The physics literature recently reported unusual convergence rates, namely N to the minus one quarter, and N to the minus three quarters, for (essentially) the mean squared error of two standard estimators of the state of a two level quantum system. This problem is the quantum analogue of the classical statistical problem of estimating the parameter p, given N outcomes of Bernoulli(p) trials. The better of the two estimators was a maximum likelihood estimator. I'll explain these rates and relate them to the quantum statistical geometry of the unit ball - the parameter space for this problem, replacing the unit interval of a classical coin toss. It seems that a 1/N rate is achievable using an adaptive estimation scheme. The work is joint with Manuel Ballester.