# Optimal Estimation of Low Rank Density Matrices

arXiv: https://arxiv.org/abs/1507.05131 · HTML: https://mlsanity.com/paper/1507.05131/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jul 2015.

## Built on by (most cited first, 6 of 6)

- [Provable Tensor-Train Format Tensor Completion by Riemannian Optimization](https://mlsanity.com/paper/2108.12163/) (2021)
- [Optimal Differentially Private PCA and Estimation for Spiked Covariance Matrices](https://mlsanity.com/paper/2401.03820/) (2024)
- [Optimal Estimation and Computational Limit of Low-rank Gaussian Mixtures](https://mlsanity.com/paper/2201.09040/) (2022)
- [Lower and Upper Bounds on the VC-Dimension of Tensor Network Models](https://mlsanity.com/paper/2106.11827/) (2021)
- [Near-Optimal differentially private low-rank trace regression with guaranteed private initialization](https://mlsanity.com/paper/2403.15999/) (2024)
- [Quantum Maximum Likelihood Prediction via Hilbert Space Embeddings](https://mlsanity.com/paper/2602.18364/) (2026)

## Abstract (arXiv)

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.