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

Abstract: Let ${\mathcal S}<em>m$ 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 $${\mathbb E}</em>{\rho}(Y_j|X_j)={\rm tr}(\rho X_j), j=1,\dots, n.$$ The design variables $X_1,\dots, X_n$ are often sampled at random from the uniform distribution in an orthonormal basis ${E_1,\dots, E_{m<sup>2}}$ of ${\mathbb H}<em>m$ (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 $$\hat Z:=\frac{m^2}{n}\sum</em>{j=1}ⁿ Y_j X_j$$ 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.