Advertisement

Brazilian Journal of Physics

, Volume 45, Issue 6, pp 575–583 | Cite as

Random Sampling of Quantum States: a Survey of Methods

And Some Issues Regarding the Overparametrized Method
  • Jonas MazieroEmail author
Atomic Physics

Abstract

The numerical generation of random quantum states (RQS) is an important procedure for investigations in quantum information science. Here, we review some methods that may be used for performing that task. We start by presenting a simple procedure for generating random state vectors, for which the main tool is the random sampling of unbiased discrete probability distributions (DPD). Afterwards, the creation of random density matrices is addressed. In this context, we first present the standard method, which consists in using the spectral decomposition of a quantum state for getting RQS from random DPDs and random unitary matrices. In the sequence, the Bloch vector parametrization method is described. This approach, despite being useful in several instances, is not in general convenient for RQS generation. In the last part of the article, we regard the overparametrized method (OPM) and the related Ginibre and Bures techniques. The OPM can be used to create random positive semidefinite matrices with unit trace from randomly produced general complex matrices in a simple way that is friendly for numerical implementations. We consider a physically relevant issue related to the possible domains that may be used for the real and imaginary parts of the elements of such general complex matrices. Subsequently, a too fast concentration of measure in the quantum state space that appears in this parametrization is noticed.

Keywords

Random quantum states Numerical generation Overparametrized method Concentration of measure 

Notes

Acknowledgments

This work was supported by the Brazilian funding agencies: Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), under processes 441875/2014-9 and 303496/2014-2, Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ), under process 2008/57856-6, and Coordenação de Desenvolvimento de Pessoal de Nível Superior (CAPES), under process 6531/2014-08. I gratefully acknowledge the hospitality of the Laser Spectroscopy Group at the Universidad de la República, Uruguay, where this article was completed.

