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A comparative study of estimation methods in quantum tomography

Acharya, Anirudh; Kypraios, Theodore; Guta, Madalin


Anirudh Acharya


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.


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.

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
Keywords Modelling and Simulation; Statistics and Probability; Mathematical Physics; General Physics and Astronomy; Statistical and Nonlinear Physics
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