Compressive Direct Measurement of the Quantum Wave Function

The direct measurement of a complex wavefunction has been recently realized by using weak-values. In this paper, we introduce a method that exploits sparsity for compressive measurement of the transverse spatial wavefunction of photons. The procedure involves a weak measurement in random projection operators in the spatial domain followed by a post-selection in the momentum basis. Using this method, we experimentally measure a 192-dimensional state with a fidelity of $90\mathrm{\%}$ using only $25$ percent of the total required measurements. Furthermore, we demonstrate measurement of a 19200 dimensional state; a task that would require an unfeasibly large acquiring time with the conventional direct measurement technique.