  The density matrices are positively semi-definite Hermitian matrices of unit
trace that describe the state of a quantum system. The goal of the paper is to
develop minimax lower bounds on error rates of estimation of low rank density
matrices in trace regression models used in quantum state tomography (in
particular, in the case of Pauli measurements) with explicit dependence of the
bounds on the rank and other complexity parameters. Such bounds are established
for several statistically relevant distances, including quantum versions of
Kullback-Leibler divergence (relative entropy distance) and of Hellinger
distance (so called Bures distance), and Schatten $p$-norm distances. Sharp
upper bounds and oracle inequalities for least squares estimator with von
Neumann entropy penalization are obtained showing that minimax lower bounds are
attained (up to logarithmic factors) for these distances.
