  Maximum likelihood estimation (MLE) is the most common approach to quantum
state tomography. In this letter, we investigate whether it is also optimal in
any sense. We show that MLE is an inadmissible estimator for most of the
commonly used metrics of accuracy, i.e., some other estimator is more accurate
for every true state. MLE is inadmissible for fidelity, mean squared error
(squared Hilbert-Schmidt distance), and relative entropy. We prove that almost
any estimator that can report both pure states and mixed states is
inadmissible. This includes MLE, compressed sensing (nuclear-norm regularized)
estimators, and constrained least squares. We provide simple examples to
illustrate why reporting pure states is suboptimal even when the true state is
itself pure, and why "hedging" away from pure states generically improves
performance.
