- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review A

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Quantifying multiparticle entanglement with randomized measurements

Sophia Ohnemus1, Heinz-Peter Breuer1,2, and Andreas Ketterer1,2,3,*

  • 1Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
  • 2EUCOR Centre for Quantum Science and Quantum Computing, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
  • 3Fraunhofer Institute for Applied Solid State Physics IAF, Tullastraße 72, 79108 Freiburg, Germany

  • *andreas.ketterer@iaf.fraunhofer.de
PDF

Phys. Rev. A 107, 042406 – Published 5 April, 2023

DOI: https://doi.org/10.1103/PhysRevA.107.042406

Abstract

Randomized measurements constitute a simple measurement primitive that exploits the information encoded in the outcome statistics of samples of local quantum measurements defined through randomly selected bases. In this work we exploit the potential of randomized measurements in order to probe the amount of entanglement contained in multiparticle quantum systems as quantified by the multiparticle concurrence. We further present a detailed statistical analysis of the underlying measurement resources required for a confident estimation of the introduced quantifiers using analytical tools from the theory of random matrices. The introduced framework is demonstrated by a series of numerical experiments analyzing the concurrence of typical multiparticle entangled states as well as of ensembles of output states produced by random quantum circuits. Finally, we examine the multiparticle entanglement of mixed states produced by noisy quantum circuits consisting of single- and two-qubit gates with nonvanishing depolarization errors, thus showing that our framework is directly applicable in the noisy intermediate-scale regime.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (77)

