  Frequency-bin qudits constitute a promising tool for quantum information
processing, but their high dimensionality can make for tedious characterization
measurements. Here we introduce and compare compressive sensing and Bayesian
mean estimation for recovering the spectral correlations of entangled photon
pairs. Using a conventional compressive sensing algorithm, we reconstruct joint
spectra with up to a 26-fold reduction in measurement time compared to the
equivalent raster scan. Applying a custom Bayesian model to the same data, we
then additionally realize reliable and consistent quantification of
uncertainty. These efficient methods of biphoton characterization should
advance our ability to use the high degree of parallelism and complexity
afforded by frequency-bin encoding.
