Skip to main content

Thank you for visiting nature.com. 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.

  • Article
  • Published:

Measuring entanglement entropy in a quantum many-body system

Nature volume 528, pages 77–83 (2015)Cite this article

Subjects

Abstract

Entanglement is one of the most intriguing features of quantum mechanics. It describes non-local correlations between quantum objects, and is at the heart of quantum information sciences. Entanglement is now being studied in diverse fields ranging from condensed matter to quantum gravity. However, measuring entanglement remains a challenge. This is especially so in systems of interacting delocalized particles, for which a direct experimental measurement of spatial entanglement has been elusive. Here, we measure entanglement in such a system of itinerant particles using quantum interference of many-body twins. Making use of our single-site-resolved control of ultracold bosonic atoms in optical lattices, we prepare two identical copies of a many-body state and interfere them. This enables us to directly measure quantum purity, Rényi entanglement entropy, and mutual information. These experiments pave the way for using entanglement to characterize quantum phases and dynamics of strongly correlated many-body systems.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Bipartite entanglement and partial measurements.
Figure 2: Measurement of quantum purity with many-body bosonic interference of quantum twins.
Figure 3: Many-body interference to probe entanglement in optical lattices.
Figure 4: Entanglement in the ground state of the Bose–Hubbard model.
Figure 5: Rényi mutual information in the ground state.
Figure 6: Entanglement dynamics in a quench.

Similar content being viewed by others

References

  1. Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)

    Article  CAS  ADS  MathSciNet  Google Scholar 

  2. Aspect, A. Bell’s inequality test: more ideal than ever. Nature 398, 189–190 (1999)

    Article  CAS  ADS  Google Scholar 

  3. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2010)

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

    Article  CAS  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Nishioka, T., Ryu, S. & Takayanagi, T. Holographic entanglement entropy: an overview. J. Phys. A 42, 504008 (2009)

    Article  MathSciNet  Google Scholar 

  7. Eisert, J., Cramer, M. & Plenio, M. B. Colloquium: area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  8. Kitaev, A. & Preskill, J. Topological entanglement entropy. Phys. Rev. Lett. 96, 110404 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  9. Levin, M. & Wen, X.-G. Detecting topological order in a ground state wave function. Phys. Rev. Lett. 96, 110405 (2006)

    Article  ADS  Google Scholar 

  10. Jiang, H.-C., Wang, Z. & Balents, L. Identifying topological order by entanglement entropy. Nature Phys. 8, 902–905 (2012)

    Article  CAS  ADS  Google Scholar 

  11. Zhang, Y., Grover, T. & Vishwanath, A. Entanglement entropy of critical spin liquids. Phys. Rev. Lett. 107, 067202 (2011)

    Article  ADS  Google Scholar 

  12. Isakov, S. V., Hastings, M. B. & Melko, R. G. Topological entanglement entropy of a Bose–Hubbard spin liquid. Nature Phys. 7, 772–775 (2011)

    Article  CAS  ADS  Google Scholar 

  13. Vidal, G., Latorre, J. I., Rico, E. & Kitaev, A. Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003)

    Article  CAS  ADS  Google Scholar 

  14. Bardarson, J. H., Pollmann, F. & Moore, J. E. Unbounded growth of entanglement in models of many-body localization. Phys. Rev. Lett. 109, 017202 (2012)

    Article  ADS  Google Scholar 

  15. Daley, A. J., Pichler, H., Schachenmayer, J. & Zoller, P. Measuring entanglement growth in quench dynamics of bosons in an optical lattice. Phys. Rev. Lett. 109, 020505 (2012)

    Article  CAS  ADS  Google Scholar 

  16. Schuch, N., Wolf, M. M., Verstraete, F. & Cirac, J. I. Entropy scaling and simulability by matrix product states. Phys. Rev. Lett. 100, 030504 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  17. Bloch, I., Dalibard, J. & Nascimbène, S. Quantum simulations with ultracold quantum gases. Nature Phys. 8, 267–276 (2012)

