- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Adaptive negativity estimation via collective measurements

Martin Zeman1,2,*, Vojtěch Trávníček2,†, Antonín Černoch2,‡, Jan Soubusta1,§, and Karel Lemr1,∥

  • *Contact author: martin.zeman01@upol.cz
  • †Contact author: travnicekv@fzu.cz
  • ‡Contact author: acernoch@fzu.cz
  • §Contact author: jan.soubusta@upol.cz
  • ∥Contact author: k.lemr@upol.cz
PDF

Phys. Rev. Research 8, 043152 – Published 10 November, 2026

DOI: https://doi.org/10.1103/jt3b-78qw

Abstract

This paper explores an efficient method for entanglement quantification in two-qubit and qubit-qutrit quantum systems based upon the framework of collective measurements in conjunction with machine learning. We introduce an adaptive measurement procedure in which measurement settings are dynamically adjusted based on prior measurement outcomes aiming to optimize the inference precision given a limited number of these measurement settings. The procedure makes use of the long short-term memory networks to recurrently process collective measurements on two copies of the investigated states. Obtained results demonstrate the tangible benefits of the adaptive measurements in comparison to previously described nonadaptive strategies.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (61)

  1. A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935).
  2. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, Experimental quantum teleportation, Nature (London) 390, 575 (1997).
  3. J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, Experimental entanglement swapping: Entangling photons that never interacted, Phys. Rev. Lett. 80, 3891 (1998).
  4. A. K. Ekert, Quantum cryptography based on Bell's theorem, Phys. Rev. Lett. 67, 661 (1991).
  5. B. C. Hiesmayr, Free versus bound entanglement, a NP-hard problem tackled by machine learning, Sci. Rep. 11, 19739 (2021).
  6. S. Gharibian, Strong NP-hardness of the quantum separability problem, Quantum Inf. Comput. 10, 343 (2010).
  7. G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  8. A. Peres, How to differentiate between non-orthogonal states, Phys. Lett. A 128, 19 (1988).
  9. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  10. G. Jaeger, Quantum Information: An Overview (Springer, New York, 2010).
  11. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  12. M. Ježek, J. Fiurášek, and Z. Hradil, Quantum inference of states and processes, Phys. Rev. A 68, 012305 (2003).
  13. A. Miranowicz, K. Bartkiewicz, J. Peřina, M. Koashi, N. Imoto, and F. Nori, Optimal two-qubit tomography based on local and global measurements: Maximal robustness against errors as described by condition numbers, Phys. Rev. A 90, 062123 (2014).
  14. A. Acín, R. Tarrach, and G. Vidal, Optimal estimation of two-qubit pure-state entanglement, Phys. Rev. A 61, 062307 (2000).
  15. F. A. Bovino, G. Castagnoli, A. Ekert, P. Horodecki, C. M. Alves, and A. V. Sergienko, Direct measurement of nonlinear properties of bipartite quantum states, Phys. Rev. Lett. 95, 240407 (2005).
  16. S. Walborn, P. S. Ribeiro, L. Davidovich, F. Mintert, and A. Buchleitner, Experimental determination of entanglement with a single measurement, Nature (London) 440, 1022 (2006).
  17. Ł. Rudnicki, P. Horodecki, and K. Życzkowski, Collective uncertainty entanglement test, Phys. Rev. Lett. 107, 150502 (2011).
  18. Ł. Rudnicki, Z. Puchała, P. Horodecki, and K. Życzkowski, Collectibility for mixed quantum states, Phys. Rev. A 86, 062329 (2012).
