Behavior of the maximum likelihood in quantum state tomography

Blume-Kohout, Robin J; Scholten, Travis L.

doi:10.1088/1367-2630/aaa7e2

Behavior of the maximum likelihood in quantum state tomography

Journal Article · Thu Feb 22 04:00:00 EST 2018 · New Journal of Physics

DOI:https://doi.org/10.1088/1367-2630/aaa7e2· OSTI ID:1335564

Blume-Kohout, Robin J ^[1]; ^[1]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Univ. of New Mexico, Albuquerque, NM (United States)

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.

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Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

DOE Contract Number:: NA0003525

OSTI ID:: 1335564

Report Number(s):: SAND--2016-8677J; 647145

Journal Information:: New Journal of Physics, Journal Name: New Journal of Physics Vol. 20; ISSN 1367-2630

Publisher:: IOP Publishing

Country of Publication:: United States

Language:: English

References (53)

Measuring the quantum state of light Leonhardt, U.; Paul, H. Progress in Quantum Electronics, Vol. 19, Issue 2 https://doi.org/10.1016/0079-6727(94)00007-L	journal	January 1995
Rank-based model selection for multiple ions quantum tomography Guţă, Mădălin; Kypraios, Theodore; Dryden, Ian New Journal of Physics, Vol. 14, Issue 10 https://doi.org/10.1088/1367-2630/14/10/105002	journal	October 2012
MPI for Python Dalcín, Lisandro; Paz, Rodrigo; Storti, Mario Journal of Parallel and Distributed Computing, Vol. 65, Issue 9 https://doi.org/10.1016/j.jpdc.2005.03.010	journal	September 2005
Bayesian measures of model complexity and fit Spiegelhalter, David J.; Best, Nicola G.; Carlin, Bradley P. Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 64, Issue 4 https://doi.org/10.1111/1467-9868.00353	journal	October 2002
Certifying Systematic Errors in Quantum Experiments Moroder, Tobias; Kleinmann, Matthias; Schindler, Philipp Physical Review Letters, Vol. 110, Issue 18 https://doi.org/10.1103/PhysRevLett.110.180401	journal	April 2013
Quantum-state estimation Hradil, Z. Physical Review A, Vol. 55, Issue 3 https://doi.org/10.1103/PhysRevA.55.R1561	journal	March 1997
Information criteria for efficient quantum state estimation Yin, J. O. S.; van Enk, S. J. Physical Review A, Vol. 83, Issue 6 https://doi.org/10.1103/PhysRevA.83.062110	journal	June 2011
Measurement of the Entanglement of Two Superconducting Qubits via State Tomography Steffen, M.; Ansmann, M.; Bialczak, R. C. Science, Vol. 313, Issue 5792 https://doi.org/10.1126/science.1130886	journal	September 2006
Random Matrices: Sharp Concentration of Eigenvalues Tao, Terence; Vu, Van Random Matrices: Theory and Applications, Vol. 02, Issue 03 https://doi.org/10.1142/S201032631350007X	journal	July 2013
Quantum tomography protocols with positivity are compressed sensing protocols Kalev, Amir; Kosut, Robert L.; Deutsch, Ivan H. npj Quantum Information, Vol. 1, Issue 1 https://doi.org/10.1038/npjqi.2015.18	journal	December 2015
On the Assumptions Used to Prove Asymptotic Normality of Maximum Likelihood Estimates LeCam, L. The Annals of Mathematical Statistics, Vol. 41, Issue 3 https://doi.org/10.1214/aoms/1177696960	journal	June 1970
Continuous-variable optical quantum-state tomography Lvovsky, A. I.; Raymer, M. G. Reviews of Modern Physics, Vol. 81, Issue 1 https://doi.org/10.1103/RevModPhys.81.299	journal	March 2009
On Some Principles of Statistical Inference: Principles of Statistical Inference Reid, Nancy; Cox, David R. International Statistical Review, Vol. 83, Issue 2 https://doi.org/10.1111/insr.12067	journal	October 2014
Efficient Method for Computing the Maximum-Likelihood Quantum State from Measurements with Additive Gaussian Noise Smolin, John A.; Gambetta, Jay M.; Smith, Graeme Physical Review Letters, Vol. 108, Issue 7 https://doi.org/10.1103/PhysRevLett.108.070502	journal	February 2012