    Article  CAS  ADS  Google Scholar 

  18. Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nature Phys. 8, 277–284 (2012)

    Article  CAS  ADS  Google Scholar 

  19. Aspuru-Guzik, A. & Walther, P. Photonic quantum simulators. Nature Phys. 8, 285–291 (2012)

    Article  CAS  ADS  Google Scholar 

  20. Houck, A. A., Türeci, H. E. & Koch, J. On-chip quantum simulation with superconducting circuits. Nature Phys. 8, 292–299 (2012)

    Article  CAS  ADS  Google Scholar 

  21. Bouwmeester, D., Pan, J.-W., Daniell, M., Weinfurter, H. & Zeilinger, A. Observation of three-photon Greenberger-Horne-Zeilinger entanglement. Phys. Rev. Lett. 82, 1345–1349 (1999)

    Article  CAS  ADS  MathSciNet  Google Scholar 

  22. Gühne, O. & Tóth, G. Entanglement detection. Phys. Rep. 474, 1–75 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  23. James, D. F. V., Kwiat, P. G., Munro, W. J. & White, A. G. Measurement of qubits. Phys. Rev. A 64, 052312 (2001)

    Article  ADS  Google Scholar 

  24. Pan, J.-W. et al. Multiphoton entanglement and interferometry. Rev. Mod. Phys. 84, 777 (2012)

    Article  ADS  Google Scholar 

  25. Häffner, H. et al. Scalable multiparticle entanglement of trapped ions. Nature 438, 643–646 (2005)

    Article  ADS  Google Scholar 

  26. Ekert, A. K. et al. Direct estimations of linear and nonlinear functionals of a quantum state. Phys. Rev. Lett. 88, 217901 (2002)

    Article  ADS  Google Scholar 

  27. Moura Alves, C. & Jaksch, D. Multipartite entanglement detection in bosons. Phys. Rev. Lett. 93, 110501 (2004)

    Article  CAS  ADS  Google Scholar 

  28. Bakr, W. S. et al. Probing the superfluid–to–Mott insulator transition at the single-atom level. Science 329, 547–550 (2010)

    Article  CAS  ADS  Google Scholar 

  29. Brun, T. A. Measuring polynomial functions of states. Quantum Inform. Comput. 4, 401–408 (2004)

    MathSciNet  MATH  Google Scholar 

  30. Bovino, F. A. et al. Direct measurement of nonlinear properties of bipartite quantum states. Phys. Rev. Lett. 95, 240407 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  31. Walborn, S. P., Ribeiro, P. S., Davidovich, L., Mintert, F. & Buchleitner, A. Experimental determination of entanglement with a single measurement. Nature 440, 1022–1024 (2006)

    Article  CAS  ADS  Google Scholar 

  32. Schmid, C. et al. Experimental direct observation of mixed state entanglement. Phys. Rev. Lett. 101, 260505 (2008)

    Article  ADS  Google Scholar 

  33. Horodecki, R. & Horodecki, M. et al. Information-theoretic aspects of inseparability of mixed states. Phys. Rev. A 54, 1838 (1996)

    Article  CAS  ADS  MathSciNet  Google Scholar 

  34. Mintert, F. & Buchleitner, A. Observable entanglement measure for mixed quantum states. Phys. Rev. Lett. 98, 140505 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  35. 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  Google Scholar 

  36. Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, 2044 (1987)

    Article  CAS  ADS  Google Scholar 

  37. Kaufman, A. M. et al. Two-particle quantum interference in tunnel-coupled optical tweezers. Science 345, 306–309 (2014)

    Article  CAS  ADS  MathSciNet  Google Scholar 

  38. Lopes, R. et al. Atomic Hong–Ou–Mandel experiment. Nature 520, 66–68 (2015)

    Article  CAS  ADS  Google Scholar 

  39. Cramer, M. et al. Spatial entanglement of bosons in optical lattices. Nat. Commun. 4, 2161 (2013)

    Article  CAS  ADS  Google Scholar 

  40. Fukuhara, T. et al. Spatially resolved detection of a spin-entanglement wave in a Bose–Hubbard chain. Phys. Rev. Lett. 115, 035302 (2015)

