The Annals of Statistics

Asymptotic equivalence of quantum state tomography and noisy matrix completion

Yazhen Wang

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Abstract

Matrix completion and quantum tomography are two unrelated research areas with great current interest in many modern scientific studies. This paper investigates the statistical relationship between trace regression in matrix completion and quantum state tomography in quantum physics and quantum information science. As quantum state tomography and trace regression share the common goal of recovering an unknown matrix, it is nature to put them in the Le Cam paradigm for statistical comparison. Regarding the two types of matrix inference problems as two statistical experiments, we establish their asymptotic equivalence in terms of deficiency distance. The equivalence study motivates us to introduce a new trace regression model. The asymptotic equivalence provides a sound statistical foundation for applying matrix completion methods to quantum state tomography. We investigate the asymptotic equivalence for sparse density matrices and low rank density matrices and demonstrate that sparsity and low rank are not necessarily helpful for achieving the asymptotic equivalence of quantum state tomography and trace regression. In particular, we show that popular Pauli measurements are bad for establishing the asymptotic equivalence for sparse density matrices and low rank density matrices.

Article information

Source
Ann. Statist., Volume 41, Number 5 (2013), 2462-2504.

Dates
First available in Project Euclid: 5 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1383661270

Digital Object Identifier
doi:10.1214/13-AOS1156

Mathematical Reviews number (MathSciNet)
MR3127872

Zentralblatt MATH identifier
1294.62006

Subjects
Primary: 62B15: Theory of statistical experiments
Secondary: 62P35: Applications to physics 62J99: None of the above, but in this section 65F10: Iterative methods for linear systems [See also 65N22] 65J20: Improperly posed problems; regularization 81P45: Quantum information, communication, networks [See also 94A15, 94A17] 81P50: Quantum state estimation, approximate cloning

Keywords
Compressed sensing deficiency distance density matrix observable Pauli matrices quantum measurement quantum probability quantum statistics trace regression fine scale trace regression low rank matrix sparse matrix

Citation

Wang, Yazhen. Asymptotic equivalence of quantum state tomography and noisy matrix completion. Ann. Statist. 41 (2013), no. 5, 2462--2504. doi:10.1214/13-AOS1156. https://projecteuclid.org/euclid.aos/1383661270


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References

  • Artiles, L. M., Gill, R. D. and Guţă, M. I. (2005). An invitation to quantum tomography. J. R. Stat. Soc. Ser. B Stat. Methodol. 67 109–134.
  • Barndorff-Nielsen, O. E., Gill, R. D. and Jupp, P. E. (2003). On quantum statistical inference (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 65 775–816.
  • Bunea, F., She, Y. and Wegkamp, M. H. (2011). Optimal selection of reduced rank estimators of high-dimensional matrices. Ann. Statist. 39 1282–1309.
  • Butucea, C., Guţă, M. and Artiles, L. (2007). Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data. Ann. Statist. 35 465–494.
  • Candès, E. J. and Plan, Y. (2009). Matrix completion with noise. Proceedings of the IEEE 98 925–936.
  • Candès, E. J. and Plan, Y. (2011). Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements. IEEE Trans. Inform. Theory 57 2342–2359.
  • Candès, E. J. and Recht, B. (2009). Exact matrix completion via convex optimization. Found. Comput. Math. 9 717–772.
  • Candès, E. J. and Tao, T. (2010). The power of convex relaxation: Near-optimal matrix completion. IEEE Trans. Inform. Theory 56 2053–2080.
  • Carter, A. V. (2002). Deficiency distance between multinomial and multivariate normal experiments. Ann. Statist. 30 708–730.
  • Donoho, D. L. (2006). Compressed sensing. IEEE Trans. Inform. Theory 52 1289–1306.
  • Gross, D. (2011). Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inform. Theory 57 1548–1566.
  • Gross, D., Liu, Y.-K., Flammia, S. T., Becker, S. and Eisert, J. (2010). Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105 150401.
  • Holevo, A. S. (1982). Probabilistic and Statistical Aspects of Quantum Theory. North-Holland Series in Statistics and Probability 1. North-Holland, Amsterdam.
  • Keshavan, R. H., Montanari, A. and Oh, S. (2010). Matrix completion from noisy entries. J. Mach. Learn. Res. 11 2057–2078.
  • Klopp, O. (2011). Rank penalized estimators for high-dimensional matrices. Electron. J. Stat. 5 1161–1183.
  • Koltchinskii, V. (2011). Von Neumann entropy penalization and low-rank matrix estimation. Ann. Statist. 39 2936–2973.
  • Koltchinskii, V., Lounici, K. and Tsybakov, A. B. (2011). Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion. Ann. Statist. 39 2302–2329.
  • Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
  • Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics: Some Basic Concepts, 2nd ed. Springer, New York.
  • Negahban, S. and Wainwright, M. J. (2011). Estimation of (near) low-rank matrices with noise and high-dimensional scaling. Ann. Statist. 39 1069–1097.
  • Nielsen, M. A. and Chuang, I. L. (2000). Quantum Computation and Quantum Information. Cambridge Univ. Press, Cambridge.
  • Recht, B. (2011). A simpler approach to matrix completion. J. Mach. Learn. Res. 12 3413–3430.
  • Rohde, A. and Tsybakov, A. B. (2011). Estimation of high-dimensional low-rank matrices. Ann. Statist. 39 887–930.
  • Sakurai, J. J. and Napolitano, J. (2010). Modern Quantum Mechanics, 2nd ed. Addison-Wesley, Reading, MA.
  • Shankar, R. (1994). Principles of Quantum Mechanics, 2nd ed. Plenum, New York.
  • Vidakovic, B. (1999). Statistical Modeling by Wavelets. Wiley, New York.
  • Wang, Y. (2002). Asymptotic nonequivalence of Garch models and diffusions. Ann. Statist. 30 754–783.
  • Wang, Y. (2011). Quantum Monte Carlo simulation. Ann. Appl. Stat. 5 669–683.
  • Wang, Y. (2012). Quantum computation and quantum information. Statist. Sci. 27 373–394.
  • Witten, D. M., Tibshirani, R. and Hastie, T. (2009). A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics 10 515–534.