We investigate a method which allows us to reconstruct low-rank quantum states from few measured expectation values of observables which can be taken from arbitrarily, even continuous, generalized bases. Building on earlier work on matrix completion, we present an algorithm which does not only suceed with very high probability but also allows for a certification of its success. One does not need any assumptions on the state but can check whether the reconstruction was successful based on the available measured data only. The algorithm is fast both in an asymptotic sense and for problem sizes of practical importance. We also discuss the issue of robustness which is vital for any real-world application and show how the performance is affected by noise and decoherence.
Joint work with David Gross, Vincent Nesme, Jens Eisert.