Behavior of the maximum likelihood in quantum state tomography
Quantum state tomography on a ddimensional 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, metricprojected 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.
 Authors:

^{[1]}
;
^{[1]}
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States); Univ. of New Mexico, Albuquerque, NM (United States)
 Publication Date:
 Report Number(s):
 SAND20180673J
Journal ID: ISSN 13672630; 660116
 Grant/Contract Number:
 NA0003525
 Type:
 Accepted Manuscript
 Journal Name:
 New Journal of Physics
 Additional Journal Information:
 Journal Volume: 20; Journal Issue: 2; Journal ID: ISSN 13672630
 Publisher:
 IOP Publishing
 Research Org:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org:
 USDOE National Nuclear Security Administration (NNSA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
 OSTI Identifier:
 1432340
Scholten, Travis L., and BlumeKohout, Robin. Behavior of the maximum likelihood in quantum state tomography. United States: N. p.,
Web. doi:10.1088/13672630/aaa7e2.
Scholten, Travis L., & BlumeKohout, Robin. Behavior of the maximum likelihood in quantum state tomography. United States. doi:10.1088/13672630/aaa7e2.
Scholten, Travis L., and BlumeKohout, Robin. 2018.
"Behavior of the maximum likelihood in quantum state tomography". United States.
doi:10.1088/13672630/aaa7e2. https://www.osti.gov/servlets/purl/1432340.
@article{osti_1432340,
title = {Behavior of the maximum likelihood in quantum state tomography},
author = {Scholten, Travis L. and BlumeKohout, Robin},
abstractNote = {Quantum state tomography on a ddimensional 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, metricprojected 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.},
doi = {10.1088/13672630/aaa7e2},
journal = {New Journal of Physics},
number = 2,
volume = 20,
place = {United States},
year = {2018},
month = {2}
}