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

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 $X1,\dots, Xn\in {\mathbb H}m$ (${\mathbb H}m$ being the space of $m\times m$ Hermitian matrices) for a quantum system identically prepared $n$ times in state $\rho.$ Outcomes $Y1,\dots, Yn$ of such measurements could be described by a trace regression model in which ${\mathbb E}{\rho}(Yj|Xj)={\rm tr}(\rho Xj), j=1,\dots, n.$ The design variables $X1,\dots, Xn$ are often sampled at random from the uniform distribution in an orthonormal basis ${E1,\dots, E{m^2}}$ of ${\mathbb H}m$ (such as Pauli basis). The goal is to estimate the unknown density matrix $\rho$ based on the data $(X1,Y1), \dots, (Xn,Yn).$ Let $$ \hat Z:=\frac{m^2}{n}\sum{j=1}ⁿ Yj Xj $$ 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.