- 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
  • Letter
  • Open Access

Lower bounds on entanglement entropy without twin copy

Yannick Meurice

PDF

Phys. Rev. Research 7, L022023 – Published 24 April, 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.L022023

Abstract

We discuss the possibility of estimating experimentally the von Neumann entanglement entropy SAvN of a symmetric bipartite quantum system AB by using the basic measurement counts (bitstrings) for a single copy of a prepared state. Using exact diagonalization and analog simulations performed with the publicly available QuEra facilities for chains and ladders of Rydberg atoms, we calculate the Shannon entropy SABX associated with the bitstrings of adiabatically prepared ground states and the reduced entropies SAX and SBX obtained from the marginal probabilities in A and B. We then calculate the classical mutual information IABX=SAX+SBX−SABX, which is a lower bound on SAvN. We show that for a broad range of lattice spacing and detuning, IABX is typically 20% below SAvN in regions where SAvN is large and a less close bound in regions where SAvN is low. We argue that this use of the easily available bitstrings provides a robust and efficient way to explore empirically the phase diagram of qubit-based quantum simulators and identify critical regions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (59)

  1. L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80, 517 (2008).
  2. J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010).
  3. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  4. J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  5. S. Ghosh, R. M. Soni, and S. P. Trivedi, On the entanglement entropy for gauge theories, J. High Energy Phys. 09 (2015) 069 .
  6. K. Van Acoleyen, N. Bultinck, J. Haegeman, M. Marien, V. B. Scholz, and F. Verstraete, The entanglement of distillation for gauge theories, Phys. Rev. Lett. 117, 131602 (2016).
  7. M. C. Bañuls, K. Cichy, J. I. Cirac, K. Jansen, and S. Kühn, Efficient basis formulation for 1+1 dimensional SU(2) lattice gauge theory: Spectral calculations with matrix product states, Phys. Rev. X 7, 041046 (2017).
  8. J. Knaute, M. Feuerstein, and E. Zohar, Entanglement and confinement in lattice gauge theory tensor networks, J. High Energy Phys. 02 (2024) 174.
  9. D. E. Kharzeev and E. M. Levin, Deep inelastic scattering as a probe of entanglement, Phys. Rev. D 95, 114008 (2017).
  10. O. K. Baker and D. E. Kharzeev, Thermal radiation and entanglement in proton-proton collisions at energies available at the CERN large hadron collider, Phys. Rev. D 98, 054007 (2018).
  11. K. Zhang, K. Hao, D. Kharzeev, and V. Korepin, Entanglement entropy production in deep inelastic scattering, Phys. Rev. D 105, 014002 (2022).
  12. S. R. Beane, D. B. Kaplan, N. Klco, and M. J. Savage, Entanglement suppression and emergent symmetries of strong interactions, Phys. Rev. Lett. 122, 102001 (2019).
  13. C. Robin, M. J. Savage, and N. Pillet, Entanglement rearrangement in self-consistent nuclear structure calculations, Phys. Rev. C 103, 034325 (2021).
  14. G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90, 227902 (2003).
  15. V. E. Korepin, Universality of entropy scaling in one dimensional gapless models, Phys. Rev. Lett. 92, 096402 (2004).
  16. P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory: A Non-technical introduction, Int. J. Quant. Inf. 04, 429 (2006).
  17. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006).
  18. D. A. Abanin and E. Demler, Measuring entanglement entropy of a generic many-body system with a quantum switch, Phys. Rev. Lett. 109, 020504 (2012).
  19. A. J. Daley, H. Pichler, J. Schachenmayer, and P. Zoller, Measuring entanglement growth in quench dynamics of bosons in an optical lattice, Phys. Rev. Lett. 109, 020505 (2012).
  20. R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Rispoli, and M. Greiner, Measuring entanglement entropy in a quantum many-body system, Nature (London) 528, 77 (2015).
  21. A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Quantum thermalization through entanglement in an isolated many-body system, Science 353, 794 (2016).
  22. J. Unmuth-Yockey, J. Zhang, P. M. Preiss, L.-P. Yang, S. W. Tsai, and Y. Meurice, Probing the conformal Calabrese-Cardy scaling with cold atoms, Phys. Rev. A 96, 023603 (2017).
  23. C. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Urbana and Chicago, 1998).
  24. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, UK, 2011).
  25. J. Preskill, Quantum Information (Cambridge University Press, Cambridge, UK, 2022), Chap. 10.
  26. J. Wurtz, A. Bylinskii, B. Braverman, J. Amato-Grill, S. H. Cantu, F. Huber, A. Lukin, F. Liu, P. Weinberg, J. Long, S.-T. Wang, N. Gemelke, and A. Keesling, Aquila: QuEra's 256-qubit neutral-atom quantum computer, arXiv:2306.11727.
  27. E. Witten, A mini-introduction to information theory, Riv. Nuovo Cim. 43, 187 (2020).
  28. J. Preskill and M. Soleimanifar (private communication).
  29. A. Kaufman, J. Corona, Z. Ozzello, M. Asaduzzaman, and Y. Meurice, Improved entanglement entropy estimates from filtered bitstring probabilities, arXiv:2411.07092.
