The Annals of Statistics

Von Neumann entropy penalization and low-rank matrix estimation

Vladimir Koltchinskii

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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.

Article information

Ann. Statist., Volume 39, Number 6 (2011), 2936-2973.

First available in Project Euclid: 24 January 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J99: None of the above, but in this section 62H12: Estimation 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60G15: Gaussian processes 81Q99: None of the above, but in this section

Low-rank matrix estimation von Neumann entropy matrix regression empirical processes noncommutative Bernstein inequality quantum state tomography Pauli basis


Koltchinskii, Vladimir. Von Neumann entropy penalization and low-rank matrix estimation. Ann. Statist. 39 (2011), no. 6, 2936--2973. doi:10.1214/11-AOS926.

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