    Article  ADS  Google Scholar 

  41. Bartlett, S. D. & Wiseman, H. M. Entanglement constrained by superselection rules. Phys. Rev. Lett. 91, 097903 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  42. Schuch, N., Verstraete, F. & Cirac, J. I. Nonlocal resources in the presence of superselection rules. Phys. Rev. Lett. 92, 087904 (2004)

    Article  CAS  ADS  Google Scholar 

  43. Palmer, R. N., Moura Alves, C. & Jaksch, D. Detection and characterization of multipartite entanglement in optical lattices. Phys. Rev. A 72, 042335 (2005)

    Article  ADS  Google Scholar 

  44. Wolf, M. M., Verstraete, F., Hastings, M. B. & Cirac, J. I. Area laws in quantum systems: mutual information and correlations. Phys. Rev. Lett. 100, 070502 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  45. Terhal, B. M., DiVincenzo, D. P. & Leung, D. W. Hiding bits in Bell states. Phys. Rev. Lett. 86, 5807–5810 (2001)

    Article  CAS  ADS  Google Scholar 

  46. Vidal, G. Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett. 93, 040502 (2004)

    Article  ADS  Google Scholar 

  47. Trotzky, S. et al. Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas. Nature Phys. 8, 325–330 (2012)

    Article  CAS  ADS  Google Scholar 

  48. Trotzky, S., Chen, Y.-A., Schnorrberger, U., Cheinet, P. & Bloch, I. Controlling and detecting spin correlations of ultracold atoms in optical lattices. Phys. Rev. Lett. 105, 265303 (2010)

    Article  ADS  Google Scholar 

  49. Pichler, H., Bonnes, L., Daley, A. J., Läuchli, A. M. & Zoller, P. Thermal versus entanglement entropy: a measurement protocol for fermionic atoms with a quantum gas microscope. New J. Phys. 15, 063003 (2013)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  CAS  ADS  Google Scholar 

  51. Zanardi, P. & Paunković, N. Ground state overlap and quantum phase transitions. Phys. Rev. E 74, 031123 (2006)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank D. Abanin, J. I. Cirac, M. Cramer, A. Daley, A. DelMaestro, E. Demler, M. Endres, S. Gopalakrishnan, M. Headrick, A. Kaufman, M. Knap, T. Monz, A. Pal, H. Pichler, S. Sachdev, B. Swingle, P. Zoller, and M. Zwierlein for useful discussions. This work was supported by grants from the Gordon and Betty Moore Foundations EPiQS Initiative (grant GBMF3795), the NSF through the Center for Ultracold Atoms, the Army Research Office with funding from the DARPA OLE programme and a MURI programme, an Air Force Office of Scientific Research MURI programme, and an NSF Graduate Research Fellowship (to M.R.).

Author information

Authors and Affiliations

  1. Department of Physics, Harvard University, Cambridge, 02138, Massachusetts, USA

    Rajibul Islam, Ruichao Ma, Philipp M. Preiss, M. Eric Tai, Alexander Lukin, Matthew Rispoli & Markus Greiner

Authors
  1. Rajibul Islam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruichao Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Philipp M. Preiss
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Eric Tai
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Alexander Lukin
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Matthew Rispoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Markus Greiner
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the construction and execution of the experiments, data analysis and the writing of the manuscript.

Corresponding author

Correspondence to Markus Greiner.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

This file contains Supplementary Text and Data, Supplementary Figures 1-6 and Supplementary References. (PDF 306 kb)

PowerPoint slides

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Islam, R., Ma, R., Preiss, P. et al. Measuring entanglement entropy in a quantum many-body system. Nature 528, 77–83 (2015). https://doi.org/10.1038/nature15750

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature15750

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

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