Optimal Estimation of Low Rank Density Matrices
This work provides foundational statistical guarantees for quantum state tomography, which is crucial for quantum computing and information processing.
The paper tackles the problem of estimating low-rank density matrices in quantum state tomography, establishing minimax lower bounds on error rates for several distances and showing that these bounds are nearly attained by a penalized least squares estimator.
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, in the case of Pauli measurements) with explicit dependence of the bounds on the rank and other complexity parameters. Such bounds are established for several statistically relevant distances, including quantum versions of Kullback-Leibler divergence (relative entropy distance) and of Hellinger distance (so called Bures distance), and Schatten $p$-norm distances. Sharp upper bounds and oracle inequalities for least squares estimator with von Neumann entropy penalization are obtained showing that minimax lower bounds are attained (up to logarithmic factors) for these distances.