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We introduce a new method to reconstruct unknown quantum states out ofincomplete and noisy information. The method is a linear convex optimizationproblem, therefore with a unique minimum, which can be efficiently solved withSemidefinite Programs. Numerical simulations indicate that the estimated statedoes not overestimate purity, and neither the expectation value of optimalentanglement witnesses. The convergence properties of the method are similar tocompressed sensing approaches, in the sense that, in order to reconstruct lowrank states, it needs just a fraction of the effort correspondig to aninformationally complete measurement.

Variational quantum tomography with incomplete information by means of semidefinite programs