Rank-penalized estimation of a quantum system
Pierre Alquier, Cristina Butucea, Mohamed Hebiri, Katia Méziani, Tomoyuki Morimae · Physical Review A · 2013
We introduce a method to reconstruct the density matrix $\ensuremath{\rho}$ 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 of minimizing the risk of a linear estimator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}$ of $\ensuremath{\rho}$ penalized by a given rank (from 1 to ${2}^{n}$), where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}$ 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.