  Maximum likelihood estimation is one of the most used methods in quantum
state tomography, where the aim is to reconstruct the density matrix of a
physical system from measurement results. One strategy to deal with positivity
and unit trace constraints is to parameterize the matrix to be reconstructed in
order to ensure that it is physical. In this case, the negative log-likelihood
function in terms of the parameters, may have several local minima. In various
papers in the field, a source of errors in this process has been associated to
the possibility that most of these local minima are not global, so that
optimization methods could be trapped in the wrong minimum, leading to a wrong
density matrix. Here we show that, for convex negative log-likelihood
functions, all local minima of the unconstrained parameterized problem are
global, thus any minimizer leads to the maximum likelihood estimation for the
density matrix. We also discuss some practical sources of errors.
