Uncertainty Quantification for Matrix Compressed Sensing and Quantum Tomography Problems

  • Alexandra Carpentier
  • Jens Eisert
  • David Gross
  • Richard NicklEmail author
Conference paper
Part of the Progress in Probability book series (PRPR, volume 74)


We construct minimax optimal non-asymptotic confidence sets for low rank matrix recovery algorithms such as the Matrix Lasso or Dantzig selector. These are employed to devise adaptive sequential sampling procedures that guarantee recovery of the true matrix in Frobenius norm after a data-driven stopping time \(\hat n\) for the number of measurements that have to be taken. With high probability, this stopping time is minimax optimal. We detail applications to quantum tomography problems where measurements arise from Pauli observables. We also give a theoretical construction of a confidence set for the density matrix of a quantum state that has optimal diameter in nuclear norm. The non-asymptotic properties of our confidence sets are further investigated in a simulation study.


Low rank recovery Quantum information Confidence sets Sequential sampling 



The work of A. Carpentier was done when she was in Cambridge and is partially supported by the Deutsche Forschungsgemeinschaft (DFG) Emmy Noether grant MuSyAD (CA 1488/1-1), by the DFG-314838170, GRK 2297 MathCoRe, by the DFG GRK 2433 DAEDALUS (384950143/GRK2433), and by the DFG CRC 1294 ‘Data Assimilation’, Project A03, and by the UFA-DFH through the French-German Doktorandenkolleg CDFA 01-18. D. Gross acknowledges support by the DFG (SPP1798 CoSIP), Germany’s Excellence Strategy—Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1—390534769, and the ARO under contract W911NF-14-1-0098 (Quantum Characterization, Verification, and Validation). J. Eisert was supported by the DFG, CRC 183, Project B01, by the DFG GRK DAEDALUS, the DFG SPP1798 CoSIP, the ERC, and the Templeton Foundation.