  30. Z. Ozzello and Y. Meurice (unpublished).
  31. H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature (London) 551, 579 (2017).
  32. A. Keesling, A. Omran, H. Levine, H. Bernien, H. Pichler, S. Choi, R. Samajdar, S. Schwartz, P. Silvi, S. Sachdev, P. Zoller, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg simulator, Nature (London) 568, 207 (2019).
  33. H. Labuhn, D. Barredo, S. Ravets, S. de Léséleuc, T. Macrì, T. Lahaye, and A. Browaeys, Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models, Nature (London) 534, 667 (2016).
  34. S. de Léséleuc, V. Lienhard, P. Scholl, D. Barredo, S. Weber, N. Lang, H. P. Büchler, T. Lahaye, and A. Browaeys, Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms, Science 365, 775 (2019).
  35. S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuletić, and M. D. Lukin, Quantum phases of matter on a 256-atom programmable quantum simulator, Nature (London) 595, 227 (2021).
  36. P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard, L.-P. Henry, T. C. Lang, T. Lahaye, A. M. Läuchli, and A. Browaeys, Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms, Nature (London) 595, 233 (2021).
  37. J. Zhang, S. H. Cantú, F. Liu, A. Bylinskii, B. Braverman, F. Huber, J. Amato-Grill, A. Lukin, N. Gemelke, A. Keesling, S.-T. Wang, Y. Meurice, and S.-W. Tsai, Probing quantum floating phases in Rydberg atom arrays, Nat. Commun. 16, 712 (2025).
  38. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.7.L022023 for details about symmetric bi-partite Hilbert spaces, features of the entropies, verification of the bound for two qubits and other bitstring proxies (Renyi and configurational).
  39. Y. Meurice, Lower bounds on entanglement entropy without twin copy [Data set], Zenodo (2025), https://doi.org/10.5281/zenodo.15103785.
  40. This was pointed out to us by Alex Lukin.
  41. A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Léonard, and M. Greiner, Probing entanglement in a many-body–localized system, Science 364, 256 (2019).
  42. G. Torlai, B. Timar, E. P. L. van Nieuwenburg, H. Levine, A. Omran, A. Keesling, H. Bernien, M. Greiner, V. Vuletić, M. D. Lukin, R. G. Melko, and M. Endres, Integrating neural networks with a quantum simulator for state reconstruction, Phys. Rev. Lett. 123, 230504 (2019).
  43. J. Zhang, J. Unmuth-Yockey, J. Zeiher, A. Bazavov, S.-W. Tsai, and Y. Meurice, Quantum simulation of the universal features of the polyakov loop, Phys. Rev. Lett. 121, 223201 (2018).
  44. F. M. Surace, P. P. Mazza, G. Giudici, A. Lerose, A. Gambassi, and M. Dalmonte, Lattice gauge theories and string dynamics in Rydberg atom quantum simulators, Phys. Rev. X 10, 021041 (2020).
  45. S. Notarnicola, M. Collura, and S. Montangero, Real-time-dynamics quantum simulation of (1+1)-dimensional lattice QED with Rydberg atoms, Phys. Rev. Res. 2, 013288 (2020).
  46. A. Celi, B. Vermersch, O. Viyuela, H. Pichler, M. D. Lukin, and P. Zoller, Emerging two-dimensional gauge theories in Rydberg configurable arrays, Phys. Rev. X 10, 021057 (2020).
  47. Y. Meurice, Theoretical methods to design and test quantum simulators for the compact Abelian Higgs model, Phys. Rev. D 104, 094513 (2021).
  48. P. Fromholz, M. Tsitsishvili, M. Votto, M. Dalmonte, A. Nersesyan, and T. Chanda, Phase diagram of Rydberg-dressed atoms on two-leg triangular ladders, Phys. Rev. B 106, 155411 (2022).
  49. D. González-Cuadra, T. V. Zache, J. Carrasco, B. Kraus, and P. Zoller, Hardware efficient quantum simulation of non-Abelian gauge theories with qudits on Rydberg platforms, Phys. Rev. Lett. 129, 160501 (2022).
  50. K. Heitritter, Y. Meurice, and S. Mrenna, Prolegomena to a hybrid classical/Rydberg simulator for hadronization (QuPYTH), arXiv:2212.02476.
  51. C. W. Bauer et al., Quantum simulation for high-energy physics, PRX Quantum 4, 027001 (2023).
  52. J. C. Halimeh, M. Aidelsburger, F. Grusdt, P. Hauke, and B. Yang, Cold-atom quantum simulators of gauge theories, arXiv:2310.12201.
  53. J. Zhang, S. W. Tsai, and Y. Meurice, Critical behavior of lattice gauge theory Rydberg simulators from effective Hamiltonians, Phys. Rev. D 110, 034513 (2024).
  54. R. D. Pisarski, V. V. Skokov, and A. M. Tsvelik, A pedagogical introduction to the Lifshitz regime, Universe 5, 48 (2019).
  55. T. Kojo, Y. Hidaka, L. McLerran, and R. D. Pisarski, Quarkyonic chiral spirals, Nucl. Phys. A 843, 37 (2010).
  56. M. Headrick and T. Takayanagi, A Holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76, 106013 (2007).
  57. P. Hayden, M. Headrick, and A. Maloney, Holographic mutual information is monogamous, Phys. Rev. D 87, 046003 (2013).
  58. T.-C. Lin and J. McGreevy, Conformal field theory ground states as critical points of an entropy function, Phys. Rev. Lett. 131, 251602 (2023).
  59. M. Asaduzzaman, S. Catterall, A. Maloney, Y. Meurice, A. Samlodia, and G. C. Toga (unpublished).

Outline

Information

Phys. Rev. Research 7, L022023– Published 24 April, 2025
  • Received 15 July 2024
  • Revised 27 October 2024
  • Accepted 1 April 2025
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