Minimax estimation of qubit states with Bures risk
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...
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| Format: | Article |
| Language: | English |
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IOP
2018
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| Online Access: | https://eprints.nottingham.ac.uk/52652/ |
| _version_ | 1848798777114099712 |
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| author | Acharya, Anirudh Guţă, Mădălin |
| author_facet | Acharya, Anirudh Guţă, Mădălin |
| author_sort | Acharya, Anirudh |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T20:25:09Z |
| format | Article |
| id | nottingham-52652 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:25:09Z |
| publishDate | 2018 |
| publisher | IOP |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-526522019-04-04T04:30:14Z https://eprints.nottingham.ac.uk/52652/ Minimax estimation of qubit states with Bures risk Acharya, Anirudh Guţă, Mădălin 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. IOP 2018-04-04 Article PeerReviewed application/pdf en https://eprints.nottingham.ac.uk/52652/7/minimax_estimation_qubit_bures.pdf Acharya, Anirudh and Guţă, Mădălin (2018) Minimax estimation of qubit states with Bures risk. Journal of Physics A: Mathematical and Theoretical, 51 (17). 175307/1-175307/22. ISSN 1751-8121 Quantum tomography; State estimation; Minimax estimation; Bures distance; Quantum relative entropy; Local asymptotic normality http://iopscience.iop.org/article/10.1088/1751-8121/aab6f2/meta doi:10.1088/1751-8121/aab6f2 doi:10.1088/1751-8121/aab6f2 |
| spellingShingle | Quantum tomography; State estimation; Minimax estimation; Bures distance; Quantum relative entropy; Local asymptotic normality Acharya, Anirudh Guţă, Mădălin Minimax estimation of qubit states with Bures risk |
| title | Minimax estimation of qubit states with Bures risk |
| title_full | Minimax estimation of qubit states with Bures risk |
| title_fullStr | Minimax estimation of qubit states with Bures risk |
| title_full_unstemmed | Minimax estimation of qubit states with Bures risk |
| title_short | Minimax estimation of qubit states with Bures risk |
| title_sort | minimax estimation of qubit states with bures risk |
| topic | Quantum tomography; State estimation; Minimax estimation; Bures distance; Quantum relative entropy; Local asymptotic normality |
| url | https://eprints.nottingham.ac.uk/52652/ https://eprints.nottingham.ac.uk/52652/ https://eprints.nottingham.ac.uk/52652/ |