  A problem of statistical estimation of a Hermitian nonnegatively definite
matrix of unit trace (for instance, a density matrix in quantum state
tomography) is studied. The approach is based on penalized least squares method
with a complexity penalty defined in terms of von Neumann entropy. A number of
oracle inequalities have been proved showing how the error of the estimator
depends on the rank and other characteristics of the oracles. The methods of
proofs are based on empirical processes theory and probabilistic inequalities
for random matrices, in particular, noncommutative versions of Bernstein
inequality.
