We significantly extend recently developed methods to faithfully reconstruct
unknown quantum states that are approximately low-rank, using only a few
measurement settings. Our new method is general enough to allow for
measurements from a continuous family, and is also applicable to
continuous-variable states. As a technical result, this work generalizes
quantum compressed sensing to the situation where the measured observables are
taken from a so-called tight frame (rather than an orthonormal basis) --- hence
covering most realistic measurement scenarios. As an application, we discuss
the reconstruction of quantum states of light from homodyne detection and other
types of measurements, and we present simulations that show the advantage of
the proposed compressed sensing technique over present methods. Finally, we
introduce a method to construct a certificate which guarantees the success of
the reconstruction with no assumption on the state, and we show how slightly
more measurements give rise to "universal" state reconstruction that is highly
robust to noise.
