  Maximum likelihood quantum state tomography yields estimators that are
consistent, provided that the likelihood model is correct, but the maximum
likelihood estimators may have bias for any finite data set. The bias of an
estimator is the difference between the expected value of the estimate and the
true value of the parameter being estimated. This paper investigates bias in
the widely used maximum likelihood quantum state tomography. Our goal is to
understand how the amount of bias depends on factors such as the purity of the
true state, the number of measurements performed, and the number of different
bases in which the system is measured. For that, we perform numerical
experiments that simulate optical homodyne tomography under various conditions,
perform tomography, and estimate bias in the purity of the estimated state. We
find that estimates of higher purity states exhibit considerable bias, such
that the estimates have lower purity than the true states.
