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Exploring large-scale entanglement in quantum simulation

Nature volume 624, pages 539–544 (2023)Cite this article

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Abstract

Entanglement is a distinguishing feature of quantum many-body systems, and uncovering the entanglement structure for large particle numbers in quantum simulation experiments is a fundamental challenge in quantum information science1. Here we perform experimental investigations of entanglement on the basis of the entanglement Hamiltonian (EH)2 as an effective description of the reduced density operator for large subsystems. We prepare ground and excited states of a one-dimensional XXZ Heisenberg chain on a 51-ion programmable quantum simulator3 and perform sample-efficient ‘learning’ of the EH for subsystems of up to 20 lattice sites4. Our experiments provide compelling evidence for a local structure of the EH. To our knowledge, this observation marks the first instance of confirming the fundamental predictions of quantum field theory by Bisognano and Wichmann5,6, adapted to lattice models that represent correlated quantum matter. The reduced state takes the form of a Gibbs ensemble, with a spatially varying temperature profile as a signature of entanglement2. Our results also show the transition from area- to volume-law scaling7 of von Neumann entanglement entropies from ground to excited states. As we venture towards achieving quantum advantage, we anticipate that our findings and methods have wide-ranging applicability to revealing and understanding entanglement in many-body problems with local interactions including higher spatial dimensions.

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Fig. 1: Learning the entanglement structure of variationally prepared ground and heated quantum many-body states.
Fig. 2: Entanglement temperature profiles for different subsystem sizes at the critical point Δ = 1.
Fig. 3: EH of disjoint five-site subsystems.

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Data availability

The experimental data presented in this manuscript can be found in the Zenodo repository, https://doi.org/10.5281/zenodo.8279583.

Code availability

Authors agree that, on request, the analysis code will be provided by the corresponding author.

References

  1. Gross, D., Sevrin, A. & Zoller, P. (eds) The Physics of Quantum Information (World Scientific, 2023).

  2. Dalmonte, M., Eisler, V., Falconi, M. & Vermersch, B. Entanglement Hamiltonians: from field theory to lattice models and experiments. Ann. Phys. 534, 2200064 (2022).

    Article  Google Scholar 

  3. Joshi, M. K. et al. Observing emergent hydrodynamics in a long-range quantum magnet. Science 376, 720 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  4. Kokail, C., van Bijnen, R., Elben, A., Vermersch, B. & Zoller, P. Entanglement Hamiltonian tomography in quantum simulation. Nat. Phys. 17, 936 (2021).

    Article  CAS  Google Scholar 

  5. Bisognano, J. J. & Wichmann, E. H. On the duality condition for a Hermitian scalar field. J. Math. Phys. 16, 985 (1975).

    Article  ADS  MathSciNet  Google Scholar 

  6. Bisognano, J. J. & Wichmann, E. H. On the duality condition for quantum fields. J. Math. Phys. 17, 303 (1976).

    Article  ADS  MathSciNet  Google Scholar 

  7. Miao, Q. & Barthel, T. Eigenstate entanglement: crossover from the ground state to volume laws. Phys. Rev. Lett. 127, 040603 (2021).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  8. Zhang, J. et al. Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature 551, 601 (2017).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  9. Bohnet, J. G. et al. Quantum spin dynamics and entanglement generation with hundreds of trapped ions. Science 352, 1297 (2016).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  10. Chen, C. et al. Continuous symmetry breaking in a two-dimensional Rydberg array. Nature 616, 691 (2023).

    Article  ADS  CAS  PubMed  Google Scholar 

  11. Semeghini, G. et al. Probing topological spin liquids on a programmable quantum simulator. Science 374, 1242 (2021).

    Article  ADS  CAS  PubMed  Google Scholar 

  12. Sompet, P. et al. Realizing the symmetry-protected Haldane phase in Fermi–Hubbard ladders. Nature 606, 484 (2022).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  13. Léonard, J. et al. Realization of a fractional quantum Hall state with ultracold atoms. Nature 619, 495 (2023).

    Article  ADS  PubMed  Google Scholar 

  14. Zhang, W.-Y. et al. Scalable multipartite entanglement created by spin exchange in an optical lattice. Phys. Rev. Lett. 131, 073401 (2023).

    Article  ADS  CAS  PubMed  Google Scholar 

  15. Cirac, J. I., Perez-Garcia, D., Schuch, N. & Verstraete, F. Matrix product states and projected entangled pair states: concepts, symmetries, theorems. Rev. Mod. Phys. 93, 045003 (2021).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  16. Ganahl, M. et al. Density matrix renormalization group with tensor processing units. PRX Quantum 4, 010317 (2023).

    Article  ADS  Google Scholar 

  17. Acharya, A., Kypraios, T. & Guţă, M. A comparative study of estimation methods in quantum tomography. J. Phys. A Math. Theor. 52, 234001 (2019).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  18. Amico, L., Fazio, R., Osterloh, A. & Vedral, V. Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  19. Dalmonte, M., Vermersch, B. & Zoller, P. Quantum simulation and spectroscopy of entanglement Hamiltonians. Nat. Phys. 14, 827 (2018).

