  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.
