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Quantum simulation and spectroscopy of entanglement Hamiltonians

Abstract

The properties of a strongly correlated many-body quantum system, from the presence of topological order to the onset of quantum criticality, leave a footprint in its entanglement spectrum. The entanglement spectrum is composed by the eigenvalues of the density matrix representing a subsystem of the whole original system, but its direct measurement has remained elusive due to the lack of direct experimental probes. Here we show that the entanglement spectrum of the ground state of a broad class of Hamiltonians becomes directly accessible via the quantum simulation and spectroscopy of a suitably constructed entanglement Hamiltonian, building on the Bisognano–Wichmann theorem of axiomatic quantum field theory. This theorem gives an explicit physical construction of the entanglement Hamiltonian, identified as the Hamiltonian of the many-body system of interest with spatially varying couplings. On this basis, we propose a scalable recipe for the measurement of a system’s entanglement spectrum via spectroscopy of the corresponding Bisognano–Wichmann Hamiltonian realized in synthetic quantum systems, including atoms in optical lattices and trapped ions. We illustrate and benchmark this scenario on a variety of models, spanning phenomena as diverse as conformal field theories, topological order and quantum phase transitions.

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Fig. 1: ES via spectroscopy.
Fig. 2: ES of Heisenberg spin-1/2 chains.
Fig. 3: ES of dipolar Ising chains.
Fig. 4: ES in 2D hopping models and topological insulators.
Fig. 5: Implementations of EHs.

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References

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Eisert, J., Cramer, M. & Plenio, M. B. Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277–306 (2010).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. 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 

  4. Peschel, I. & Eisler, V. Reduced density matrices and entanglement entropy in free lattice models. J. Phys. A 42, 504003 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  5. Haag, R. Local Quantum Physics: Fields, Particles, Algebras (Springer, Berlin, 2012).

  6. Regnault, N. Entanglement spectroscopy and its application to the quantum Hall effects. Preprint at https://arxiv.org/abs/1510.07670 (2015).

  7. Pollmann, F., Berg, E., Turner, A. M. & Oshikawa, M. Phys. Rev. B 81, 064439 (2010).

    Article  ADS  Google Scholar 

  8. Fidkowski, L. Entanglement spectrum of topological insulators and superconductors. Phys. Rev. Lett. 104, 130502 (2010).

    Article  ADS  Google Scholar 

  9. Calabrese, P. & Lefevre, A. Entanglement spectrum in one-dimensional systems. Phys. Rev. A 78, 032329 (2008).

    Article  ADS  Google Scholar 

  10. Cirac, J. I., Poilblanc, D., Schuch, N. & Verstraete, F. Entanglement spectrum and boundary theories with projected entangled-pair states. Phys. Rev. B 83, 245134 (2011).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  12. Chiara, G. D., Lepori, L., Lewenstein, M. & Sanpera, A. Entanglement spectrum, critical exponents and order parameters in quantum spin chains. Phys. Rev. Lett. 109, 237208 (2012).

    Article  ADS  Google Scholar 

  13. Läuchli, A. M. Operator content of real-space entanglement spectra at conformal critical points. Preprint at https://arxiv.org/abs/1303.0741 (2013).

  14. Schollwöck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326, 96–192 (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Pichler, H., Zhu, G., Seif, A., Zoller, P. & Hafezi, M. Measurement protocol for the entanglement spectrum of cold atoms. Phys. Rev. X 6, 041033 (2016).

    Google Scholar 

  16. Beverland, M. E. et al. Spectrum estimation of density operators with alkaline-earth atoms. Phys. Rev. Lett. 120, 025301 (2018).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  19. Casini, H., Huerta, M. & Myers, R. C. Towards a derivation of holographic entanglement entropy. J. High Energy Phys. 1105, 036 (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Swingle, B. & McGreevy, J. Area law for gapless states from local entanglement thermodynamics. Phys. Rev. B 93, 205120 (2016).

    Article  ADS  Google Scholar 

  21. Wong, G., Klich, I., Zayas, L. A. P. & Vaman, D. Entanglement temperature and entanglement entropy of excited states. J. High Energy Phys. 12, 20 (2013).

    Article  ADS  Google Scholar 

  22. Sewell, G. L. Quantum fields on manifolds: PCT and gravitationally induced thermal states. Ann. Phys. 141, 201–224 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  23. Casini, H., Huerta, M. & Rosabal, J. A. Remarks on entanglement entropy for gauge fields. Phys. Rev. D 89, 085012 (2014).