    Article  CAS  Google Scholar 

  20. Giudici, G., Mendes-Santos, T., Calabrese, P. & Dalmonte, M. Entanglement Hamiltonians of lattice models via the Bisognano-Wichmann theorem. Phys. Rev. B 98, 134403 (2018).

    Article  ADS  CAS  Google Scholar 

  21. Zhu, W., Huang, Z., He, Y.-C. & Wen, X. Entanglement Hamiltonian of many-body dynamics in strongly correlated systems. Phys. Rev. Lett. 124, 100605 (2020).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  22. Casini, H., Huerta, M. & Myers, R. C. Towards a derivation of holographic entanglement entropy. J. High Energy Phys. 2011, 1 (2011).

    Article  ADS  MathSciNet  Google Scholar 

  23. Cardy, J. & Tonni, E. Entanglement Hamiltonians in two-dimensional conformal field theory. J. Stat. Mech. 2016, 123103 (2016).

    Article  MathSciNet  Google Scholar 

  24. Anshu, A., Arunachalam, S., Kuwahara, T. & Soleimanifar, M. Sample-efficient learning of interacting quantum systems. Nat. Phys. 17, 931 (2021).

    Article  CAS  Google Scholar 

  25. Garrison, J. R. & Grover, T. Does a single eigenstate encode the full Hamiltonian? Phys. Rev. X 8, 021026 (2018).

    CAS  Google Scholar 

  26. Calabrese, P. & Cardy, J. Entanglement entropy and quantum field theory. J. Stat. Mech. 2004, P06002 (2004).

    Article  Google Scholar 

  27. Kokail, C. et al. Self-verifying variational quantum simulation of lattice models. Nature 569, 355 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  28. Alba, V. Eigenstate thermalization hypothesis and integrability in quantum spin chains. Phys. Rev. B 91, 155123 (2015).

    Article  ADS  Google Scholar 

  29. Alba, V., Fagotti, M. & Calabrese, P. Entanglement entropy of excited states. J. Stat. Mech. 2009, P10020 (2009).

    Article  MathSciNet  Google Scholar 

  30. Eisler, V., Giulio, G. D., Tonni, E. & Peschel, I. Entanglement Hamiltonians for non-critical quantum chains. J. Stat. Mech. 2020, 103102 (2020).

    Article  MathSciNet  Google Scholar 

  31. Elben, A. et al. Cross-platform verification of intermediate scale quantum devices. Phys. Rev. Lett. 124, 010504 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  32. Elben, A. et al. The randomized measurement toolbox. Nat. Rev. Phys. 5, 9 (2023).

    Article  Google Scholar 

  33. Casini, H. & Huerta, M. Reduced density matrix and internal dynamics for multicomponent regions. Class. Quantum Gravity 26, 185005 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  34. Eisler, V., Tonni, E. & Peschel, I. Local and non-local properties of the entanglement Hamiltonian for two disjoint intervals. J. Stat. Mech. 2022, 083101 (2022).

    Article  MathSciNet  Google Scholar 

  35. Sels, D. & Wouters, M. Quantum statistical gravity: time dilation due to local information in many-body quantum systems. J. Phys. Conf. Ser. 880, 012047 (2017).

  36. Zache, T. V., Kokail, C., Sundar, B. & Zoller, P. Entanglement spectroscopy and probing the Li–Haldane conjecture in topological quantum matter. Quantum 6, 702 (2022).

    Article  Google Scholar 

  37. Li, H. & Haldane, F. D. M. Entanglement spectrum as a generalization of entanglement entropy: identification of topological order in non-Abelian fractional quantum hall effect states. Phys. Rev. Lett. 101, 010504 (2008).

    Article  ADS  PubMed  Google Scholar 

  38. Chandran, A., Khemani, V. & Sondhi, S. L. How universal is the entanglement spectrum? Phys. Rev. Lett. 113, 060501 (2014).

    Article  ADS  PubMed  Google Scholar 

  39. Murciano, S., Vitale, V., Dalmonte, M. & Calabrese, P. Negativity Hamiltonian: an operator characterization of mixed-state entanglement. Phys. Rev. Lett. 128, 140502 (2022).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  40. Kim, I. H., Shi, B., Kato, K. & Albert, V. V. Modular commutator in gapped quantum many-body systems. Phys. Rev. B 106, 075147 (2022).

    Article  ADS  CAS  Google Scholar 

  41. Ryu, S. & Takayanagi, T. Holographic derivation of entanglement entropy from the anti–de Sitter space/conformal field theory correspondence. Phys. Rev. Lett. 96, 181602 (2006).

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  42. Periwal, A. et al. Programmable interactions and emergent geometry in an array of atom clouds. Nature 600, 630 (2021).

    Article  ADS  CAS  PubMed  Google Scholar 

  43. Kranzl, F. et al. Controlling long ion strings for quantum simulation and precision measurements. Phys. Rev. A 105, 052426 (2022).

    Article  ADS  CAS  Google Scholar 

  44. Jurcevic, P. et al. Quasiparticle engineering and entanglement propagation in a quantum many-body system. Nature 511, 202 (2014).

    Article  ADS  CAS  PubMed  Google Scholar 

  45. Porras, D. & Cirac, J. I. Effective quantum spin systems with trapped ions. Phys. Rev. Lett. 92, 207901 (2004).

    Article  ADS  CAS  PubMed  Google Scholar 

  46. Liang, Y.-C. et al. Quantum fidelity measures for mixed states. Rep. Prog. Phys. 82, 076001 (2019).

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  47. Flammia, S. T. & Liu, Y.-K. Direct fidelity estimation from few Pauli measurements. Phys. Rev. Lett. 106, 230501 (2011).

    Article  ADS  PubMed  Google Scholar 

  48. da Silva, M. P., Landon-Cardinal, O. & Poulin, D. Practical characterization of quantum devices without tomography. Phys. Rev. Lett. 107, 210404 (2011).

    Article  PubMed  Google Scholar 

  49. Cerezo, M. et al. Variational quantum algorithms. Nat. Rev. Phys. 3, 625 (2021).

    Article  Google Scholar 

  50. Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242 (2017).

    Article  ADS  CAS  PubMed  Google Scholar 

  51. Rasmussen, C. E. & Williams, C. K. I. Gaussian Processes for Machine Learning (MIT Press, 2005); https://doi.org/10.7551/mitpress/3206.001.0001.

  52. Sopena, A., Gordon, M. H., García-Martín, D., Sierra, G. & López, E. Algebraic Bethe circuits. Quantum 6, 796 (2022).

    Article  Google Scholar 

  53. Fishman, M., White, S. R. & Stoudenmire, E. M. The ITensor software library for tensor network calculations. SciPost Phys. Codebases https://doi.org/10.21468/SciPostPhysCodeb.4 (2022).

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Acknowledgements

C.K. and P.Z. thank D. Sels for discussions. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 101113690 (PASQuanS2.1). C.K., R.v.B., T.V.Z. and P.Z. were supported by the US Air Force Office of Scientific Research through IOE grant no. FA9550-19-1-7044 LASCEM, the Austrian Research Promotion Agency contract 884471 (R.v.B.) and the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440, P.Z.). M.K.J., F.K., C.F.R. and R.B. acknowledge financial support for the experiment from the Austrian Science Fund through the SFB BeyondC (F7110) and the Institut für Quanteninformation GmbH. This research was supported in part by the National Science Foundation under grant no. NSF PHY-1748958. The computational results presented have been achieved (in part) using the high-performance computer infrastructure LEO of the University of Innsbruck. Simulations were performed using iTensor53.

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Author notes

  1. These authors contributed equally: Manoj K. Joshi, Christian Kokail, Rick van Bijnen

Authors and Affiliations

  1. Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck, Austria

    Manoj K. Joshi, Christian Kokail, Rick van Bijnen, Florian Kranzl, Torsten V. Zache, Rainer Blatt, Christian F. Roos & Peter Zoller

  2. University of Innsbruck, Institute for Experimental Physics, Innsbruck, Austria

    Manoj K. Joshi, Florian Kranzl, Rainer Blatt & Christian F. Roos

  3. University of Innsbruck, Institute for Theoretical Physics, Innsbruck, Austria

    Christian Kokail, Rick van Bijnen, Torsten V. Zache & Peter Zoller

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Contributions

M.K.J. and F.K. developed and conducted the experiment under the guidance of R.B. and C.F.R. C.K., R.v.B., T.V.Z. and P.Z. proposed the research and developed the quantum protocols. C.K., R.v.B. and M.K.J. performed the data analysis. C.K., R.v.B., T.V.Z. and P.Z. wrote the manuscript, and M.K.J. contributed texts on experimental setups. All authors contributed to the discussion of the results.

Corresponding author

Correspondence to Peter Zoller.

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Nature thanks Anurag Anshu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Extended data figures and tables

Extended Data Fig. 1 Effective interactions and spin dynamics for a 51-ion chain.

(a) Experimentally measured nearest-neighbor interaction terms Ji,i+1, compared to theoretical calculations (solid line). (b) The quench dynamics of a single spin initialized to a spin-up state in the middle of the ion chain, while other spins initialized to a spin-down state, under the engineered flip-flop type interaction plotted in discs and solid lines are numerical results. (c) Theoretically calculated interaction matrix for the experimental parameters.

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Joshi, M.K., Kokail, C., van Bijnen, R. et al. Exploring large-scale entanglement in quantum simulation. Nature 624, 539–544 (2023). https://doi.org/10.1038/s41586-023-06768-0

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