References

  1. 1.
    P. Benioff, The computer as a physical system: a microscopic quantum mechanical hamiltonian model of computers as represented by Turing machines. J. Stat. Phys. 22, 563 (1980)MathSciNetCrossRefADSGoogle Scholar
  2. 2.
    P. Benioff, Quantum mechanical models of Turing machines that dissipate no energy. Phys. Rev. Lett. 48, 1581 (1982)MathSciNetCrossRefADSGoogle Scholar
  3. 3.
    R.P. Feynman, Simulating physics with computers. Int. J. Theor. Phys. 21, 467 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    R.P. Feynman, Quantum mechanical computers. Opt. News. 11, 11 (1985)CrossRefGoogle Scholar
  5. 5.
    C.H. Bennett, D.P. DiVincenzo, Quantum information and computation. Nature. 404, 247 (2000)CrossRefADSGoogle Scholar
  6. 6.
    T.D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, J.L. O Brien, Quantum computers. Nature. 464, 45 (2010)CrossRefADSGoogle Scholar
  7. 7.
    I.M. Georgescu, S. Ashhab, F. Nori, Quantum simulation. Rev. Mod. Phys. 86, 153 (2014)CrossRefADSGoogle Scholar
  8. 8.
    A. Ekert, R. Renner, The ultimate physical limits of privacy. Nature. 507, 443 (2014)CrossRefADSGoogle Scholar
  9. 9.
    N. Lambert, Y.-N. Chen, Y.-C. Cheng, C.-M. Li, G.-Y. Chen, F. Nori, Quantum biology. Nat. Phys. 9, 10 (2013)CrossRefGoogle Scholar
  10. 10.
    C. Jarzynski, Diverse phenomena, common themes. Nat. Phys. 11, 105 (2015)CrossRefGoogle Scholar
  11. 11.
    M. Schuld, I. Sinayskiy, F. Petruccione, An introduction to quantum machine learning. Contemp. Phys. 56, 172 (2015)CrossRefADSGoogle Scholar
  12. 12.
    S. Trotzky, Y-A. Chen, A. Flesch, I.P. McCulloch, U. Schollwöck, J. Eisert, I. Bloch, Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas. Nat. Phys. 8, 325 (2012)CrossRefGoogle Scholar
  13. 13.
    J. Preskill, Quantum information and physics: some future directions. J. Mod. Opt. 47, 127 (2000)MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    S. Aaronson, How might quantum information transform our future? https://www.bigquestionsonline.com/content/how-might-quantum-information-transform-our-future (2014)
  15. 15.
    J. Grondalski, D.M. Etlinger, D.F.V. James, The fully entangled fraction as an inclusive measure of entanglement applications. Phys. Lett. A. 300, 573 (2002)zbMATHMathSciNetCrossRefADSGoogle Scholar
  16. 16.
    R.V. Ramos, Numerical algorithms for use in quantum information. J. Comput. Phys. 192, 95 (2003)zbMATHCrossRefADSGoogle Scholar
  17. 17.
    D. Girolami, G. Adesso, Quantum discord for general two-qubit states: analytical progress. Phys. Rev. A. 83, 052108 (2011)CrossRefADSGoogle Scholar
  18. 18.
    J. Batle, M. Casas, A.R. Plastino, A. Plastino, Entanglement, mixedness, and q-entropies. Phys. Lett. A. 296, 251 (2002)zbMATHMathSciNetCrossRefADSGoogle Scholar
  19. 19.
    M. Roncaglia, A. Montorsi, M. Genovese, Bipartite entanglement of quantum states in a pair basis. Phys. Rev. A. 90, 062303 (2014)CrossRefADSGoogle Scholar
  20. 20.
    S. Vinjanampathy, A.R.P. Rau, Quantum discord for qubit-qudit systems. J. Phys. A Math. Theor. 45, 095303 (2012)MathSciNetCrossRefADSGoogle Scholar
  21. 21.
    X.-M. Lu, J. Ma, Z. Xi, X. Wang, Optimal measurements to access classical correlations of two-qubit states. Phys. Rev. A. 83, 012327 (2011)CrossRefADSGoogle Scholar
  22. 22.
    F.M. Miatto, K. Piché, T. Brougham, R.W Boyd, The optimal bound of quantum erasure with limited means. arXiv:2313.1410
  23. 23.
    F.M. Miatto, K. Piché, T. Brougham, R.W Boyd, Recovering full coherence in a qubit by measuring half of its environment. arXiv:1502.07030
  24. 24.
    W.K. Wootters, Random quantum states. Found. Phys. 20, 1365 (1990)MathSciNetCrossRefADSGoogle Scholar
  25. 25.
    M.J.W. Hall, Random quantum correlations, density operator distributions. Phys. Lett. A. 242, 123 (1998)zbMATHMathSciNetCrossRefADSGoogle Scholar
  26. 26.
    I. Nechita, Asymptotics of random density matrices. Ann. Henri Poincaré. 8, 1521 (2007)zbMATHMathSciNetCrossRefADSGoogle Scholar
  27. 27.
    C. Nadal, S.N. Majumdar, M. Vergassola, Statistical distribution of quantum entanglement for a random bipartite state. J. Stat. Phys. 142, 403 (2011)zbMATHMathSciNetCrossRefADSGoogle Scholar
  28. 28.
    A. Hamma, S. Santra, P. Zanardi, Quantum entanglement in random physical states. Phys. Rev. Lett. 109, 040502 (2012)CrossRefADSGoogle Scholar
  29. 29.
    S. Agarwal, S.M.H. Rafsanjani, Maximizing genuine multipartite entanglement of n mixed qubits. Int. J. Quant. Inf. 11, 1350043 (2013)CrossRefGoogle Scholar
  30. 30.
    F.D. Cunden, P. Facchi, G. Florio, Polarized ensembles of random pure states. J. Phys A: Math. Theor. 46, 315306 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    M.B. Hastings, Superadditivity of communication capacity using entangled inputs. arXiv:0809.3972
  32. 32.
    E.T. Jaynes. Theory Probability: The Logic of Science (Cambridge University Press, New York, 2003)CrossRefGoogle Scholar
