2019/01/23 by Anirudh Acharya, Theodore Kypraios, Mădălin Guţă +1
Computer Science · Mathematics · Physics and Astronomy · #Covariant transformation #Estimation theory #Estimator #Focus (optics) #Key (lock) #Least-squares function approximation #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum tomography #math-ph #math.MP #math.ST #quant-ph #stat.TH #stochastic dynamics and bifurcation
paper · pdf · doi:10.1088/1751-8121/ab1958
40 pages, 11 figures
arxiv created 2019/01/23 · openalex created_date 2019/02/21 · openalex publication_date 2019/04/15 · arxiv updated 2019/05/22 · openalex updated_date 2026/08/06
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 Guta et al (2018 (arXiv:1809.11162)) 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.