skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Behavior of the Maximum Likelihood in Quantum State Tomography.

Abstract

Abstract not provided.

Authors:
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1426603
Report Number(s):
SAND2017-2633C
Journal ID: ISSN 1367--2630; 651629
DOE Contract Number:
AC04-94AL85000
Resource Type:
Conference
Resource Relation:
Journal Volume: 20; Journal Issue: 2; Conference: Proposed for presentation at the APS March Meeting held March 13-17, 2017 in New Orleans, Louisiana.
Country of Publication:
United States
Language:
English

Citation Formats

Scholten, Travis L. Behavior of the Maximum Likelihood in Quantum State Tomography.. United States: N. p., 2017. Web. doi:10.1088/1367-2630/aaa7e2.
Scholten, Travis L. Behavior of the Maximum Likelihood in Quantum State Tomography.. United States. doi:10.1088/1367-2630/aaa7e2.
Scholten, Travis L. Wed . "Behavior of the Maximum Likelihood in Quantum State Tomography.". United States. doi:10.1088/1367-2630/aaa7e2. https://www.osti.gov/servlets/purl/1426603.
@article{osti_1426603,
title = {Behavior of the Maximum Likelihood in Quantum State Tomography.},
author = {Scholten, Travis L.},
abstractNote = {Abstract not provided.},
doi = {10.1088/1367-2630/aaa7e2},
journal = {},
number = 2,
volume = 20,
place = {United States},
year = {Wed Mar 01 00:00:00 EST 2017},
month = {Wed Mar 01 00:00:00 EST 2017}
}

Conference:
Other availability
Please see Document Availability for additional information on obtaining the full-text document. Library patrons may search WorldCat to identify libraries that hold this conference proceeding.

Save / Share:
  • 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) shouldmore » 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.« less
  • 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) shouldmore » 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.« less
  • Two distinctly different methods have been used to improve images produced in positron-emission tomography. The first method is to measure the differential time of flight of the photon pairs which are detected; the second is to use an iterative algorithm which computes maximum likelihood estimates of radioactivity distributions. The authors have quantified the performance of algorithms which include neither, one or the other, or both methods of improvement by performing a repetitive simulation experiment using the Hoffman brain phantom as the underlying distribution of radioactivity. The authors' simulations show that all of the algorithms yield unbiased estimates of the desiredmore » image. The algorithm which computes maximum-likelihood estimates using time-of-flight information reconstructs images with the lowest variance. The algorithm which uses neither of these methods (filtered backprojection) reconstructs images with the highest variance.« less
  • The frequency spectral characteristics, bias and variance of images reconstructed from real Positron Emission Tomography (PET) data have been studied. Feasible images obtained from statistically based reconstruction methods have been compared to Filtered Backprojection (FBP) images. Feasible images have been described as those images that are compatible with the measured data by consideration of the Poisson nature of the emission process. The results show that the spectral characteristics of reconstructions obtained by statistically based methods are at least as good as those obtained by the FBP methods. With some exceptions, statistically based reconstructions do not exhibit abnormal amounts of bias.more » The most significant difference between the two groups of reconstructions is in the image variance, where the statistically based methods yield substantially smaller variances in the regions with smaller image intensity than the FBP images. 14 refs., 12 figs., 3 tabs.« less
  • A maximum-likelihood algorithm (ML) has been previously proposed for the reconstruction of positron emission tomography (PET) images. Herein, the authors compare the relative performance of this new algorithm to the filtered back projectin (FBP) technique for the PETT VI system with measured and simulated phantom data. Using point source data from PETT VI, in low resolution mode, the ML reconstructed image had a resolution which was approximately half of the full-width half-maximum (FWHM) detector resolution. Using PETT VI brain images (o-15 water bolus) ML yielded images with visibly better resolution than FBP. Simulations of data collections on PETT VI weremore » used to quantitate the relationship between image resolution and noise: 1) A pie-phantom containing six uniform wedges and one million events was reconstructed with ML and FBP. The image resolution and standard deviation (SD) of pixel values within the regions of uniform activity were measured. 2) A brain-phantom containing detailed structure, and one million events was simulated for 20 realizations, all of which were reconstructed using ML and FBP. Image resolution and SD of lcm/sup 2/ region values, across the 20 realizations, were measured. In both simulations ML had little to no advantage over FBP for reconstruction of low resolution and low regional SD images. However, if images with higher SD were accepted, the difference in resolution attained between ML and FBP became substantial: FWHM(ML)/FWHM(FBP)approx. =0.70. These results indicate that ML may be preferred over FBP for high resolution reconstruction, approaching, or even less than the detector resolution.« less