  The estimation of high dimensional quantum states is an important statistical
problem arising in current quantum technology applications. A key example is
the tomography of multiple ions states, employed in the validation of state
preparation in ion trap experiments \cite{Haffner2005}. Since full tomography
becomes unfeasible even for a small number of ions, there is a need to
investigate lower dimensional statistical models which capture prior
information about the state, and to devise estimation methods tailored to such
models. In this paper we propose several new methods aimed at the efficient
estimation of low rank states in multiple ions tomography. All methods consist
in first computing the least squares estimator, followed by its truncation to
an appropriately chosen smaller rank. The latter is done by setting eigenvalues
below a certain "noise level" to zero, while keeping the rest unchanged, or
normalising them appropriately. We show that (up to logarithmic factors in the
space dimension) the mean square error of the resulting estimators scales as
$r\cdot d/N$ where $r$ is the rank, $d=2^k$ is the dimension of the Hilbert
space, and $N$ is the number of quantum samples. Furthermore we establish a
lower bound for the asymptotic minimax risk which shows that the above scaling
is optimal. The performance of the estimators is analysed in an extensive
simulations study, with emphasis on the dependence on the state rank, and the
number of measurement repetitions. We find that all estimators perform
significantly better that the least squares, with the "physical estimator"
(which is a bona fide density matrix) slightly outperforming the other
estimators.