From Steiner Formulas for Cones to Concentration of Intrinsic Volumes McCoy, Michael B.; Tropp, Joel A. Discrete & Computational Geometry, Vol. 51, Issue 4 https://doi.org/10.1007/s00454-014-9595-4	journal	April 2014
The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses Wilks, S. S. The Annals of Mathematical Statistics, Vol. 9, Issue 1 https://doi.org/10.1214/aoms/1177732360	journal	March 1938
Measurement of the quantum states of squeezed light Breitenbach, G.; Schiller, S.; Mlynek, J. Nature, Vol. 387, Issue 6632 https://doi.org/10.1038/387471a0	journal	May 1997
Entanglement Verification with Finite Data Blume-Kohout, Robin; Yin, Jun O. S.; van Enk, S. J. Physical Review Letters, Vol. 105, Issue 17 https://doi.org/10.1103/PhysRevLett.105.170501	journal	October 2010
Simple Pulses for Elimination of Leakage in Weakly Nonlinear Qubits Motzoi, F.; Gambetta, J. M.; Rebentrost, P. Physical Review Letters, Vol. 103, Issue 11 https://doi.org/10.1103/PhysRevLett.103.110501	journal	September 2009
Photonic State Tomography Altepeter, J. B.; Jeffrey, E. R.; Kwiat, P. G. Advances In Atomic, Molecular, and Optical Physics https://doi.org/10.1016/S1049-250X(05)52003-2	book	May 2005
Errors in quantum tomography: diagnosing systematic versus statistical errors Langford, Nathan K. New Journal of Physics, Vol. 15, Issue 3 https://doi.org/10.1088/1367-2630/15/3/035003	journal	March 2013
Fidelity and Leakage of Josephson Qubits Fazio, Rosario; Palma, G. Massimo; Siewert, Jens Physical Review Letters, Vol. 83, Issue 25 https://doi.org/10.1103/PhysRevLett.83.5385	journal	December 1999
On the Distribution of the Roots of Certain Symmetric Matrices Wigner, Eugene P. The Annals of Mathematics, Vol. 67, Issue 2 https://doi.org/10.2307/1970008	journal	March 1958
Maximum-likelihood coherent-state quantum process tomography Anis, Aamir; Lvovsky, A. I. New Journal of Physics, Vol. 14, Issue 10 https://doi.org/10.1088/1367-2630/14/10/105021	journal	October 2012
Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase Vogel, K.; Risken, H. Physical Review A, Vol. 40, Issue 5 https://doi.org/10.1103/PhysRevA.40.2847	journal	September 1989
Gradient-based stopping rules for maximum-likelihood quantum-state tomography Glancy, S.; Knill, E.; Girard, M. New Journal of Physics, Vol. 14, Issue 9 https://doi.org/10.1088/1367-2630/14/9/095017	journal	September 2012
Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators Flammia, Steven T.; Gross, David; Liu, Yi-Kai New Journal of Physics, Vol. 14, Issue 9 https://doi.org/10.1088/1367-2630/14/9/095022	journal	September 2012
mwaskom/seaborn: v0.11.1 (December 2020) Waskom, Michael; Gelbart, Maoz; Botvinnik, Olga Zenodo https://doi.org/10.5281/zenodo.592845	software	December 2020
Bayes Factors Kass, Robert E.; Raftery, Adrian E. Journal of the American Statistical Association, Vol. 90, Issue 430 https://doi.org/10.1080/01621459.1995.10476572	journal	June 1995
Minimax Quantum Tomography: Estimators and Relative Entropy Bounds Ferrie, Christopher; Blume-Kohout, Robin Physical Review Letters, Vol. 116, Issue 9 https://doi.org/10.1103/PhysRevLett.116.090407	journal	March 2016
A new look at the statistical model identification Akaike, H. IEEE Transactions on Automatic Control, Vol. 19, Issue 6 https://doi.org/10.1109/TAC.1974.1100705	journal	December 1974
On the Problem of the Most Efficient Tests of Statistical Hypotheses Neyman, J.; Pearson, E. S. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 231, Issue 694-706 https://doi.org/10.1098/rsta.1933.0009	journal	January 1933
Experimental procedures for entanglement verification van Enk, S. J.; Lütkenhaus, N.; Kimble, H. J. Physical Review A, Vol. 75, Issue 5 https://doi.org/10.1103/PhysRevA.75.052318	journal	May 2007
Estimating the Dimension of a Model Schwarz, Gideon The Annals of Statistics, Vol. 6, Issue 2 https://doi.org/10.1214/aos/1176344136	journal	March 1978
