Rank penalized estimation of a quantum system

TL;DR

This paper presents a new efficient method for reconstructing quantum system density matrices and estimating their rank from measurement data, providing theoretical error bounds and practical validation.

Contribution

It introduces a rank-penalized estimator for quantum state tomography that simultaneously estimates the density matrix and its rank, with proven error bounds and real data applications.

Findings

Error bound of order $dn(4/3)^n /m$ for the estimator

Consistent estimation of the density matrix rank

Method demonstrated on experimental quantum data

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.