Behavior of the maximum likelihood in quantum state tomography
Quantum state tomography on a d -dimensional system demands resources that grow rapidly with d . They may be reduced by using model selection to tailor the number of parameters in the model (i.e., the size of the density matrix). Most model selection methods typically rely on a test statistic and a...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2018-01-01
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| Series: | New Journal of Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1088/1367-2630/aaa7e2 |
| _version_ | 1797750661966725120 |
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| author | Travis L Scholten Robin Blume-Kohout |
| author_facet | Travis L Scholten Robin Blume-Kohout |
| author_sort | Travis L Scholten |
| collection | DOAJ |
| description | Quantum state tomography on a d -dimensional system demands resources that grow rapidly with d . They may be reduced by using model selection to tailor the number of parameters in the model (i.e., the size of the density matrix). Most model selection methods typically rely on a test statistic and a null theory that describes its behavior when two models are equally good. Here, we consider the loglikelihood ratio. Because of the positivity constraint ρ ≥ 0, quantum state space does not generally satisfy local asymptotic normality (LAN), meaning the classical null theory for the loglikelihood ratio (the Wilks theorem) should not be used. Thus, understanding and quantifying how positivity affects the null behavior of this test statistic is necessary for its use in model selection for state tomography. We define a new generalization of LAN, metric-projected LAN, show that quantum state space satisfies it, and derive a replacement for the Wilks theorem. In addition to enabling reliable model selection, our results shed more light on the qualitative effects of the positivity constraint on state tomography. |
| first_indexed | 2024-03-12T16:36:04Z |
| format | Article |
| id | doaj.art-04746b37fd134216b9f404f9cbd1b38e |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:36:04Z |
| publishDate | 2018-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-04746b37fd134216b9f404f9cbd1b38e2023-08-08T14:51:00ZengIOP PublishingNew Journal of Physics1367-26302018-01-0120202305010.1088/1367-2630/aaa7e2Behavior of the maximum likelihood in quantum state tomographyTravis L Scholten0https://orcid.org/0000-0001-9334-4248Robin Blume-Kohout1Center for Computing Research (CCR) , Sandia National Laboratories, United States of America; Center for Quantum Information and Control (CQuIC), University of New Mexico , United States of AmericaCenter for Computing Research (CCR) , Sandia National Laboratories, United States of America; Center for Quantum Information and Control (CQuIC), University of New Mexico , United States of AmericaQuantum state tomography on a d -dimensional system demands resources that grow rapidly with d . They may be reduced by using model selection to tailor the number of parameters in the model (i.e., the size of the density matrix). Most model selection methods typically rely on a test statistic and a null theory that describes its behavior when two models are equally good. Here, we consider the loglikelihood ratio. Because of the positivity constraint ρ ≥ 0, quantum state space does not generally satisfy local asymptotic normality (LAN), meaning the classical null theory for the loglikelihood ratio (the Wilks theorem) should not be used. Thus, understanding and quantifying how positivity affects the null behavior of this test statistic is necessary for its use in model selection for state tomography. We define a new generalization of LAN, metric-projected LAN, show that quantum state space satisfies it, and derive a replacement for the Wilks theorem. In addition to enabling reliable model selection, our results shed more light on the qualitative effects of the positivity constraint on state tomography.https://doi.org/10.1088/1367-2630/aaa7e2quantum state tomographymodel selectioncompressed sensing |
| spellingShingle | Travis L Scholten Robin Blume-Kohout Behavior of the maximum likelihood in quantum state tomography New Journal of Physics quantum state tomography model selection compressed sensing |
| title | Behavior of the maximum likelihood in quantum state tomography |
| title_full | Behavior of the maximum likelihood in quantum state tomography |
| title_fullStr | Behavior of the maximum likelihood in quantum state tomography |
| title_full_unstemmed | Behavior of the maximum likelihood in quantum state tomography |
| title_short | Behavior of the maximum likelihood in quantum state tomography |
| title_sort | behavior of the maximum likelihood in quantum state tomography |
| topic | quantum state tomography model selection compressed sensing |
| url | https://doi.org/10.1088/1367-2630/aaa7e2 |
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