><i>d</i>+1 experimental settings. To this end, we identify a family of bases with the property that the outcomes of a projective measurement are <i>spatially multiplexed</i> on the interference pattern of the projected state. Using the proposed scheme, we performed, as a proof-of-principle, an experimental reconstruction of <i>d</i>=6-dimensional states, for which a complete set of mutually unbiased bases does not exist. We obtained fidelity values above 0.97 for both pure and mixed states, reducing the number of experimental settings from 42 to only 7.