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Minimax estimation of qubit states with Bures risk

Acharya, Anirudh; Guţă, Mădălin

Authors

Anirudh Acharya



Abstract

The central problem of quantum statistics is to devise measurement schemes for the estimation of an unknown state, given an ensemble of n independent identically prepared systems. For locally quadratic loss functions, the risk of standard procedures has the usual scaling of 1/n. However, it has been noticed that for fidelity based metrics such as the Bures distance, the risk of conventional (non-adaptive) qubit tomography schemes scales as 1/√n for states close to the boundary of the Bloch sphere. Several proposed estimators appear to improve this scaling, and our goal is to analyse the problem from the perspective of the maximum risk over all states.

We propose qubit estimation strategies based on separate adaptive measurements, and collective measurements, that achieve 1/n scalings for the maximum Bures risk. The estimator involving local measurements uses a fixed fraction of the available resource n to estimate the Bloch vector direction; the length of the Bloch vector is then estimated from the remaining copies by measuring in the estimator eigenbasis. The estimator based on collective measurements uses local asymptotic normality techniques which allows us to derive upper and lower bounds to its maximum Bures risk. We also discuss how to construct a minimax optimal estimator in this setup. Finally, we consider quantum relative entropy and show that the risk of the estimator based on collective measurements achieves a rate O(n-1 log n) under this loss function. Furthermore, we show that no estimator can achieve faster rates, in particular the 'standard' rate n −1.

Citation

Acharya, A., & Guţă, M. (2018). Minimax estimation of qubit states with Bures risk. Journal of Physics A: Mathematical and Theoretical, 51(17), 1-27. https://doi.org/10.1088/1751-8121/aab6f2

Journal Article Type Article
Acceptance Date Mar 15, 2018
Online Publication Date Apr 4, 2018
Publication Date Apr 4, 2018
Deposit Date Jun 28, 2018
Publicly Available Date Apr 5, 2019
Journal Journal of Physics A: Mathematical and Theoretical
Print ISSN 1751-8113
Electronic ISSN 1751-8121
Publisher IOP Publishing
Peer Reviewed Peer Reviewed
Volume 51
Issue 17
Article Number 175307
Pages 1-27
DOI https://doi.org/10.1088/1751-8121/aab6f2
Keywords Quantum tomography; State estimation; Minimax estimation;Bures distance; Quantum relative entropy; Local asymptotic normality
Public URL http://eprints.nottingham.ac.uk/id/eprint/52652
Publisher URL http://iopscience.iop.org/article/10.1088/1751-8121/aab6f2/meta
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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