Minimax mean estimator for the trine

TL;DR

This paper investigates a quantum state estimator for a qubit in the x-z plane using the trine measurement, comparing a minimax mean estimator with simpler methods, and finds the simpler estimator often performs better.

Contribution

It introduces a minimax mean estimator for quantum state estimation and compares its performance with existing estimators, highlighting limitations of classical intuition in quantum problems.

Findings

Minimax mean estimator outperforms maximum-likelihood in worst-case error

Simpler adapted estimator from previous work remains more effective

Classical intuition may not always translate well to quantum estimation

Abstract

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.