  Precise reconstruction of unknown quantum states from measurement data, a
process commonly called quantum state tomography, is a crucial component in the
development of quantum information processing technologies. Many different
tomography methods have been proposed over the years. Maximum likelihood
estimation is a prominent example, being the most popular method for a long
period of time. Recently, more advanced neural network methods have started to
emerge. Here, we go back to basics and present a method for continuous variable
state reconstruction that is both conceptually and practically simple, based on
convex optimization. Convex optimization has been used for process tomography
and qubit state tomography, but seems to have been overlooked for continuous
variable quantum state tomography. We demonstrate high-fidelity reconstruction
of an underlying state from data corrupted by thermal noise and imperfect
detection, for both homodyne and heterodyne measurements. A major advantage
over other methods is that convex optimization algorithms are guaranteed to
converge to the optimal solution.
