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Measuring entanglement entropy in a quantum many-body system

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.

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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.

References

  1. 1

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

    CAS  ADS  MathSciNet  Article  Google Scholar 

  2. 2

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

    CAS  ADS  Article  Google Scholar 

  3. 3

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

  4. 4

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

    CAS  ADS  MathSciNet  Article  Google Scholar 

  5. 5

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

    MathSciNet  Article  Google Scholar 

  6. 6

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

    MathSciNet  Article  Google Scholar 

  7. 7

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

    ADS  MathSciNet  Article  Google Scholar 

  8. 8

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

    ADS  MathSciNet  Article  Google Scholar 

  9. 9

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

    ADS  Article  Google Scholar 

  10. 10

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

    CAS  ADS  Article  Google Scholar 

  11. 11

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

    ADS  Article  Google Scholar 

  12. 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)

    CAS  ADS  Article  Google Scholar 

  13. 13

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

    CAS  ADS  Article  Google Scholar 

  14. 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)

    ADS  Article  Google Scholar 

  15. 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)

    CAS  ADS  Article  Google Scholar 

  16. 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)

    ADS  MathSciNet  Article  Google Scholar 

  17. 17

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

    CAS  ADS  Article  Google Scholar 

  18. 18

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

    CAS  ADS  Article  Google Scholar 

  19. 19

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

    CAS  ADS  Article  Google Scholar 

  20. 20

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

    CAS  ADS  Article  Google Scholar 

  21. 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)

    CAS  ADS  MathSciNet  Article  Google Scholar 

  22. 22

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

    ADS  MathSciNet  Article  Google Scholar 

  23. 23

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

    ADS  Article  Google Scholar 

  24. 24

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

    ADS  Article  Google Scholar 

  25. 25

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

    ADS  Article  Google Scholar 

  26. 26

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

    ADS  Article  Google Scholar 

  27. 27

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

    CAS  ADS  Article  Google Scholar 

  28. 28

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

    CAS  ADS  Article  Google Scholar 

  29. 29

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

    MathSciNet  MATH  Google Scholar 

  30. 30

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

    ADS  MathSciNet  Article  Google Scholar 

  31. 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)

    CAS  ADS  Article  Google Scholar 

  32. 32

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

    ADS  Article  Google Scholar 

  33. 33

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

    CAS  ADS  MathSciNet  Article  Google Scholar 

  34. 34

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

    ADS  MathSciNet  Article  Google Scholar 

  35. 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)

    ADS  Article  Google Scholar 

  36. 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)

    CAS  ADS  Article  Google Scholar 

  37. 37

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

    CAS  ADS  MathSciNet  Article  Google Scholar 

  38. 38

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

    CAS  ADS  Article  Google Scholar 

  39. 39

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

    CAS  ADS  Article  Google Scholar 

  40. 40

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

    ADS  Article  Google Scholar 

  41. 41

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

    ADS  MathSciNet  Article  Google Scholar 

  42. 42

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

    CAS  ADS  Article  Google Scholar 

  43. 43

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

    ADS  Article  Google Scholar 

  44. 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)

    ADS  MathSciNet  Article  Google Scholar 

  45. 45

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

    CAS  ADS  Article  Google Scholar 

  46. 46

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

    ADS  Article  Google Scholar 

  47. 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)

    CAS  ADS  Article  Google Scholar 

  48. 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)

    ADS  Article  Google Scholar 

  49. 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)

    ADS  MathSciNet  Article  Google Scholar 

  50. 50

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

    CAS  ADS  Article  Google Scholar 

  51. 51

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

    ADS  MathSciNet  Article  Google Scholar 

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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.).

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

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Correspondence to Markus Greiner.

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The authors declare no competing financial interests.

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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

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