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December 2011 Von Neumann entropy penalization and low-rank matrix estimation
Vladimir Koltchinskii
Ann. Statist. 39(6): 2936-2973 (December 2011). DOI: 10.1214/11-AOS926


We study a problem of estimation of a Hermitian nonnegatively definite matrix ρ of unit trace (e.g., a density matrix of a quantum system) based on n i.i.d. measurements (X1, Y1), …, (Xn, Yn), where

Yj = tr(ρXj) + ξj, j = 1, …, n,

{Xj} being random i.i.d. Hermitian matrices and {ξj} being i.i.d. random variables with ${\mathbb{E}}(\xi_{j}|X_{j})=0$. The estimator $$\hat{\rho}^{\varepsilon }:=\mathop{\arg\min}_{S\in{\mathcal{S}}}\Biggl[n^{-1}\sum_{j=1}^{n}\bigl(Y_{j}-\operatorname{tr}(SX_{j})\bigr)^{2}+\varepsilon \operatorname{tr}(S\log S)\Biggr] $$ is considered, where ${\mathcal{S}}$ is the set of all nonnegatively definite Hermitian m × m matrices of trace 1. The goal is to derive oracle inequalities showing how the estimation error depends on the accuracy of approximation of the unknown state ρ by low-rank matrices.


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Vladimir Koltchinskii. "Von Neumann entropy penalization and low-rank matrix estimation." Ann. Statist. 39 (6) 2936 - 2973, December 2011.


Published: December 2011
First available in Project Euclid: 24 January 2012

zbMATH: 1246.62138
MathSciNet: MR3012397
Digital Object Identifier: 10.1214/11-AOS926

Primary: 60B20 , 60G15 , 62H12 , 62J99 , 81Q99

Keywords: Empirical processes , low-rank matrix estimation , matrix regression , noncommutative Bernstein inequality , Pauli basis , quantum state tomography , von Neumann entropy

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 6 • December 2011
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