Von Neumann entropy penalization and low-rank matrix estimation
The Annals of Statistics · 2011 · Volume 39 · Issue 6
Abstract
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.
Keywords
60B20, 60G15, 62H12, 62J99, 81Q99, Empirical processes, low-rank matrix estimation, matrix regression, noncommutative Bernstein inequality, Pauli basis, quantum state tomography, von Neumann entropy
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