A remark on low rank matrix recovery and noncommutative Bernstein type inequalities

Abstract

A problem of estimation of a large Hermitian nonnegatively definite matrix of trace 1 (a density matrix of a quantum system) motivated by quantum state tomography is studied. The estimator is based on a modified least squares method suitable in the case of models with random design with known design distributions. The bounds on Hilbert-Schmidt error of the estimator, including low rank oracle inequalities, have been proved. The proofs rely on Bernstein type inequalities for sums of independent random matrices.