Statistically efficient tomography of low rank states with incomplete measurements
The construction of physically relevant low dimensional state models, and the design of appropriate measurements are key issues in tackling quantum state tomography for large dimensional systems. We consider the statistical problem of estimating low rank states in the set-up of multiple ions tomogra...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Published: |
IOP Publishing
2016
|
| Online Access: | https://eprints.nottingham.ac.uk/46912/ |
| _version_ | 1848797425853005824 |
|---|---|
| author | Acharya, Anirudh Kypraios, Theodore Guţă, Mădălin |
| author_facet | Acharya, Anirudh Kypraios, Theodore Guţă, Mădălin |
| author_sort | Acharya, Anirudh |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | The construction of physically relevant low dimensional state models, and the design of appropriate measurements are key issues in tackling quantum state tomography for large dimensional systems. We consider the statistical problem of estimating low rank states in the set-up of multiple ions tomography, and investigate how the estimation error behaves with a reduction in the number of measurement settings, compared with the standard ion tomography setup. We present extensive simulation results showing that the error is robust with respect to the choice of states of a given rank, the random selection of settings, and that the number of settings can be significantly reduced with only a negligible increase in error. We present an argument to explain these findings based on a concentration inequality for the Fisher information matrix. In the more general setup of random basis measurements we use this argument to show that for certain rank r states it suffices to measure in $O(r\mathrm{log}d)$ bases to achieve the average Fisher information over all bases. We present numerical evidence for random states of up to eight atoms, which suggests that a similar behaviour holds in the case of Pauli bases measurements, for randomly chosen states. The relation to similar problems in compressed sensing is also discussed. |
| first_indexed | 2025-11-14T20:03:41Z |
| format | Article |
| id | nottingham-46912 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T20:03:41Z |
| publishDate | 2016 |
| publisher | IOP Publishing |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-469122020-05-04T17:46:47Z https://eprints.nottingham.ac.uk/46912/ Statistically efficient tomography of low rank states with incomplete measurements Acharya, Anirudh Kypraios, Theodore Guţă, Mădălin The construction of physically relevant low dimensional state models, and the design of appropriate measurements are key issues in tackling quantum state tomography for large dimensional systems. We consider the statistical problem of estimating low rank states in the set-up of multiple ions tomography, and investigate how the estimation error behaves with a reduction in the number of measurement settings, compared with the standard ion tomography setup. We present extensive simulation results showing that the error is robust with respect to the choice of states of a given rank, the random selection of settings, and that the number of settings can be significantly reduced with only a negligible increase in error. We present an argument to explain these findings based on a concentration inequality for the Fisher information matrix. In the more general setup of random basis measurements we use this argument to show that for certain rank r states it suffices to measure in $O(r\mathrm{log}d)$ bases to achieve the average Fisher information over all bases. We present numerical evidence for random states of up to eight atoms, which suggests that a similar behaviour holds in the case of Pauli bases measurements, for randomly chosen states. The relation to similar problems in compressed sensing is also discussed. IOP Publishing 2016-04-13 Article PeerReviewed Acharya, Anirudh, Kypraios, Theodore and Guţă, Mădălin (2016) Statistically efficient tomography of low rank states with incomplete measurements. New Journal of Physics, 18 (4). 043018. ISSN 1367-2630 http://iopscience.iop.org/article/10.1088/1367-2630/18/4/043018/meta doi:10.1088/1367-2630/18/4/043018 doi:10.1088/1367-2630/18/4/043018 |
| spellingShingle | Acharya, Anirudh Kypraios, Theodore Guţă, Mădălin Statistically efficient tomography of low rank states with incomplete measurements |
| title | Statistically efficient tomography of low rank states with incomplete measurements |
| title_full | Statistically efficient tomography of low rank states with incomplete measurements |
| title_fullStr | Statistically efficient tomography of low rank states with incomplete measurements |
| title_full_unstemmed | Statistically efficient tomography of low rank states with incomplete measurements |
| title_short | Statistically efficient tomography of low rank states with incomplete measurements |
| title_sort | statistically efficient tomography of low rank states with incomplete measurements |
| url | https://eprints.nottingham.ac.uk/46912/ https://eprints.nottingham.ac.uk/46912/ https://eprints.nottingham.ac.uk/46912/ |