We introduce a method to reconstruct the density matrix ρ of a system of n qubits and estimate its rank d from data obtained by quantum-state-tomography measurements repeated m times. The procedure consists of minimizing the risk of a linear estimator ˆρ of ρ penalized by a given rank (from 1 to 2n), where ˆρ is previously obtained by the moment method. We obtain simultaneously an estimator of the rank and the resulting density matrix associated to this rank. We establish an upper bound for the error of the penalized estimator, evaluated with the Frobenius norm, which is of order dn(4/3)n/m and consistent for the estimator of the rank. The proposed methodology is computationally efficient and is illustrated with some example states and real experimental data sets.
ALQUIER, P., BUTUCEA, C., HEBIRI, M., MEZIANI, K. et MORIMAE, T. (2013). Rank-penalized estimation of a quantum system. Physical Review A, 88(3).