  The need to perform quantum state tomography on ever larger systems has
spurred a search for methods that yield good estimates from incomplete data. We
study the performance of compressed sensing (CS) and least squares (LS)
estimators in a fast protocol based on continuous measurement on an ensemble of
cesium atomic spins. Both efficiently reconstruct nearly pure states in the
16-dimensional ground manifold, reaching average fidelities FCS = 0.92 and FLS
= 0.88 using similar amounts of incomplete data. Surprisingly, the main
advantage of CS in our protocol is an increased robustness to experimental
imperfections.
