Data processing for qubit state tomography: An information geometric approach

TL;DR

This paper introduces an information geometric approach to qubit state tomography, modeling the state space as a Riemannian manifold and proposing an efficient maximum likelihood data processing algorithm.

Contribution

It presents a novel geometric framework for qubit tomography and develops an efficient maximum likelihood estimation algorithm based on this approach.

Findings

The state space is modeled as a Riemannian manifold with a specific metric.

Maximum likelihood estimation corresponds to orthogonal projection in this geometric space.

An efficient algorithm for maximum likelihood estimation is proposed.

Abstract

A statistically feasible data post-processing method for the conventional qubit state tomography is studied from an information geometrical point of view. It is shown that the space $(-1,1)^3$ of the Stokes parameters $(\xi_1, \xi_2,\xi_3)$ that specify qubit states should be regarded as a Riemannian manifold endowed with a metric $g_{ij}:=\delta_{ij}/(1-(\xi_i)^2)$, and that the data processing based on the maximum likelihood method is realized by the orthogonal projection from the empirical distribution onto the Bloch sphere with respect to the metric $g_{ij}$. An efficient algorithm for computing the maximum likelihood estimate is also proposed.