  19. K. Lemr, K. Bartkiewicz, and A. Černoch, Experimental measurement of collective nonlinear entanglement witness for two qubits, Phys. Rev. A 94, 052334 (2016).
  20. P. Horodecki and A. Ekert, Method for direct detection of quantum entanglement, Phys. Rev. Lett. 89, 127902 (2002).
  21. R. Filip, Overlap and entanglement-witness measurements, Phys. Rev. A 65, 062320 (2002).
  22. J. Fiurášek, Structural physical approximations of unphysical maps and generalized quantum measurements, Phys. Rev. A 66, 052315 (2002).
  23. K. Bartkiewicz, G. Chimczak, and K. Lemr, Direct method for measuring and witnessing quantum entanglement of arbitrary two-qubit states through Hong-Ou-Mandel interference, Phys. Rev. A 95, 022331 (2017).
  24. J. Roik, K. Bartkiewicz, A. Černoch, and K. Lemr, Accuracy of entanglement detection via artificial neural networks and human-designed entanglement witnesses, Phys. Rev. Appl. 15, 054006 (2021).
  25. J. Roik, K. Bartkiewicz, A. Černoch, and K. Lemr, Entanglement quantification from collective measurements processed by machine learning, Phys. Lett. A 446, 128270 (2022).
  26. K. Hornik, M. Stinchcombe, and H. White, Multilayer feedforward networks are universal approximators, Neural Netw. 2, 359 (1989).
  27. H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, Quantum repeaters: The role of imperfect local operations in quantum communication, Phys. Rev. Lett. 81, 5932 (1998).
  28. B. C. Jacobs, T. B. Pittman, and J. D. Franson, Quantum relays and noise suppression using linear optics, Phys. Rev. A 66, 052307 (2002).
  29. R. Bassoli, H. Boche, C. Deppe, R. Ferrara, F. H. P. Fitzek, G. Janssen, and S. Saeedinaeeni, Quantum Communication Networks (Springer, Nature Cham, 2021).
  30. V. Trávníček, K. Bartkiewicz, A. Černoch, and K. Lemr, Experimental diagnostics of entanglement swapping by a collective entanglement test, Phys. Rev. Appl. 14, 064071 (2020).
  31. L. O. Conlon, T. Vogl, C. D. Marciniak, I. Pogorelov, S. K. Yung, F. Eilenberger, D. W. Berry, F. S. Santana, R. Blatt, T. Monz, P. K. Lam, and S. M. Assad, Approaching optimal entangling collective measurements on quantum computing platforms, Nat. Phys. 19, 351 (2023).
  32. V. Trávníček, K. Bartkiewicz, A. Černoch, and K. Lemr, Experimental measurement of the Hilbert-Schmidt distance between two-qubit states as a means for reducing the complexity of machine learning, Phys. Rev. Lett. 123, 260501 (2019).
  33. V. Trávníček, J. Roik, K. Bartkiewicz, A. Černoch, P. Horodecki, and K. Lemr, Sensitivity versus selectivity in entanglement detection via collective witnesses, Phys. Rev. Res. 6, 033056 (2024).
  34. S. Abo, J. Soubusta, K. Jiráková, K. Bartkiewicz, A. Černoch, K. Lemr, and A. Miranowicz, Experimental hierarchy of two-qubit quantum correlations without state tomography, Sci. Rep. 13, 8564 (2023).
  35. J. Soubusta, A. Černoch, and K. Lemr, Deep-learned classification of quantum correlations from collective measurements, Phys. Lett. A 559, 130911 (2025).
  36. A. Hentschel and B. C. Sanders, Machine learning for adaptive quantum measurement, in 2010 Seventh International Conference on Information Technology: New Generations (IEEE, Las Vegas, NV, USA, 2010), pp. 506–511.
  37. D. G. Fischer, S. H. Kienle, and M. Freyberger, Quantum-state estimation by self-learning measurements, Phys. Rev. A 61, 032306 (2000).
  38. T. Hannemann, D. Reiss, C. Balzer, W. Neuhauser, P. E. Toschek, and C. Wunderlich, Self-learning estimation of quantum states, Phys. Rev. A 65, 050303 (2002).
  39. C. J. Happ and M. Freyberger, Adaptive estimation of qubits by symmetry measurements, Phys. Rev. A 78, 064303 (2008).
