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In the past few years, physicists and engineers have demonstrated the possibility of utilizing multiple degrees of freedom of the photon to perform information processing tasks for a wide variety of applications. Furthermore, complex states of light offer the possibility of encoding and processing many bits of information in a single photon. However, the challenges involved in the process of extracting large amounts of information, encoded in photonic states, impose practical limitations to realistic quantum technologies. Here, we demonstrate characterization of quantum correlated photon pairs in the spatial and spectral degrees of freedom. Our technique utilizes a series of random projective measurements in the spatial basis that do not perturb the spectral properties of the photon. The sparsity in the spatial properties of downconverted photons allows us to exploit the potential of compressive sensing to reduce the number of measurements to reconstruct spatial and spectral properties of correlated photon pairs at telecom wavelength. We demonstrate characterization of a photonic state with 12 × 109 dimensions using only 20% of the measurements with respect to the conventional raster scan technique. Our characterization technique opens the possibility of increasing and exploiting the complexity and dimensionality of quantum protocols that utilize multiple degrees of freedom of light with high efficiency.

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