Rank penalized estimation of a quantum system

Abstract

We introduce a new method to reconstruct the density matrix $\rho$ 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 $\hat{\rho }$ of $\rho$ penalized by given rank (from 1 to $2^n$), where $\hat{\rho }$ 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 penalized estimator, evaluated with the Frobenius norm, which is of order $dn(4/3{)}^n/m$ and consistency for the estimator of the rank. The proposed methodology is computationaly efficient and is illustrated with some example states and real experimental data sets.

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Cited by in corpus (11)

• ROP: Matrix recovery via rank-one projections
• Rank-based model selection for multiple ions quantum tomography
• Bayesian methods for low-rank matrix estimation: short survey and theoretical study
• Optimal large-scale quantum state tomography with Pauli measurements
• Pseudo-Bayesian Quantum Tomography with Rank-adaptation
• Spectral thresholding quantum tomography for low rank states
• Statistical analysis of low rank tomography with compressive random measurements
• Estimation of low rank density matrices by Pauli measurements
• An efficient adaptive MCMC algorithm for Pseudo-Bayesian quantum tomography
• Estimation of low rank density matrices: bounds in Schatten norms and other distances
• Quantum Tomography by Regularized Linear Regression