Rank penalized estimation of a quantum system

We introduce a new method to reconstruct the quantum matrix $\bar{\rho}$ 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 in minimizing the risk of a linear estimator $\hat{\bar{\rho}}$ of $\rho$ penalized by given rank (from 1 to $2^n$), where $\hat{\bar{\rho}}$ is previously obtained by the moment method. We obtain simultaneously an estimator of the rank and the resulting state matrix associated to this rank. We establish an upper bound for the error of penalized estimator, evaluated with the Frobenius norm, which is of order $dn(3/4)^n /m$ and consistency for the estimator of the rank. The proposed methodology is computationnaly efficient and is illustrated with synthetic and real data sets.