We introduce a new method to reconstruct the density matrix $\rho$ of a
system of $n$-qubits and estimate its rank $d$ from data obtained by quantum
state tomography measurements repeated $m$ times. The procedure consists in
minimizing the risk of a linear estimator $\hat{\rho}$ of $\rho$ penalized by
given rank (from 1 to $2^n$), where $\hat{\rho}$ is previously obtained by the
moment method. We obtain simultaneously an estimator of the rank and the
resulting density matrix associated to this rank. We establish an upper bound
for the error of penalized estimator, evaluated with the Frobenius norm, which
is of order $dn(4/3)^n /m$ and consistency for the estimator of the rank. The
proposed methodology is computationaly efficient and is illustrated with some
example states and real experimental data sets.