Error models in quantum computation: An application of model selection Schwarz, Lucia; van Enk, S. J. Physical Review A, Vol. 88, Issue 3 https://doi.org/10.1103/PhysRevA.88.032318	journal	September 2013
Error regions in quantum state tomography: computational complexity caused by geometry of quantum states Suess, Daniel; Rudnicki, Łukasz; maciel, Thiago O. New Journal of Physics, Vol. 19, Issue 9 https://doi.org/10.1088/1367-2630/aa7ce9	journal	September 2017
When quantum tomography goes wrong: drift of quantum sources and other errors van Enk, S. J.; Blume-Kohout, Robin New Journal of Physics, Vol. 15, Issue 2 https://doi.org/10.1088/1367-2630/15/2/025024	journal	February 2013
Quantum State Reconstruction of the Single-Photon Fock State Lvovsky, A. I.; Hansen, H.; Aichele, T. Physical Review Letters, Vol. 87, Issue 5 https://doi.org/10.1103/PhysRevLett.87.050402	journal	July 2001
Quantum state estimation with informationally overcomplete measurements Zhu, Huangjun Physical Review A, Vol. 90, Issue 1 https://doi.org/10.1103/PhysRevA.90.012115	journal	July 2014
Strictly-complete measurements for bounded-rank quantum-state tomography Baldwin, Charles H.; Deutsch, Ivan H.; Kalev, Amir Physical Review A, Vol. 93, Issue 5 https://doi.org/10.1103/PhysRevA.93.052105	journal	May 2016
A tomographic approach to Wigner's function Bertrand, J.; Bertrand, P. Foundations of Physics, Vol. 17, Issue 4 https://doi.org/10.1007/BF00733376	journal	April 1987
IPython: A System for Interactive Scientific Computing Perez, Fernando; Granger, Brian E. Computing in Science & Engineering, Vol. 9, Issue 3 https://doi.org/10.1109/MCSE.2007.53	journal	January 2007
SymPy: symbolic computing in Python Meurer, Aaron; Smith, Christopher P.; Paprocki, Mateusz PeerJ Computer Science, Vol. 3 https://doi.org/10.7717/peerj-cs.103	journal	January 2017
Variational Analysis Rockafellar, R. Tyrrell; Wets, Roger J. B. Grundlehren der mathematischen Wissenschaften https://doi.org/10.1007/978-3-642-02431-3	book	January 1998
Matplotlib: A 2D Graphics Environment Hunter, John D. Computing in Science & Engineering, Vol. 9, Issue 3 https://doi.org/10.1109/MCSE.2007.55	journal	January 2007
Measurement of qubits James, Daniel F. V.; Kwiat, Paul G.; Munro, William J. Physical Review A, Vol. 64, Issue 5 https://doi.org/10.1103/PhysRevA.64.052312	journal	October 2001
Living on the edge: phase transitions in convex programs with random data Amelunxen, D.; Lotz, M.; McCoy, M. B. Information and Inference, Vol. 3, Issue 3 https://doi.org/10.1093/imaiai/iau005	journal	June 2014
The pauli problem, state reconstruction and quantum-real numbers Corbett, J. V. Reports on Mathematical Physics, Vol. 57, Issue 1 https://doi.org/10.1016/S0034-4877(06)80008-X	journal	February 2006
Bootstrap Methods: Another Look at the Jackknife Efron, B. The Annals of Statistics, Vol. 7, Issue 1 https://doi.org/10.1214/aos/1176344552	journal	January 1979
Multimodel Inference: Understanding AIC and BIC in Model Selection Burnham, Kenneth P.; Anderson, David R. Sociological Methods & Research, Vol. 33, Issue 2 https://doi.org/10.1177/0049124104268644	journal	November 2004
Python for Scientific Computing Oliphant, Travis E. Computing in Science & Engineering, Vol. 9, Issue 3 https://doi.org/10.1109/MCSE.2007.58	journal	January 2007
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? Candes, Emmanuel J.; Tao, Terence IEEE Transactions on Information Theory, Vol. 52, Issue 12, p. 5406-5425 https://doi.org/10.1109/TIT.2006.885507	journal	January 2006
The NumPy Array: A Structure for Efficient Numerical Computation van der Walt, Stéfan; Colbert, S. Chris; Varoquaux, Gaël Computing in Science & Engineering, Vol. 13, Issue 2 https://doi.org/10.1109/MCSE.2011.37	journal	March 2011

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