Statistical analysis of compressive low rank tomography with random measurements
We consider the statistical problem of 'compressive' estimation of low rank states (r«d ) with random basis measurements, where r, d are the rank and dimension of the state respectively. We investigate whether for a fixed sample size N, the estimation error associated with a 'compress...
| Main Authors: | , |
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| Format: | Article |
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IOP Publishing
2017
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| Online Access: | https://eprints.nottingham.ac.uk/41454/ |
| _version_ | 1848796276726956032 |
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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 | We consider the statistical problem of 'compressive' estimation of low rank states (r«d ) with random basis measurements, where r, d are the rank and dimension of the state respectively. We investigate whether for a fixed sample size N, the estimation error associated with a 'compressive' measurement setup is 'close' to that of the setting where a large number of bases are measured. We generalise and extend previous results, and show that the mean square error (MSE) associated with the Frobenius norm attains the optimal rate rd/N with only O(rlogd) random basis measurements for all states. An important tool in the analysis is the concentration of the Fisher information matrix (FIM). We demonstrate that although a concentration of the MSE follows from a concentration of the FIM for most states, the FIM fails to concentrate for states with eigenvalues close to zero.
We analyse this phenomenon in the case of a single qubit and demonstrate a concentration of the MSE about its optimal despite a lack of concentration of the FIM for states close to the boundary of the Bloch sphere. We also consider the estimation error in terms of a different metric–the quantum infidelity. We show that a concentration in the mean infidelity (MINF) does not exist uniformly over all states, highlighting the importance of loss function choice. Specifically, we show that for states that are nearly pure, the MINF scales as 1/√N but the constant converges to zero as the number of settings is increased. This demonstrates a lack of 'compressive' recovery for nearly pure states in this metric. |
| first_indexed | 2025-11-14T19:45:25Z |
| format | Article |
| id | nottingham-41454 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:45:25Z |
| publishDate | 2017 |
| publisher | IOP Publishing |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-414542020-05-04T18:41:00Z https://eprints.nottingham.ac.uk/41454/ Statistical analysis of compressive low rank tomography with random measurements Acharya, Anirudh Guţă, Mădălin We consider the statistical problem of 'compressive' estimation of low rank states (r«d ) with random basis measurements, where r, d are the rank and dimension of the state respectively. We investigate whether for a fixed sample size N, the estimation error associated with a 'compressive' measurement setup is 'close' to that of the setting where a large number of bases are measured. We generalise and extend previous results, and show that the mean square error (MSE) associated with the Frobenius norm attains the optimal rate rd/N with only O(rlogd) random basis measurements for all states. An important tool in the analysis is the concentration of the Fisher information matrix (FIM). We demonstrate that although a concentration of the MSE follows from a concentration of the FIM for most states, the FIM fails to concentrate for states with eigenvalues close to zero. We analyse this phenomenon in the case of a single qubit and demonstrate a concentration of the MSE about its optimal despite a lack of concentration of the FIM for states close to the boundary of the Bloch sphere. We also consider the estimation error in terms of a different metric–the quantum infidelity. We show that a concentration in the mean infidelity (MINF) does not exist uniformly over all states, highlighting the importance of loss function choice. Specifically, we show that for states that are nearly pure, the MINF scales as 1/√N but the constant converges to zero as the number of settings is increased. This demonstrates a lack of 'compressive' recovery for nearly pure states in this metric. IOP Publishing 2017-04-06 Article PeerReviewed Acharya, Anirudh and Guţă, Mădălin (2017) Statistical analysis of compressive low rank tomography with random measurements. Journal of Physics A: Mathematical and Theoretical, 50 (19). p. 195301. ISSN 1751-8121 http://iopscience.iop.org/article/10.1088/1751-8121/aa682e/meta doi:10.1088/1751-8121/aa682e doi:10.1088/1751-8121/aa682e |
| spellingShingle | Acharya, Anirudh Guţă, Mădălin Statistical analysis of compressive low rank tomography with random measurements |
| title | Statistical analysis of compressive low rank tomography with random measurements |
| title_full | Statistical analysis of compressive low rank tomography with random measurements |
| title_fullStr | Statistical analysis of compressive low rank tomography with random measurements |
| title_full_unstemmed | Statistical analysis of compressive low rank tomography with random measurements |
| title_short | Statistical analysis of compressive low rank tomography with random measurements |
| title_sort | statistical analysis of compressive low rank tomography with random measurements |
| url | https://eprints.nottingham.ac.uk/41454/ https://eprints.nottingham.ac.uk/41454/ https://eprints.nottingham.ac.uk/41454/ |