A comparative study of estimation methods in quantum tomography

Acharya, Anirudh; Kypraios, Theodore; Guta, Madalin

Abstract

As quantum tomography is becoming a key component of the quantum engineering toolbox, there is a need for a deeper understanding of the multitude of estimation methods available. Here we investigate and compare several such methods: maximum likelihood, least squares, generalised least squares, positive least squares, thresholded least squares and projected least squares. The common thread of the analysis is that each estimator projects the measurement data onto a parameter space with respect to a specific metric, thus allowing us to study the relationships between different estimators. The asymptotic behaviour of the least squares and the projected least squares estimators is studied in detail for the case of the covariant measurement and a family of states of varying ranks. This gives insight into the rank-dependent risk reduction for the projected estimator, and uncovers an interesting non-monotonic behaviour of the Bures risk. These asymptotic results complement recent non-asymptotic concentration bounds of [36] which point to strong optimality properties, and high computational efficiency of the projected linear estimators. To illustrate the theoretical methods we present results of an extensive simulation study. An app running the different estimators has been made available online.

Citation

Acharya, A., Kypraios, T., & Guta, M. (2019). A comparative study of estimation methods in quantum tomography. Journal of Physics A: Mathematical and Theoretical, 52(23), 1-36. https://doi.org/10.1088/1751-8121/ab1958

• Journal Article Type: Article
• Acceptance Date: Apr 15, 2019
• Online Publication Date: May 7, 2019
• Publication Date: May 7, 2019
• Deposit Date: Apr 11, 2019
• Publicly Available Date: May 8, 2020
• Journal: Journal of Physics A: Mathematical and Theoretical
• Print ISSN: 1751-8113
• Electronic ISSN: 1751-8121
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 52
• Issue: 23
• Article Number: 234001
• Pages: 1-36
• DOI: https://doi.org/10.1088/1751-8121/ab1958
• Keywords: Modelling and Simulation; Statistics and Probability; Mathematical Physics; General Physics and Astronomy; Statistical and Nonlinear Physics
• Public URL: https://nottingham-repository.worktribe.com/output/1777687
• Publisher URL: https://iopscience.iop.org/article/10.1088/1751-8121/ab1958
• Contract Date: Apr 11, 2019