VON NEUMANN ENTROPY PENALIZATION AND LOW-RANK MATRIX ESTIMATION
VON NEUMANN ENTROPY PENALIZATION AND LOW-RANK MATRIX ESTIMATION
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DOI：
10.1214/11-aos926
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发表时间：
2011-12-01
影响因子：
4.5
通讯作者：
Koltchinskii, Vladimir
中科院分区：
文献类型：
--
作者：
Koltchinskii, Vladimir
We study a problem of estimation of a Hermitian nonnegatively definite matrix rho of unit trace (e. g., a density matrix of a quantum system) based on n i.i.d. measurements (X-1, Y-1), ... , (X-n, Y-n), whereY-j = tr(rho X-j)+ xi(j), j = 1, ... , n,{X-j} being random i.i.d. Hermitian matrices and {xi(j)} being i.i.d. random variables with E(xi(j) vertical bar X-j) = 0. The estimator(rho) over cap (epsilon) := S is an element of S-argmin [n(-1) Sigma(n)(j=1) (Y-j - tr(SXj))(2) + epsilon tr(S log S)]is considered, where S is the set of all nonnegatively definite Hermitian m x 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 rho by low-rank matrices.