  33. 33.
    D.P. Landau, K. Binder. A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, Cambridge, 2009)zbMATHCrossRefGoogle Scholar
  34. 34.
    T.M. Cover, J.A. Thomas. Elements of Information Theory (John Wiley, New Jersey, 2006)zbMATHGoogle Scholar
  35. 35.
    M.A. Carlton, J.L. Devore. Probability with Applications in Engineering, Science, and Technology (Springer, New York , 2014)zbMATHCrossRefGoogle Scholar
  36. 36.
    E. Brüning, H. Mäkelä, A. Messina, F. Petruccione, Parametrizations of density matrices. J. Mod. Opt. 59, 1 (2012)CrossRefADSGoogle Scholar
  37. 37.
    T. Radtke, S. Fritzsche, Simulation of n-qubit quantum systems. IV. Parametrizations of quantum states, matrices and probability distributions. Comput. Phys. Commun. 179, 647 (2008)zbMATHMathSciNetCrossRefADSGoogle Scholar
  38. 38.
    V. Vedral, M.B. Plenio, Entanglement measures and purification procedures. Phys. Rev. A. 57, 1619 (1998)CrossRefADSGoogle Scholar
  39. 39.
    J. Maziero, Generating pseudo-random discrete probability distributions. Braz. J. Phys. 45, 377 (2015)CrossRefADSGoogle Scholar
  40. 40.
    M.A. Nielsen, I.L. Chuang. Quantum Computation and Quantum Information (Cambridge University Press, Cambridge , 2000)zbMATHGoogle Scholar
  41. 41.
    M.M. Wilde. Quantum Information Theory (Cambridge University Press, Cambridge, 2013)zbMATHCrossRefGoogle Scholar
  42. 42.
    G.W. Stewart, The efficient generation of random orthogonal matrices with an application to condition estimators. SIAM J. Numer. Anal. 17, 403 (1980)zbMATHMathSciNetCrossRefADSGoogle Scholar
  43. 43.
    K. życzkowski, M. Kuś, Random unitary matrices. J. Phys. A: Math. Gen. 27, 4235 (1994)CrossRefADSGoogle Scholar
  44. 44.
    J. Emerson. Y.S. Weinstein, M. Saraceno, S. Lloyd, D.G. Cory, Pseudo-random unitary operators for quantum information processing. Science. 302, 2098 (2003)zbMATHMathSciNetCrossRefADSGoogle Scholar
  45. 45.
    J. Shang, Y.-L. Seah, H.K. Ng, D.J. Nott, B.-G. Englert, Monte Carlo sampling from the quantum state space. I. New J. Phys. 17, 043017 (2015)CrossRefADSGoogle Scholar
  46. 46.
    Y.-L. Seah, J. Shang, H.K. Ng, D.J. Nott, B.-G. Englert, Monte Carlo sampling from the quantum state space. II. New J. Phys. 17, 043018 (2015)CrossRefADSGoogle Scholar
  47. 47.
    J. Maziero, Distribution of mutual information in multipartite states. Braz. J. Phys. 44, 194 (2014)CrossRefADSGoogle Scholar
  48. 48.
    L. Aolita, F. de Melo, L. Davidovich, Open-system dynamics of entanglement: a key issues review. Rep. Prog. Phys. 78, 042001 (2015)CrossRefADSGoogle Scholar
  49. 49.
    L.C. Céleri, J. Maziero, R.M. Serra, Theoretical and experimental aspects of quantum discord and related measures. Int. J. Quant. Inf. 9, 1837 (2011)zbMATHCrossRefGoogle Scholar
  50. 50.
    T. Baumgratz, M. Cramer, M.B. Plenio, Quantifying coherence. Phys Rev. Lett. 113, 140401 (2014)CrossRefADSGoogle Scholar
  51. 51.
    D. Girolami, Observable measure of quantum coherence in finite dimensional systems. Phys. Rev. Lett. 113, 170401 (2014)CrossRefADSGoogle Scholar
  52. 52.
    F. Caruso, V. Giovannetti, C. Lupo, S. Mancini, Quantum channels and memory effects. Rev. Mod. Phys. 86, 1203 (2014)CrossRefADSGoogle Scholar
  53. 53.
    M. Matsumoto, T. Nishimura, Mersenne Twister: a 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Trans Model. Comput. Sim. 8, 3 (1998)zbMATHCrossRefGoogle Scholar
  54. 54.
    E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen. LAPACK Users’ Guide, 3rd (Society for Industrial and Applied Mathematics, Philadelphia, 1999)CrossRefGoogle Scholar
  55. 55.
    J.A. Miszczak, Generating and using random quantum states in Mathematica. Comput. Phys. Commun. 183, 118 (2012)zbMATHCrossRefADSGoogle Scholar
  56. 56.
    M. Ledoux, The concentration of measure phenomenon. Mathematical Surveys and Monographs of the American Mathematical Society. 89 (2001)Google Scholar
  57. 57.
    P. Hayden, in Concentration of measure effects in quantum information. Proceedings of Symposia in Applied Mathematics, Vol. 68, (2010), p. 3Google Scholar
  58. 58.
    K. życzkowski, K.A. Penson, I. Nechita, B. Collins, Generating random density matrices. J. Math. Phys. 52, 062201 (2011)MathSciNetCrossRefADSGoogle Scholar
  59. 59.
    I. Bengtsson, K. życzkowski. Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, 2007)Google Scholar
  60. 60.
    V. Al Osipov, H.-J. Sommers, K. życzkowski, Random Bures mixed states and the distribution of their purity. J. Phys. A: Math. Theor. 43, 055302 (2010)CrossRefADSGoogle Scholar

Copyright information

© Sociedade Brasileira de Física 2015

Authors and Affiliations

  1. 1.Departamento de Física, Centro de Ciências Naturais e ExatasUniversidade Federal de Santa MariaSanta MariaBrazil
  2. 2.Instituto de Física, Facultad de IngenieríaUniversidad de la RepúblicaMontevideoUruguay

Personalised recommendations