Maximal qubit tomography

TL;DR

This paper introduces a novel qubit tomography method using a continuous POVM supported by pure states, providing a simple unbiased estimator and analyzing its variance, with comparisons to existing protocols.

Contribution

It presents a new symmetry-invariant qubit tomography protocol and a practical estimator, expanding the toolkit for quantum state estimation.

Findings

The proposed estimator is unbiased.

Variance of the estimator is explicitly calculated.

Comparison with minimal qubit tomography highlights advantages.

Abstract

A new qubit tomography protocol is introduced, based on a continuous positive operator valued measure, which is supported by the set of pure states, and equivariant under the symmetry group SO(3,R) of the qubit state space. Thus the sample data for the tomography protocol is a discrete set of points on the Bloch sphere obtained by a series of independent measurements on identical replicas of the qubit. Although the maximum likelihood estimator uniquely exists, it cannot be explicitly calculated. Instead, we present another convenient, simple estimator, show that it is unbiased and calculate its variance. Furthermore, a detailed account is given about known properties of related discrete qubit state estimation protocols (like ``minimal qubit tomography''), and the results are compared, discussed.