  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, New York, 2000).
  2. H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, Rev. Mod. Phys. 82, 665 (2010).
  3. F. Arute et al., Nature (London) 574, 505 (2019).
  4. M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Annu. Rev. Condens. Matter Phys. 11, 369 (2020).
  5. F. Graselli, G. Murta, J. de Jong, F. Hahn, D. Bruß, H. Kampermann, and A. Pappa, PRX Quantum 3, 040306 (2022).
  6. Z. Ren, W. Li, A. Smerzi, and M. Gessner, Phys. Rev. Lett. 126, 080502 (2021).
  7. M. Van den Nest, Phys. Rev. Lett. 110, 060504 (2013).
  8. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
  9. O. Gühne and G. Tóth, Phys. Rep. 474, 1 (2009).
  10. J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Nat. Rev. Phys. 2, 382 (2020).
  11. R. Blume-Kohout, New J. Phys. 12, 043034 (2010).
  12. I. Šupić and J. Bowles, Quantum 4, 337 (2020).
  13. M. Gluza, M. Kliesch, J. Eisert, and L. Aolita, Phys. Rev. Lett. 120, 190501 (2018).
  14. D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, Phys. Rev. Lett. 105, 150401 (2010).
  15. C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, Nat. Commun. 8, 15305 (2017).
  16. J. Carrasquilla, G. Torlai, R. G. Melko, and L. Aolita, Nat. Mach. Intell. 1, 155 (2019).
  17. Y.-C. Liang, N. Harrigan, S. D. Bartlett, and T. Rudolph, Phys. Rev. Lett. 104, 050401 (2010).
  18. S. T. Flammia and Y.-K. Liu, Phys. Rev. Lett. 106, 230501 (2011).
  19. P. Shadbolt, T. Vértesi, Y.-C. Liang, C. Branciard, N. Brunner, and J. L. O'Brien, Sci. Rep. 2, 470 (2012).
  20. S. J. van Enk and C. W. J. Beenakker, Phys. Rev. Lett. 108, 110503 (2012).
  21. M. C. Tran, B. Dakić, F. Arnault, W. Laskowski, and T. Paterek, Phys. Rev. A 92, 050301(R) (2015).
  22. M. C. Tran, B. Dakić, W. Laskowski, and T. Paterek, Phys. Rev. A 94, 042302 (2016).
  23. M. Walschaers, J. Kuipers, J.-D. Urbina, K. Mayer, M. C. Tichy, K. Richter, and A. Buchleitner, New J. Phys. 18, 032001 (2016).
  24. T. Giordani et al., Nat. Photon. 12, 173 (2018).
  25. A. Ketterer, N. Wyderka, and O. Gühne, Phys. Rev. Lett. 122, 120505 (2019).
  26. A. Ketterer, N. Wyderka, and O. Gühne, Quantum 4, 325 (2020).
  27. M. Krebsbach, M.Sc. thesis, Albert-Ludwigs-Universität Freiburg, 2019, available at https://freidok.uni-freiburg.de/data/150706.
  28. A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 120, 050406 (2018).
  29. B. Vermersch, A. Elben, M. Dalmonte, J. I. Cirac, and P. Zoller, Phys. Rev. A 97, 023604 (2018).
  30. T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Science 364, 260 (2019).
  31. A. Elben, B. Vermersch, C. F. Roos, and P. Zoller, Phys. Rev. A 99, 052323 (2019).
  32. V. Saggio, A. Dimić, C. Greganti, L. A. Rozema, P. Walther, and B. Dakić, Nat. Phys. 15, 935 (2019).
  33. L. Knips, J. Dziewior, W. Kłobus, W. Laskowski, T. Paterek, P. J. Shadbolt, H. Weinfurter, and J. D. A. Meinecke, npj Quantum Inf. 6, 51 (2020).
  34. A. Elben, B. Vermersch, R. van Bijnen, C. Kokail, T. Brydges, C. Maier, M. K. Joshi, R. Blatt, C. F. Roos, and P. Zoller, Phys. Rev. Lett. 124, 010504 (2020).
  35. A. Elben, R. Kueng, H.-Y. R. Huang, R. van Bijnen, C. Kokail, M. Dalmonte, P. Calabrese, B. Kraus, J. Preskill, P. Zoller, and B. Vermersch, Phys. Rev. Lett. 125, 200501 (2020).
  36. S.-X. Yang, G. N. Tabia, P.-S. Lin, and Y.-C. Liang, Phys. Rev. A 102, 022419 (2020).
  37. S. Imai, N. Wyderka, A. Ketterer, and O. Gühne, Phys. Rev. Lett. 126, 150501 (2021).
  38. A. Ketterer, S. Imai, N. Wyderka, and O. Gühne, Phys. Rev. A 106, L010402 (2022).
  39. L. Knips, Quantum Views 4, 47 (2020).
  40. N. Wyderka and A. Ketterer, arXiv:2211.09610.
  41. S. Liu, Q. He, M. Huber, O. Gühne, and G. Vitagliano, arXiv:2211.09614.
  42. N. Wyderka, A. Ketterer, S. Imai, J. L. Bönsel, D. E. Jones, B. T. Kirby, X.-D. Yu, and O. Gühne, arXiv:2212.07894.
  43. S. Aaronson, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (ACM, New York, 2018).
  44. H.-Y. Huang, R. Kueng, and J. Preskill, Nat. Phys. 16, 1050 (2020).
  45. M. Paini, A. Kalev, D. Padilha, and B. Ruck, Quantum 5, 413 (2021).
  46. F. Mintert, M. Kuś, and A. Buchleitner, Phys. Rev. Lett. 92, 167902 (2004).
  47. A. R. R. Carvalho, F. Mintert, and A. Buchleitner, Phys. Rev. Lett. 93, 230501 (2004).
  48. F. Mintert, A. R. R. Carvalho, M. Kuś, and A. Buchleitner, Phys. Rep. 415, 207 (2005).
  49. L. Aolita and F. Mintert, Phys. Rev. Lett. 97, 050501 (2006).
  50. F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98, 140505 (2007).
  51. L. Aolita, A. Buchleitner, and F. Mintert, Phys. Rev. A 78, 022308 (2008).
  52. N. Wyderka and O. Gühne, J. Phys. A: Math. Theor. 53, 345302 (2020).
  53. S. Ohnemus, M.Sc. thesis, Albert-Ludwigs-Universität Freiburg, 2021, available at https://doi.org/10.6094/UNIFR/227071.
  54. Z. Liu, P. Zeng, Y. Zhou, and M. Gu, Phys. Rev. A 105, 022407 (2022).
  55. A. Borras, A. P. Majtey, A. R. Plastino, M. Casas, and A. Plastino, Phys. Rev. A 79, 022112 (2009).
  56. C. Dankert, M.Sc. thesis, University of Waterloo, 2005.
  57. P. D. Seymour and T. Zaslavsky, Adv. Math. 52, 213 (1984).
  58. F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, Phys. Rev. Lett. 116, 170502 (2016).
  59. Y. Nakata, C. Hirche, M. Koashi, and A. Winter, Phys. Rev. X 7, 021006 (2017).
  60. J. Haferkamp, F. Montealegre-Mora, M. Heinrich, J. Eisert, D. Gross, and I. Roth, Commun. Math. Phys. 397, 995 (2023).
  61. Z. Webb, Quantum Inf. Comput. 16, 1379 (2016).
  62. H. Zhu, R. Kueng, M. Grassl, and D. Gross, arXiv:1609.08172.
  63. K. D. Schmidt, Maßund Wahrscheinlichkeit (Springer, Heidelberg, 2011).
  64. N. Ullah, Nucl. Phys. 58, 65 (1964).
  65. D. Petz and J. Réffy, Period. Math. Hung. 49, 103 (2004).
  66. B. Collins and P. Sniady, Commun. Math. Phys. 264, 773 (2006).
  67. D. Weingarten, J. Math. Phys. 19, 999 (1978).
  68. M. Ledoux, The Concentration of Measure Phenomenon (American Mathematical Society, Providence, 2001).
  69. M. Tiersch, F. de Melo, and A. Buchleitner, J. Phys. A: Math. Theor. 46, 085301 (2013).
  70. R. Jozsa and A. Miyake, Proc. R. Soc. A 464, 3089 (2008).
  71. D. J. Brod and A. M. Childs, Quantum Inf. Comput. 14, 901 (2014).
  72. B. M. Terhal and D. P. DiVincenzo, Phys. Rev. A 65, 032325 (2002).
  73. D. Shepherd and M. J. Bremner, Proc. R. Soc. A 465, 1413 (2009).
  74. M. J. Bremner, R. Josza, and D. Shepherd, Proc. R. Soc. A 467, 459 (2011).
  75. Y. Nakata and M. Murao, Eur. Phys. J. Plus 129, 152 (2014).
  76. R. O. Vallejos, F. de Melo, and G. G. Carlo, Phys. Rev. A 104, 012602 (2021).
  77. K. Fujii and T. Morimae, New J. Phys. 19, 033003 (2017).

Outline

Information

Phys. Rev. A 107, 042406– Published 5 April, 2023
  • Received 13 December 2022
  • Accepted 13 March 2023
Crossmark - Check for updates button

©2023 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation