Behavior of the maximum likelihood in quantum state tomography
Abstract
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:

 Sandia National Lab. (SNLNM), Albuquerque, NM (United States); Univ. of New Mexico, Albuquerque, NM (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1432340
 Report Number(s):
 SAND20180673J
Journal ID: ISSN 13672630; 660116; TRN: US1802658
 Grant/Contract Number:
 NA0003525
 Resource 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
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Citation Formats
Scholten, Travis L., and BlumeKohout, Robin. Behavior of the maximum likelihood in quantum state tomography. United States: N. p., 2018.
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. Thu .
"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}
}
Works referenced in this record:
The pauli problem, state reconstruction and quantumreal numbers
journal, February 2006
 Corbett, J. V.
 Reports on Mathematical Physics, Vol. 57, Issue 1
Measurement of the Entanglement of Two Superconducting Qubits via State Tomography
journal, September 2006
 Steffen, M.; Ansmann, M.; Bialczak, R. C.
 Science, Vol. 313, Issue 5792
Entanglement Verification with Finite Data
journal, October 2010
 BlumeKohout, Robin; Yin, Jun O. S.; van Enk, S. J.
 Physical Review Letters, Vol. 105, Issue 17
Experimental procedures for entanglement verification
journal, May 2007
 van Enk, S. J.; Lütkenhaus, N.; Kimble, H. J.
 Physical Review A, Vol. 75, Issue 5
Maximumlikelihood coherentstate quantum process tomography
journal, October 2012
 Anis, Aamir; Lvovsky, A. I.
 New Journal of Physics, Vol. 14, Issue 10
On Some Principles of Statistical Inference: Principles of Statistical Inference
journal, October 2014
 Reid, Nancy; Cox, David R.
 International Statistical Review, Vol. 83, Issue 2
A tomographic approach to Wigner's function
journal, April 1987
 Bertrand, J.; Bertrand, P.
 Foundations of Physics, Vol. 17, Issue 4
Continuousvariable optical quantumstate tomography
journal, March 2009
 Lvovsky, A. I.; Raymer, M. G.
 Reviews of Modern Physics, Vol. 81, Issue 1
Measurement of the quantum states of squeezed light
journal, May 1997
 Breitenbach, G.; Schiller, S.; Mlynek, J.
 Nature, Vol. 387, Issue 6632
Measuring the quantum state of light
journal, January 1995
 Leonhardt, U.; Paul, H.
 Progress in Quantum Electronics, Vol. 19, Issue 2
Simple Pulses for Elimination of Leakage in Weakly Nonlinear Qubits
journal, September 2009
 Motzoi, F.; Gambetta, J. M.; Rebentrost, P.
 Physical Review Letters, Vol. 103, Issue 11
Fidelity and Leakage of Josephson Qubits
journal, December 1999
 Fazio, Rosario; Palma, G. Massimo; Siewert, Jens
 Physical Review Letters, Vol. 83, Issue 25
Multimodel Inference: Understanding AIC and BIC in Model Selection
journal, November 2004
 Burnham, Kenneth P.; Anderson, David R.
 Sociological Methods & Research, Vol. 33, Issue 2
NearOptimal Signal Recovery From Random Projections: Universal Encoding Strategies?
journal, January 2006
 Candes, Emmanuel J.; Tao, Terence
 IEEE Transactions on Information Theory, Vol. 52, Issue 12, p. 54065425
Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators
journal, September 2012
 Flammia, Steven T.; Gross, David; Liu, YiKai
 New Journal of Physics, Vol. 14, Issue 9
Error regions in quantum state tomography: computational complexity caused by geometry of quantum states
journal, September 2017
 Suess, Daniel; Rudnicki, Łukasz; maciel, Thiago O.
 New Journal of Physics, Vol. 19, Issue 9
Error models in quantum computation: An application of model selection
journal, September 2013
 Schwarz, Lucia; van Enk, S. J.
 Physical Review A, Vol. 88, Issue 3
Rankbased model selection for multiple ions quantum tomography
journal, October 2012
 Guţă, Mădălin; Kypraios, Theodore; Dryden, Ian
 New Journal of Physics, Vol. 14, Issue 10
When quantum tomography goes wrong: drift of quantum sources and other errors
journal, February 2013
 van Enk, S. J.; BlumeKohout, Robin
 New Journal of Physics, Vol. 15, Issue 2
Errors in quantum tomography: diagnosing systematic versus statistical errors
journal, March 2013
 Langford, Nathan K.
 New Journal of Physics, Vol. 15, Issue 3
Information criteria for efficient quantum state estimation
journal, June 2011
 Yin, J. O. S.; van Enk, S. J.
 Physical Review A, Vol. 83, Issue 6
Certifying Systematic Errors in Quantum Experiments
journal, April 2013
 Moroder, Tobias; Kleinmann, Matthias; Schindler, Philipp
 Physical Review Letters, Vol. 110, Issue 18
A new look at the statistical model identification
journal, December 1974
 Akaike, H.
 IEEE Transactions on Automatic Control, Vol. 19, Issue 6
Measurement of qubits
journal, October 2001
 James, Daniel F. V.; Kwiat, Paul G.; Munro, William J.