  1. 1.
    L. Artiles, R. Gill, M. Guta, An invitation to quantum tomography. J. R. Stat. Soc. 67, 109 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    K. Audenaert, S. Scheel, Quantum tomographic reconstruction with error bars: a Kalman filter approach. New J. Phys. 11(2), 023028 (2009)CrossRefGoogle Scholar
  3. 3.
    P.J. Bickel, Y. Ritov, A.B. Tsybakov, Simultaneous analysis of lasso and Dantzig selector. Ann. Stat. 37(4), 1705–1732 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Blume-Kohout, Robust error bars for quantum tomography (2012). arXiv:1202.5270Google Scholar
  5. 5.
    P. Bühlmann, S. van de Geer, Statistics for High-Dimensional Data: Methods, Theory and Applications (Springer, Berlin, 2011)CrossRefGoogle Scholar
  6. 6.
    E.J. Candès, Y. Plan, Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements. IEEE Trans. Inform. Theory 57(4), 2342–2359 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    E.J. Candès, T. Tao, The Dantzig selector: statistical estimation when p is much larger than n. Ann. Stat. 35(6), 2313–2351 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Carpentier, A. Kim, An iterative hard thresholding estimator for low rank matrix recovery with explicit limiting distribution (2015). arxiv preprint 1502.04654Google Scholar
  9. 9.
    A. Carpentier, O. Klopp, M. Löffler, R. Nickl, et al., Adaptive confidence sets for matrix completion. Bernoulli 24(4A), 2429–2460 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    M. Christandl, R. Renner, Reliable quantum state tomography. Phys. Rev. Lett. 109, 120403 (2012)CrossRefGoogle Scholar
  11. 11.
    R. De Wolf, A brief introduction to Fourier analysis on the Boolean cube. Theory Comput. Libr. Grad. Surv. 1, 1–20 (2008)Google Scholar
  12. 12.
    S.T. Flammia, D. Gross, Y.-K. Liu, J. Eisert, Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators. New J. Phys. 14(9), 095022 (2012)CrossRefGoogle Scholar
  13. 13.
    E. Giné, R. Nickl, Confidence bands in density estimation. Ann. Stat. 38, 1122–1170 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    E. Giné, R. Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models (Cambridge University Press, Cambridge, 2015)zbMATHGoogle Scholar
  15. 15.
    D. Gross, Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inf. Theory 57(3), 1548–1566 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    D. Gross, Y.-K. Liu, S.T. Flammia, S. Becker, J. Eisert, Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105(15), 150401 (2010)Google Scholar
  17. 17.
    M. Guta, T. Kypraios, I. Dryden, Rank-based model selection for multiple ions quantum tomography. New J. Phys. 14, 105002 (2012)CrossRefGoogle Scholar
  18. 18.
    H. Haeffner, W. Haensel, C.F. Roos, J. Benhelm, D.C. al kar, M. Chwalla, T. Koerber, U.D. Rapol, M. Riebe, P.O. Schmidt, C. Becher, O. Gühne, W. Dür, R. Blatt, Scalable multi-particle entanglement of trapped ions. Nature 438, 643 (2005)CrossRefGoogle Scholar
  19. 19.
    M. Hoffmann, R. Nickl, On adaptive inference and confidence bands. Ann. Stat. 39, 2382–2409 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    A.S. Holevo, Statistical Structure of Quantum Theory (Springer, Berlin, 2001)CrossRefGoogle Scholar
  21. 21.
    Y.I. Ingster, A.B. Tsybakov, N. Verzelen, Detection boundary in sparse regression. Electron. J. Stat. 4, 1476–1526 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Javanmard, A. Montanari, Confidence intervals and hypothesis testing for high-dimensional regression. J. Mach. Learn. Res. 15(1), 2869–2909 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    A. Kalev, R.L. Kosut, I.H. Deutsch, Informationally complete measurements from compressed sensing methodology. arXiv:1502.00536Google Scholar
  24. 24.
    V. Koltchinskii, Von Neumann entropy penalization and low-rank matrix estimation. Ann. Stat. 39(6), 2936–2973 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    V. Koltchinskii, K. Lounici, A.B. Tsybakov, Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion. Ann. Stat. 39(5), 2302–2329 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    H. Leeb, B.M. Pötscher, Can one estimate the conditional distribution of post-model-selection estimators? Ann. Stat. 34(5), 2554–2591 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    U. Leonhardt, Measuring the Quantum State of Light (Cambridge University Press, Cambridge, 2005)zbMATHGoogle Scholar
  28. 28.
    Y.-K. Liu, Universal low-rank matrix recovery from Pauli measurements. Adv. Neural Inf. Process. Syst. 24, 1638–1646 (2011)Google Scholar
  29. 29.
    V. Mnih, C. Szepesvári, J.Y. Audibert, Empirical bernstein stopping, in Proceedings of the 25th International Conference on Machine Learning (ACM, New York, 2008), pp. 672–679Google Scholar
  30. 30.
    R. Nickl, S. van de Geer, Confidence sets in sparse regression. Ann. Stat. 41(6), 2852–2876 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)zbMATHGoogle Scholar
  32. 32.
    A. Peres, Quantum Theory (Springer, Berlin, 1995)zbMATHGoogle Scholar
  33. 33.
    B. Recht, M. Fazel, P.A. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52, 471 (2010)MathSciNetCrossRefGoogle Scholar
  34. 34.
    J. Shang, H.K. Ng, A. Sehrawat, X. Li, B.-G. Englert, Optimal error regions for quantum state estimation. New J. Phys. 15(12), 123026 (2013)MathSciNetCrossRefGoogle Scholar
  35. 35.
    K. Temme, F. Verstraete, Quantum chi-squared and goodness of fit testing. J. Math. Phys. 56(1), 012202 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    S. van de Geer, P. Bühlmann, Y. Ritov, R. Dezeure, On asymptotically optimal confidence regions and tests for high-dimensional models. Ann. Stat. 42(3), 1166–1202 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alexandra Carpentier
    • 1
  • Jens Eisert
    • 2
  • David Gross
    • 3
  • Richard Nickl
    • 4
    Email author
  1. 1.Institut für Mathematische StochastikOtto von Guericke Universität MagdeburgMagdeburgGermany
  2. 2.Department of Physics, Dahlem Center for Complex Quantum SystemsFreie Universität zu BerlinBerlinGermany
  3. 3.Institute of Theoretical PhysicsUniversität KölnKölnGermany
  4. 4.Statistical LaboratoryUniversity of CambridgeCambridgeUK

Personalised recommendations