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.

  • Letter
  • Published:

Scaling of entanglement close to a quantum phase transition

Nature volume 416, pages 608–610 (2002)Cite this article

Abstract

Classical phase transitions occur when a physical system reaches a state below a critical temperature characterized by macroscopic order1. Quantum phase transitions occur at absolute zero; they are induced by the change of an external parameter or coupling constant2, and are driven by quantum fluctuations. Examples include transitions in quantum Hall systems3, localization in Si-MOSFETs (metal oxide silicon field-effect transistors; ref. 4) and the superconductor–insulator transition in two-dimensional systems5,6. Both classical and quantum critical points are governed by a diverging correlation length, although quantum systems possess additional correlations that do not have a classical counterpart. This phenomenon, known as entanglement, is the resource that enables quantum computation and communication8. The role of entanglement at a phase transition is not captured by statistical mechanics—a complete classification of the critical many-body state requires the introduction of concepts from quantum information theory9. Here we connect the theory of critical phenomena with quantum information by exploring the entangling resources of a system close to its quantum critical point. We demonstrate, for a class of one-dimensional magnetic systems, that entanglement shows scaling behaviour in the vicinity of the transition point.

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: The change in the ground-state wavefunction in the critical region is analysed considering dC(1)/dλ as a function of the reduced coupling strength λ.
Figure 2: The finite size scaling is performed for the case of logarithmic divergences22.
Figure 3: As in the case of the nearest-neighbour concurrence, data collapse is also obtained for the next-nearest-neighbour concurrence C(2).
Figure 4: The universality hypothesis for the entanglement is checked by considering the model hamiltonian, defined in equation (1), for a different value of γ.

Similar content being viewed by others

References

  1. Goldenfeld, N. Lectures on Phase Transitions and the Renormalization Group (Addison Wesley, New York, 1992).

    MATH  Google Scholar 

  2. Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, Cambridge, 2000).

    Book  Google Scholar 

  3. Sondhi, S. I., Girvin, S. M., Carini, J. P. & Shahar, D. Continuous phase transitions. Rev. Mod. Phys. 69, 315–333 (1997).

    Article  ADS  Google Scholar 

  4. Kravchenko, S. V., Kravchenko, G. V., Furneaux, J. E., Pudalov, V. M. & D'Iorio, M. Possible metal-insulator transition at B=0 in two dimensions. Phys. Rev. B 50, 8039–8042 (1994).

    Article  ADS  CAS  Google Scholar 

  5. Haviland, D. B., Liu, Y. & Goldman, A. M. Onset of superconductivity in the two-dimensional limit. Phys. Rev. Lett. 62, 2180–2183 (1989).

    Article  ADS  CAS  Google Scholar 

  6. van der Zant, H. S. J., Fritschy, F. C., Elion, W. E., Geerligs, L. J. & Mooij, J. E. Field-induced superconductor-to-insulator transition in Josephson junction arrays. Phys. Rev. Lett. 69, 2971–2974 (1992).

    Article  ADS  CAS  Google Scholar 

  7. Bell, J. S. Speakable and Unspeakable in Quantum Mechanics (Cambridge Univ. Press, Cambridge, 1987).

    MATH  Google Scholar 

  8. Nielsen, M. & Chuang, I. Quantum Computation and Quantum Communication (Cambridge Univ. Press, Cambridge, 2000).

    MATH  Google Scholar 

  9. Preskill, J. Quantum information and physics: some future directions. J. Mod. Opt. 47, 127–137 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  10. Brooke, J., Bitko, D., Rosenbaum, T. F. & Aeppli, G. Quantum annealing of a disordered spin system. Science 284, 779–781 (1999).

    Article  ADS  CAS  Google Scholar 

  11. O'Connors, K. M. & Wootters, W. K. Entangled rings. Phys. Rev. A 63, 052302-1–052302-9 (2001).

    ADS  Google Scholar 

  12. Dür, W., Vidal, G. & Cirac, J. I. Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314-1–062314-12 (2000).

    ADS  MathSciNet  Google Scholar 

  13. Wang, X. Entanglement in the quantum Heisenberg XY model. Phys. Rev. A 64, 012313-1–012313-7 (2001).

    ADS  Google Scholar 

  14. Arnesen, M. C., Bose, S. & Vedral, V. Natural thermal and magnetic entanglement in the 1D Heisenberg model. Phys. Rev. Lett. 87, 017901-1–017901-4 (2001).

    Article  ADS  Google Scholar 

  15. Gunlycke, D., Bose, S., Kendon, V. M. & Vedral, V. Thermal concurrence mixing in a 1D Ising model. Phys. Rev. A 64, 042302-1–042302-7 (2001).

    Article  ADS  Google Scholar 

  16. Osborne, T. J. & Nielsen, M. A. Entanglement, quantum phase transitions, and density matrix renormalization.Preprint quant-ph/0109024 at 〈http://xxx.lanl.gov〉 (2001).

  17. Pfeuty, P. The one-dimensional Ising model with a transverse field. Ann. Phys. 57, 79–90 (1970).

    Article  ADS  CAS  Google Scholar 

  18. Barouch, E. & McCoy, B. M. Statistical mechanics of the XY model. II Spin-correlation functions. Phys. Rev. A 3, 786–804 (1971).

    Article  ADS  Google Scholar 

  19. Bennett, C. J., Bernstein, H. J., Popescu, S. & Schumacher, B. Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046–2052 (1996).

    Article  ADS  CAS  Google Scholar 

  20. Vedral, V., Plenio, M. B., Rippin, M. A. & Knight, P. Quantifying entanglement. Phys. Rev. Lett. 78, 2275–2279 (1997).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  21. Bennett, C. H., DiVincenzo, D. P., Smolin, J. & Wootters, W. K. Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  22. Wootters, W. K. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998).

    Article  ADS  CAS  Google Scholar 

  23. Lieb, E., Schultz, T. & Mattis, D. Two soluble models in antiferromagnetic chain. Ann. Phys. 60, 407–466 (1961).

    Article  ADS  Google Scholar 

  24. Barber, M. N. in Phase Transitions and Critical Phenomena Vol. 8 (eds Domb, C. & Leibovitz, J. L.) 146–259 (Academic, London, 1983).

    Google Scholar 

Download references

Acknowledgements

We thank G.M. Palma, F. Plastina and J. Siewert for discussions. This work was supported by the European Community (IST-SQUBIT) and by INFM-PRA-SSQI.

Author information

Authors and Affiliations

  1. Dipartimento di Metodologie Fisiche e Chimiche (DMFCI), viale A. Doria 6, Catania, 95125, Italy

    A. Osterloh, Luigi Amico & G. Falci

  2. NEST-INFM, Piazza dei Cavaliezi 7, Pisa, I-56126, Italy

    A. Osterloh, Luigi Amico, G. Falci & Rosario Fazio

  3. Scuola Normale Superiore, Piazza dei Cavaliezi 7, Pisa, I-56126, Italy

    Rosario Fazio

Authors
  1. A. Osterloh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luigi Amico
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Falci
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rosario Fazio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rosario Fazio.

Ethics declarations

Competing interests

The authors declare that they have no competing financial interests

Rights and permissions

Reprints and permissions

About this article

Cite this article

Osterloh, A., Amico, L., Falci, G. et al. Scaling of entanglement close to a quantum phase transition. Nature 416, 608–610 (2002). https://doi.org/10.1038/416608a

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/416608a

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