- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Complexity of energy eigenstates as a mechanism for equilibration

Lluís Masanes1, Augusto J. Roncaglia1,2, and Antonio Acín1,3

  • 1ICFO–Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
  • 2Departamento de Física, FCEyN, UBA and IFIBA, CONICET, Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina
  • 3ICREA–Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain

PDF

Phys. Rev. E 87, 032137 – Published 18 March, 2013

DOI: https://doi.org/10.1103/PhysRevE.87.032137

Abstract

Understanding the mechanisms responsible for the equilibration of isolated quantum many-body systems is a long-standing open problem. In this work we obtain a statistical relationship between the equilibration properties of Hamiltonians and the complexity of their eigenvectors, provided that a conjecture about the incompressibility of quantum circuits holds. We quantify the complexity by the size of the smallest quantum circuit mapping the local basis onto the energy eigenbasis. Specifically, we consider the set of all Hamiltonians having complexity C, and show that almost all such Hamiltonians equilibrate if C is superquadratic in the system size, which includes the fully random Hamiltonian case in the limit C→∞, and do not equilibrate if C is sublinear. We also provide a simple formula for the equilibration time scale in terms of the Fourier transform of the level density. Our results are statistical and, therefore, do not apply to specific Hamiltonians. Yet they establish a fundamental link between equilibration and complexity theory.

View figure in article

Article Text

References (33)

  1. H. B. Callen, Thermodynamics and an Introduction to Thermostatistics (John Wiley & Sons, New York, 1985).
  2. V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (W. A. Benjamin, New York, 1968).
  3. E. Schrödinger, Ann. Phys. (Leipzig) 388, 956 (1927).
  4. J. Von Neumann, Z. Phys. A 57, 30 (1929).
  5. J. Gemmer, M. Michel, and G. Mahler, Quantum Thermodynamics (Springer, Berlin, 2004).
  6. M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) 452, 854 (2008); M. Rigol, Phys. Rev. Lett. 103, 100403 (2009); A. C. Cassidy, C. W. Clark, and M. Rigol, ibid. 106, 140405 (2011).
  7. M. C. Bañuls, J. I. Cirac, and M. B. Hastings, Phys. Rev. Lett. 106, 050405 (2011).
  8. N. Linden, S. Popescu, A. J. Short, and A. Winter, Phys. Rev. E 79, 061103 (2009).
  9. M. Cramer and J. Eisert, New J. Phys. 12, 055020 (2010).
  10. C. Gogolin, M. P. Mueller, and J. Eisert, Phys. Rev. Lett. 106, 040401 (2011).
  11. I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).
  12. T. Kinoshita, T. Wenger, and D. S. Weiss, Nature (London) 440, 900 (2006).
  13. S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Nature (London) 449, 324 (2007).
  14. S. Goldstein, J. L. Lebowitz, C. Mastrodonato, R. Tumulka, and N. Zanghi, Proc. R. Soc. London, Sect. A 466, 3203 (2010).
  15. P. Reimann, Phys. Rev. Lett. 101, 190403 (2008).
  16. A Hamiltonian has no degenerate gap whenever its spectrum is such that if Em−En=Em′−En′ then m=n and m′=n′, or m=m′ and n=n′.
  17. A. J. Short and T. C. Farrelly, arXiv:1110.5759.
  18. M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Phys. Rev. Lett. 100, 030602 (2008).
  19. A. R. Usha Devi and A. K. Rajagopal, Phys. Rev. E 80, 011136 (2009).
  20. Z.-X. Gong and L.-M. Duan, arXiv:1109.4696.
  21. M. A. Nielsen and I. L. Chuang, Quantum Information and Quantum Computation (Cambridge University Press, Cambridge, 2000).
  22. F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, arXiv:1208.0692
  23. The trace norm of a matrix A is ∥A∥1=trA†A. The probability of discriminating between ρ1 and ρ2 is p=12+14∥ρ1−ρ2∥1.
  24. A. Haar, Ann. Math. 34, 147 (1933).
  25. M. L. Metha, Random Matrices, 2nd ed. (Academic Press, New York, 1990).
  26. E. Wigner, Ann. Math. 62, 548 (1955); N. Rosenzweig and C. E. Porter, Phys. Rev. 120, 1698 (1960).
  27. W. G. Brown and L. Viola, Phys. Rev. Lett. 104, 250501 (2010).
  28. Recall that the number of circuits with C gates is finite; hence there is no need to define a measure over them. Also, different circuits can give rise to the same unitary or Hamiltonian; hence, the bound on the proportion ε cannot be directly translated to the set of Hamiltonians.
  29. M. Kastner, Phys. Rev. Lett. 106, 130601 (2011).
  30. J.-S. Caux and J. Mossel, J. Stat. Mech. Theory Exp. (2011) P02023.
  31. Vinayak and M. Znidaric, J. Phys. A 45, 125204 (2012).
  32. F. G. S. L. Brandão, P. Ćwikliński, M. Horodecki, P. Horodecki, J. Korbicz, and M. Mozrzymas, Phys. Rev. E 86, 031101 (2012).
  33. W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics (Springer, New York, 2004).

Outline

Information

Phys. Rev. E 87, 032137– Published 18 March, 2013
  • Received 19 October 2011

©2013 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation