Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Entanglement Hamiltonian tomography in quantum simulation


Entanglement is the crucial ingredient of quantum many-body physics, and characterizing and quantifying entanglement in the closed-system dynamics of quantum simulators remains a challenge in today’s era of intermediate-scale quantum devices. Here we discuss an efficient tomographic protocol for reconstructing reduced density matrices and entanglement spectra for spin systems. The key step is a parametrization of the reduced density matrix in terms of an entanglement Hamiltonian involving only quasilocal few-body terms. This ansatz is fitted to, and can be independently verified from, a small number of randomized measurements. By analysing data from trapped-ion quantum simulators for quench dynamics of a one-dimensional long-range Ising model, we demonstrate the ability of the protocol to measure the time evolution of the entanglement spectrum, in agreement with theoretical expectations. Furthermore, we develop the protocol as a testbed for predictions of entanglement structure in quantum field theories, which we illustrate for conformal field theory in quench dynamics, as well as the Bisognano–Wichmann theorem for ground states. In theoretical simulations, we demonstrate favourable scaling of sampling efficiency with subsystem size. Although the post-processing might ultimately be exponential, our protocol addresses the bottleneck of exponential sampling complexity in the investigation of entanglement structure in quantum simulation, and brings subsystems of tens of spins into reach for present experiments

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: EHT protocol and its application in experimental quench dynamics.
Fig. 2: Simulation of EHT for ground states of a long-range transverse-field Ising model.
Fig. 3: Simulation of EHT for a global quench in the critical Ising model.
Fig. 4: Scaling and sampling efficiency of EHT.
Fig. 5: Experimental verification of EHT in quench dynamics on 10- and 20-spin trapped-ion quantum simulators.

Data availability

All data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


  1. 1.

    Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153–185 (2014).

    ADS  Article  Google Scholar 

  2. 2.

    Browaeys, A. & Lahaye, T. Many-body physics with individually controlled Rydberg atoms. Nat. Phys. 16, 132–142 (2020).

    Article  Google Scholar 

  3. 3.

    Monroe, C. et al. Programmable quantum simulations of spin systems with trapped ions. Rev. Mod. Phys. 93, 025001 (2021).

    ADS  Article  Google Scholar 

  4. 4.

    Brydges, T. et al. Probing Rényi entanglement entropy via randomized measurements. Science 364, 260–263 (2019).

    ADS  Article  Google Scholar 

  5. 5.

    Kokail, C. et al. Self-verifying variational quantum simulation of lattice models. Nature 569, 355–360 (2019).

    ADS  Article  Google Scholar 

  6. 6.

    Wilkinson, S. A. & Hartmann, M. J. Superconducting quantum many-body circuits for quantum simulation and computing. Appl. Phys. Lett. 116, 230501 (2020).

    ADS  Article  Google Scholar 

  7. 7.

    King, A. D. et al. Observation of topological phenomena in a programmable lattice of 1,800 qubits. Nature 560, 456–460 (2018).

    ADS  Article  Google Scholar 

  8. 8.

    Cirac, J. I. & Zoller, P. Goals and opportunities in quantum simulation. Nat. Phys. 8, 264–266 (2012).

    Article  Google Scholar 

  9. 9.

    Amico, L., Fazio, R., Osterloh, A. & Vedral, V. Entanglement in many-body systems. Rev. Mod. Phys. 80, 517–576 (2008).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Zeng, B., Chen, X., Zhou, D.-L. & Wen, X.-G. Quantum Information Meets Quantum Matter (Springer, 2019)

  11. 11.

    Regnault, N. in Topological Aspects of Condensed Matter Physics (eds Chamon, C. et al.) 165–210 (Oxford Univ. Press, 2017).

  12. 12.

    Dalmonte, M., Vermersch, B. & Zoller, P. Quantum simulation and spectroscopy of entanglement Hamiltonians. Nat. Phys. 14, 827–831 (2018).

    Article  Google Scholar 

  13. 13.

    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).

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Chang, P.-Y., Chen, X., Gopalakrishnan, S. & Pixley, J. H. Evolution of entanglement spectra under generic quantum dynamics. Phys. Rev. Lett. 123, 190602 (2019).

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    Calabrese, P. & Cardy, J. Quantum quenches in 1 + 1 dimensional conformal field theories. J. Stat. Mech. Theor. Exp. 2016, 064003 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Wen, X., Ryu, S. & Ludwig, A. W. Entanglement Hamiltonian evolution during thermalization in conformal field theory. J. Stat. Mech. Theor. Exp. 2018, 113103 (2018).

    Article  Google Scholar 

  17. 17.

    Orús, R. Tensor networks for complex quantum systems. Nat. Rev. Phys. 1, 538–550 (2019).

    Article  Google Scholar 

  18. 18.

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

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Gross, D., Liu, Y.-K., Flammia, S. T., Becker, S. & Eisert, J. Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105, 150401 (2010).

    ADS  Article  Google Scholar 

  20. 20.

    Cramer, M. et al. Efficient quantum state tomography. Nat. Commun. 1, 149 (2010).

    ADS  Article  Google Scholar 

  21. 21.

    Smolin, J. A., Gambetta, J. M. & Smith, G. Efficient method for computing the maximum-likelihood quantum state from measurements with additive Gaussian noise. Phys. Rev. Lett. 108, 070502 (2012).

    ADS  Article  Google Scholar 

  22. 22.

    Torlai, G. et al. Neural-network quantum state tomography. Nat. Phys. 14, 447–450 (2018).

    Article  Google Scholar 

  23. 23.

    Choo, K., von Keyserlingk, C. W., Regnault, N. & Neupert, T. Measurement of the entanglement spectrum of a symmetry-protected topological state using the IBM quantum computer. Phys. Rev. Lett. 121, 086808 (2018).

    ADS  Article  Google Scholar 

  24. 24.

    Haah, J., Harrow, A. W., Ji, Z., Wu, X. & Yu, N. Sample-optimal tomography of quantum states. IEEE Trans. Inf. Theory 63, 5628–5641 (2017).

    MathSciNet  MATH  Google Scholar 

  25. 25.

    O’Donnell, R. & Wright, J. Efficient quantum tomography. In Proc. Forty-eighth Annual ACM Symposium on Theory of Computing, STOC ’16, 899–912 (ACM, 2016);

  26. 26.

    Brando, F. G. S. L., Kueng, R. & Frana, D. S. Fast and robust quantum state tomography from few basis measurements. Preprint at (2020).

  27. 27.

    Anshu, A., Arunachalam, S., Kuwahara, T. & Soleimanifar, M. Sample-efficient learning of quantum many-body systems. Preprint at (2020).

  28. 28.

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

    ADS  Article  Google Scholar 

  29. 29.

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

    ADS  MathSciNet  Article  Google Scholar 

  30. 30.

    Bisognano, J. J. & Wichmann, E. H. On the duality condition for a Hermitian scalar field. J. Math. Phys. 16, 985–1007 (1975).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Bisognano, J. J. & Wichmann, E. H. On the duality condition for quantum fields. J. Math. Phys. 17, 303–321 (1976).

    ADS  MathSciNet  Article  Google Scholar 

  32. 32.

    Calabrese, P. & Cardy, J. Entanglement entropy and conformal field theory. J. Phys. A 42, 504005 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Swingle, B. & Senthil, T. Geometric proof of the equality between entanglement and edge spectra. Phys. Rev. B 86, 045117 (2012).

    ADS  Article  Google Scholar 

  34. 34.

    Alba, V., Haque, M. & Läuchli, A. M. Boundary-locality and perturbative structure of entanglement spectra in gapped systems. Phys. Rev. Lett. 108, 227201 (2012).

    ADS  Article  Google Scholar 

  35. 35.

    Eisler, V. & Peschel, I. Analytical results for the entanglement Hamiltonian of a free-fermion chain. J. Phys. A 50, 284003 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Itoyama, H. & Thacker, H. B. Lattice Virasoro algebra and corner transfer matrices in the Baxter eight-vertex model. Phys. Rev. Lett. 58, 1395–1398 (1987).

    ADS  MathSciNet  Article  Google Scholar 

  37. 37.

    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).

    ADS  Article  Google Scholar 

  38. 38.

    Hislop, P. D. & Longo, R. Modular structure of the local algebras associated with the free massless scalar field theory. Commun. Math. Phys. 84, 71–85 (1982).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  39. 39.

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

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Elben, A. et al. Mixed-state entanglement from local randomized measurements. Phys. Rev. Lett. 125, 200501 (2020).

    ADS  Article  Google Scholar 

  41. 41.

    Deutsch, J. M. Quantum statistical mechanics in a closed system. Phys. Rev. A 43, 2046–2049 (1991).

    ADS  Article  Google Scholar 

  42. 42.

    Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E 50, 888–901 (1994).

    ADS  Article  Google Scholar 

  43. 43.

    Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008).

    ADS  Article  Google Scholar 

  44. 44.

