  We explore the question of state estimation for a qubit restricted to the
$x$-$z$ plane of the Bloch sphere, with the trine measurement. In our earlier
work [H. K. Ng and B.-G. Englert, eprint arXiv:1202.5136[quant-ph] (2012)],
similarities between quantum tomography and the tomography of a classical die
motivated us to apply a simple modification of the classical estimator for use
in the quantum problem. This worked very well. In this article, we adapt a
different aspect of the classical estimator to the quantum problem. In
particular, we investigate the mean estimator, where the mean is taken with a
weight function identical to that in the classical estimator but now with
quantum constraints imposed. Among such mean estimators, we choose an optimal
one with the smallest worst-case error-the minimax mean estimator-and compare
its performance with that of other estimators. Despite the natural
generalization of the classical approach, this minimax mean estimator does not
work as well as one might expect from the analogous performance in the
classical problem. While it outperforms the often-used maximum-likelihood
estimator in having a smaller worst-case error, the advantage is not
significant enough to justify the more complicated procedure required to
construct it. The much simpler adapted estimator introduced in our earlier work
is still more effective. Our previous work emphasized the similarities between
classical and quantum state estimation; in contrast, this paper highlights how
intuition gained from classical problems can sometimes fail in the quantum
arena.