    Article  ADS  Google Scholar 

  24. Pretko, M. & Senthil, T. Entanglement entropy of U(1) quantum spin liquids. Phys. Rev. B 94, 125112 (2016).

    Article  ADS  Google Scholar 

  25. Montvay, I. & Muenster, G. Quantum Fields on a Lattice (Cambridge Univ. Press, Cambridge, 1994).

    Book  Google Scholar 

  26. White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863–2866 (1992).

    Article  ADS  Google Scholar 

  27. Haldane, F. Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Néel state. Phys. Rev. Lett. 50, 1153–1156 (1983).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Schulz, H. Phase diagrams and correlation exponents for quantum spin chains of arbitrary spin quantum number. Phys. Rev. B 34, 6372–6385 (1986).

    Article  ADS  Google Scholar 

  30. Affleck, I., Kennedy, T., Lieb, E. H. & Tasaki, H. Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett. 59, 799–802 (1987).

    Article  ADS  Google Scholar 

  31. Cho, G. Y., Ludwig, A. W. W. & Ryu, S. Universal entanglement spectra of gapped one-dimensional field theories. Phys. Rev. B 95, 115122 (2017).

    Article  ADS  Google Scholar 

  32. Qi, X.-L., Hughes, T. L. & Zhang, S.-C. Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  36. van Bijnen, R. M. W. & Pohl, T. Quantum magnetism and topological ordering via Rydberg dressing near Förster resonances. Phys. Rev. Lett. 114, 243002 (2015).

    Article  ADS  Google Scholar 

  37. Glaetzle, A. W. et al. Designing frustrated quantum magnets with laser-dressed Rydberg atoms. Phys. Rev. Lett. 114, 173002 (2015).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  39. Jaksch, D. & Zoller, P. Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms. New J. Phys. 5, 56–66 (2003).

    Article  ADS  Google Scholar 

  40. Gerbier, F. & Dalibard, J. Gauge fields for ultracold atoms in optical superlattices. New J. Phys. 12, 033007 (2010).

    Article  ADS  Google Scholar 

  41. Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017).

    Article  ADS  Google Scholar 

  42. de Léséleuc, S. et al. Accurate mapping of multilevel Rydberg atoms on interacting spin-1/2 particles for the quantum simulation of Ising models. Phys. Rev. Lett. 120, 113602 (2018).

    Article  ADS  Google Scholar 

  43. Lahaye, T., Menotti, C., Santos, L., Lewenstein, M. & Pfau, T. The physics of dipolar bosonic quantum gases. Rep. Prog. Phys. 72, 126401 (2009).

    Article  ADS  Google Scholar 

  44. Cai, J., Retzker, A., Jelezko, F. & Plenio, M. B. A large-scale quantum simulator on a diamond surface at room temperature. Nat. Phys. 9, 168–173 (2013).

    Article  Google Scholar 

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

    Article  Google Scholar 

  46. Senko, C. et al. Coherent imaging spectroscopy of a quantum many-body spin system. Science 345, 430–433 (2014).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  48. Koffel, T., Lewenstein, M. & Tagliacozzo, L. Entanglement entropy for the long-range Ising chain in a transverse field. Phys. Rev. Lett. 109, 267203 (2012).

    Article  ADS  Google Scholar 

  49. Saffman, M., Walker, T. G. & Mølmer, K. Quantum information with Rydberg atoms. Rev. Mod. Phys. 82, 2313–2363 (2010).

    Article  ADS  Google Scholar 

  50. Johansson, J. R., Nation, P. D. & Nori, F. QuTiP 2: A Python framework for the dynamics of open quantum systems. Comput. Phys. Commun. 184, 1234–1240 (2013).

    Article  ADS  Google Scholar 

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Acknowledgements

We thank V. Alba, P. Calabrese, L. Chomaz, R. Fazio, C. Roos, E. Tonni and R. van Bijnen for useful discussions. M.D. thanks M. Falconi for useful discussions and clarifications. Work in Innsbruck was supported in part by the ERC Synergy Grant UQUAM, SIQS and the SFB FoQuS (FWF project no. F4016-N23). Work in Trieste was supported in part by the ERC Starting grant AGEnTh.

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Correspondence to M. Dalmonte.

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Dalmonte, M., Vermersch, B. & Zoller, P. Quantum simulation and spectroscopy of entanglement Hamiltonians. Nature Phys 14, 827–831 (2018). https://doi.org/10.1038/s41567-018-0151-7

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