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

    Google Scholar 

  45. 45.

    Guta, M., Kahn, J., Kueng, R. & Tropp, J. A. Fast state tomography with optimal error bounds. J. Phys. A 53, 204001 (2020).

    ADS  MathSciNet  Article  Google Scholar 

  46. 46.

    Huang, H.-Y., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020).

    Article  Google Scholar 

  47. 47.

    Riofrío, C. A. et al. Experimental quantum compressed sensing for a seven-qubit system. Nat. Commun. 8, 15305 (2017).

    ADS  Article  Google Scholar 

  48. 48.

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

    ADS  Article  Google Scholar 

  49. 49.

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

    ADS  Article  Google Scholar 

  50. 50.

    Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).

    ADS  Article  Google Scholar 

  51. 51.

    Dankert, C., Cleve, R., Emerson, J. & Livine, E. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 80, 012304 (2009).

    ADS  Article  Google Scholar 

  52. 52.

    Qi, X.-L. & Ranard, D. Determining a local Hamiltonian from a single eigenstate. Quantum 3, 159 (2019).

    Article  Google Scholar 

  53. 53.

    Bairey, E., Arad, I. & Lindner, N. H. Learning a local Hamiltonian from local measurements. Phys. Rev. Lett. 122, 020504 (2019).

    ADS  Article  Google Scholar 

  54. 54.

    Wang, J. et al. Experimental quantum Hamiltonian learning. Nat. Phys. 13, 551–555 (2017).

    Article  Google Scholar 

  55. 55.

    Elben, A., Vermersch, B., Dalmonte, M., Cirac, J. I. & Zoller, P. Rényi entropies from random quenches in atomic Hubbard and spin models. Phys. Rev. Lett. 120, 050406 (2018).

    ADS  Article  Google Scholar 

  56. 56.

    LaRose, R., Tikku, A., O’Neel-Judy, É., Cincio, L. & Coles, P. J. Variational quantum state diagonalization. npj Quantum Inf. 5, 57 (2019).

    Article  Google Scholar 

  57. 57.

    Bravo-Prieto, C., García-Martín, D. & Latorre, J. I. Quantum singular value decomposer. Phys. Rev. A 101, 062310 (2020).

    ADS  MathSciNet  Article  Google Scholar 

  58. 58.

    Sugiyama, T., Turner, P. S. & Murao, M. Precision-guaranteed quantum tomography. Phys. Rev. Lett. 111, 160406 (2013).

    ADS  Article  Google Scholar 

  59. 59.

    Elben, A., Vermersch, B., Roos, C. F. & Zoller, P. Statistical correlations between locally randomized measurements: a toolbox for probing entanglement in many-body quantum states. Phys. Rev. A 99, 052323 (2019).

    ADS  MathSciNet  Article  Google Scholar 

  60. 60.

    Fishman, M., White, S. R. & Stoudenmire, E. M. The ITensor software library for tensor network calculations. Preprint at (2020).

Download references


We thank P. Calabrese, M. Dalmonte, G. Giudici, L.K. Joshi, B. Kraus, R. Kueng, C. Roos, L. Sieberer, J. Yu and W. Zhu for discussions, and members of the Innsbruck trapped-ion group for generously sharing the experimental data of ref. 4. Work at Innsbruck is supported by the European Union programme Horizon 2020 under grants 817482 (PASQuanS) and 731473 (FWF QuantERA via QTFLAG I03769), the US Air Force Office of Scientific Research (AFOSR) via IOE grant FA9550-19-1-7044 LASCEM and by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440, P.Z.). B.V. acknowledges funding from the Austrian Science Foundation (FWF, P 32597 N), and the French National Research Agency (ANR-20-CE47-0005 JCJC QRand). The computational results presented here have been achieved (in part) using the LEO HPC infrastructure of the University of Innsbruck. Numerical calculations were performed (in part) using the ITensor library60.

Author information




The research topic was developed by C.K., R.v.B., A.E. and B.V., following suggestions by P.Z. C.K., R.v.B. and A.E. developed the theoretical protocols. C.K., R.v.B., A.E. and P.Z. wrote the manuscript. All authors contributed to the discussion of the results and the manuscript.

Corresponding authors

Correspondence to Christian Kokail, Rick van Bijnen, Andreas Elben, Benoît Vermersch or Peter Zoller.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary information

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kokail, C., van Bijnen, R., Elben, A. et al. Entanglement Hamiltonian tomography in quantum simulation. Nat. Phys. 17, 936–942 (2021).

Download citation


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing