  The ability to completely characterize the state of a quantum system is an
essential element for the emerging quantum technologies. Here, we present a
compressed-sensing inspired method to ascertain any rank-deficient qudit state,
which we experimentally encode in photonic orbital angular momentum. We
efficiently reconstruct these qudit states from a few scans with an intensified
CCD camera. Since it requires only a few intensity measurements, our technique
would provide an easy and accurate way to identify quantum sources, channels,
and systems.
