Errors in quantum tomography: diagnosing systematic versus statistical errors

TL;DR

This paper investigates the limitations of the chi-squared goodness-of-fit test in quantum tomography, especially near physical boundaries, and proposes a heuristic correction to improve its diagnostic accuracy.

Contribution

It identifies the failure modes of the chi-squared test in quantum state reconstruction and introduces a simple heuristic method to enhance its reliability.

Findings

Chi-squared test deviates near state space boundaries.

Heuristic correction improves goodness-of-fit assessment.

Method aids in diagnosing technical noise in quantum experiments.

Abstract

A prime goal of quantum tomography is to provide quantitatively rigorous characterisation of quantum systems, be they states, processes or measurements, particularly for the purposes of trouble-shooting and benchmarking experiments in quantum information science. A range of techniques exist to enable the calculation of errors, such as Monte-Carlo simulations, but their quantitative value is arguably fundamentally flawed without an equally rigorous way of authenticating the quality of a reconstruction to ensure it provides a reasonable representation of the data, given the known noise sources. A key motivation for developing such a tool is to enable experimentalists to rigorously diagnose the presence of technical noise in their tomographic data. In this work, I explore the performance of the chi-squared goodness-of-fit test statistic as a measure of reconstruction quality. I show that its behaviour deviates noticeably from expectations for states lying near the boundaries of physical state space, severely undermining its usefulness as a quantitative tool precisely in the region which is of most interest in quantum information processing tasks. I suggest a simple, heuristic approach to compensate for these effects and present numerical simulations showing that this approach provides substantially improved performance.