  40. F. Huszár and N. M. T. Houlsby, Adaptive Bayesian quantum tomography, Phys. Rev. A 85, 052120 (2012).
  41. G. I. Struchalin, I. A. Pogorelov, S. S. Straupe, K. S. Kravtsov, I. V. Radchenko, and S. P. Kulik, Experimental adaptive quantum tomography of two-qubit states, Phys. Rev. A 93, 012103 (2016).
  42. Y. Quek, S. Fort, and H. K. Ng, Adaptive quantum state tomography with neural networks, npj Quantum Inf. 7, 105 (2021).
  43. A. Gaikwad, O. Bihani, Arvind, and K. Dorai, Neural-network-assisted quantum state and process tomography using limited data sets, Phys. Rev. A 109, 012402 (2024).
  44. D. T. Pope, H. M. Wiseman, and N. K. Langford, Adaptive phase estimation is more accurate than nonadaptive phase estimation for continuous beams of light, Phys. Rev. A 70, 043812 (2004).
  45. N. Linden and R. de Wolf, Adaptive quantum phase estimation can be better than non-adaptive, arXiv:2511.05372.
  46. J. G. Smith, C. H. W. Barnes, and D. R. M. Arvidsson-Shukur, Adaptive Bayesian quantum algorithm for phase estimation, Phys. Rev. A 109, 042412 (2024).
  47. S. Lerch and A. Stefanov, Adaptive quantum state estimation of an entangled qubit state, Opt. Lett. 39, 5399 (2014).
  48. M. Weilenmann, E. A. Aguilar, and M. Navascués, Analysis and optimization of quantum adaptive measurement protocols with the framework of preparation games, Nat. Commun. 12, 4553 (2021).
  49. J. Pauwels, S. Pironio, E. Z. Cruzeiro, and A. Tavakoli, Adaptive advantage in entanglement-assisted communications, Phys. Rev. Lett. 129, 120504 (2022).
  50. S. Hochreiter and J. Schmidhuber, Long short-term memory, Neural Comput. 9, 1735 (1997).
  51. R. Pascanu, Y. Yere, T. Mikolov, and Y. Rachmad, On the difficulty of training recurrent neural networks, in 30th International Conference on Machine Learning, ICML 2013, Cycle 3 (Proceedings of Machine Learning Research (PMLR), 2012), 28, pp. 1310–1318.
  52. J. Maziero, Random sampling of quantum states: A survey of methods, Braz. J. Phys. 45, 575 (2015).
  53. M. Zeman, V. Trávníček, A. Černoch, J. Soubusta, and K. Lemr, Programming codes and neural network models (checkpoints), Figshare repository, 2026, https://doi.org/10.6084/m9.figshare.31861405.
  54. S. L. Smith, P.-J. Kindermans, C. Ying, and Q. V. Le, Don't decay the learning rate, increase the batch size, arXiv:1711.00489.
  55. A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al., PyTorch: An imperative style, high-performance deep learning library, arXiv:1912.01703.
  56. K. Bartkiewicz, K. Lemr, A. Černoch, and A. Miranowicz, Bell nonlocality and fully entangled fraction measured in an entanglement-swapping device without quantum state tomography, Phys. Rev. A 95, 030102(R) (2017).
  57. B. C. Sanctuary, Separation of Bell states, arXiv:0705.3657.
  58. E. R. Kaločová, Quantum tomography and density matrix estimation, thesis, Palacky University, 2026.
  59. A. Černoch, K. Bartkiewicz, K. Lemr, and J. Soubusta, Experimental tests of coherence and entanglement conservation under unitary evolutions, Phys. Rev. A 97, 042305 (2018).
  60. K. Jiráková, A. Barasiński, A. Černoch, K. Lemr, and J. Soubusta, Measuring concurrence in qubit Werner states without an aligned reference frame, Phys. Rev. Appl. 16, 054042 (2021).
  61. J. Kadlec, A. Barasiński, and K. Lemr, Resource-efficient entanglement detection in high-dimensional states via two-qubit witnesses, Phys. Rev. Res. 8, 013195 (2026).

Outline

Information

Phys. Rev. Research 8, 043152– Published 10 November, 2026
  • Received 26 March 2026
  • Accepted 7 October 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation