Estimation of low rank density matrices: Bounds in Schatten norms and other distances

Abstract

Let $\mathcal{S}_{m}$ be the set of all $m\times m$ density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix $\rho\in\mathcal{S}_{m}$ based on outcomes of $n$ measurements of observables $X_{1},\dots,X_{n}\in\mathbb{H}_{m}$ ($\mathbb{H}_{m}$ being the space of $m\times m$ Hermitian matrices) for a quantum system identically prepared $n$ times in state $\rho.$ Outcomes $Y_{1},\dots,Y_{n}$ of such measurements could be described by a trace regression model in which $\mathbb{E}_{\rho}(Y_{j}|X_{j})={\rm tr}(\rho X_{j}),j=1,\dots,n.$ The design variables $X_{1},\dots,X_{n}$ are often sampled at random from the uniform distribution in an orthonormal basis $\{E_{1},\dots,E_{m^{2}}\}$ of $\mathbb{H}_{m}$ (such as Pauli basis). The goal is to estimate the unknown density matrix $\rho$ based on the data $(X_{1},Y_{1}),\dots,(X_{n},Y_{n}).$ Let \[\hat{Z}:=\frac{m^{2}}{n}\sum_{j=1}^{n}Y_{j}X_{j}\] and let $\check{\rho}$ be the projection of $\hat{Z}$ onto the convex set $\mathcal{S}_{m}$ of density matrices. It is shown that for estimator $\check{\rho}$ the minimax lower bounds in classes of low rank density matrices (established earlier) are attained up logarithmic factors for all Schatten $p$-norm distances, $p\in[1,\infty]$ and for Bures version of quantum Hellinger distance. Moreover, for a slightly modified version of estimator $\check{\rho}$ the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.