 Physical Review A, Vol. 64, Issue 5
On the Problem of the Most Efficient Tests of Statistical Hypotheses
journal, January 1933
 Neyman, J.; Pearson, E. S.
 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 231, Issue 694706
The LargeSample Distribution of the Likelihood Ratio for Testing Composite Hypotheses
journal, March 1938
 Wilks, S. S.
 The Annals of Mathematical Statistics, Vol. 9, Issue 1
On the Assumptions Used to Prove Asymptotic Normality of Maximum Likelihood Estimates
journal, June 1970
 LeCam, L.
 The Annals of Mathematical Statistics, Vol. 41, Issue 3
Estimating the Dimension of a Model
journal, March 1978
 Schwarz, Gideon
 The Annals of Statistics, Vol. 6, Issue 2
Bayes Factors
journal, June 1995
 Kass, Robert E.; Raftery, Adrian E.
 Journal of the American Statistical Association, Vol. 90, Issue 430
Bayesian measures of model complexity and fit
journal, October 2002
 Spiegelhalter, David J.; Best, Nicola G.; Carlin, Bradley P.
 Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 64, Issue 4
Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase
journal, September 1989
 Vogel, K.; Risken, H.
 Physical Review A, Vol. 40, Issue 5
Quantum state estimation with informationally overcomplete measurements
journal, July 2014
 Zhu, Huangjun
 Physical Review A, Vol. 90, Issue 1
Minimax Quantum Tomography: Estimators and Relative Entropy Bounds
journal, March 2016
 Ferrie, Christopher; BlumeKohout, Robin
 Physical Review Letters, Vol. 116, Issue 9
From Steiner Formulas for Cones to Concentration of Intrinsic Volumes
journal, April 2014
 McCoy, Michael B.; Tropp, Joel A.
 Discrete & Computational Geometry, Vol. 51, Issue 4
Living on the edge: phase transitions in convex programs with random data
journal, June 2014
 Amelunxen, D.; Lotz, M.; McCoy, M. B.
 Information and Inference, Vol. 3, Issue 3
Efficient Method for Computing the MaximumLikelihood Quantum State from Measurements with Additive Gaussian Noise
journal, February 2012
 Smolin, John A.; Gambetta, Jay M.; Smith, Graeme
 Physical Review Letters, Vol. 108, Issue 7
On the Distribution of the Roots of Certain Symmetric Matrices
journal, March 1958
 Wigner, Eugene P.
 The Annals of Mathematics, Vol. 67, Issue 2
Random Matrices: Sharp Concentration of Eigenvalues
journal, July 2013
 Tao, Terence; Vu, Van
 Random Matrices: Theory and Applications, Vol. 02, Issue 03
Quantum State Reconstruction of the SinglePhoton Fock State
journal, July 2001
 Lvovsky, A. I.; Hansen, H.; Aichele, T.
 Physical Review Letters, Vol. 87, Issue 5
Gradientbased stopping rules for maximumlikelihood quantumstate tomography
journal, September 2012
 Glancy, S.; Knill, E.; Girard, M.
 New Journal of Physics, Vol. 14, Issue 9
Bootstrap Methods: Another Look at the Jackknife
journal, January 1979
 Efron, B.
 The Annals of Statistics, Vol. 7, Issue 1
Strictlycomplete measurements for boundedrank quantumstate tomography
journal, May 2016
 Baldwin, Charles H.; Deutsch, Ivan H.; Kalev, Amir
 Physical Review A, Vol. 93, Issue 5
Quantum tomography protocols with positivity are compressed sensing protocols
journal, December 2015
 Kalev, Amir; Kosut, Robert L.; Deutsch, Ivan H.
 npj Quantum Information, Vol. 1, Issue 1
IPython: A System for Interactive Scientific Computing
journal, January 2007
 Perez, Fernando; Granger, Brian E.
 Computing in Science & Engineering, Vol. 9, Issue 3
Matplotlib: A 2D Graphics Environment
journal, January 2007
 Hunter, John D.
 Computing in Science & Engineering, Vol. 9, Issue 3
MPI for Python
journal, September 2005
 Dalcín, Lisandro; Paz, Rodrigo; Storti, Mario
 Journal of Parallel and Distributed Computing, Vol. 65, Issue 9
The NumPy Array: A Structure for Efficient Numerical Computation
journal, March 2011
 van der Walt, Stéfan; Colbert, S. Chris; Varoquaux, Gaël
 Computing in Science & Engineering, Vol. 13, Issue 2
Python for Scientific Computing
journal, January 2007
 Oliphant, Travis E.
 Computing in Science & Engineering, Vol. 9, Issue 3
SymPy: symbolic computing in Python
journal, January 2017
 Meurer, Aaron; Smith, Christopher P.; Paprocki, Mateusz
 PeerJ Computer Science, Vol. 3
Works referencing / citing this record:
Quantum gate teleportation between separated qubits in a trappedion processor
journal, May 2019
 Wan, Yong; Kienzler, Daniel; Erickson, Stephen D.
 Science, Vol. 364, Issue 6443
Quantum gate teleportation between separated qubits in a trappedion processor
journal, May 2019
 Wan, Yong; Kienzler, Daniel; Erickson, Stephen D.
 Science, Vol. 364, Issue 6443