Home | About Journal | Web Links | E-mail Alerts | RSS RSS Icon | Browse
Previous Article Next Article

Emergence of equilibrium thermodynamic properties in quantum pure states. I. Theory

Source: J. Chem. Phys. 133, 034509 (2010); doi:10.1063/1.3455998

Published 20 July 2010

EDITORIALLY RELATED
  1. Emergence of equilibrium thermodynamic properties in quantum pure states. II. Analysis of a spin model system
    Barbara Fresch et al.
    J. Chem. Phys. 133, 034510 (2010)
KEYWORDS and PACS
Keywords
PACS
  • 05.30.Ch
    Quantum ensemble theory
  • 05.70.Ce
    Thermodynamic functions and equations of state
  • 03.65.Ge
    Solutions of wave equations: bound states in quantum mechanics
  • 02.50.Ng
    Distribution theory and Monte Carlo studies
  • 03.67.-a
    Quantum information
  • YEAR: 2010
RELATED DATABASES

To view database links for this article,
you need to log in.
To view database links for this article,
you need to log in.
PUBLICATION DATA
Publisher:
AIP is a member of CrossRef AIP
Barbara Fresch and Giorgio J. Moro
Department of Chemical Science, University of Padova, Via Marzolo 1, Padova 35131, Italy
Investigation on foundational aspects of quantum statistical mechanics recently entered a renaissance period due to novel intuitions from quantum information theory and to increasing attention on the dynamical aspects of single quantum systems. In the present contribution a simple but effective theoretical framework is introduced to clarify the connections between a purely mechanical description and the thermodynamic characterization of the equilibrium state of an isolated quantum system. A salient feature of our approach is the very transparent distinction between the statistical aspects and the dynamical aspects in the description of isolated quantum systems. Like in the classical statistical mechanics, the equilibrium distribution of any property is identified on the basis of the time evolution of the considered system. As a consequence equilibrium properties of quantum system appear to depend on the details of the initial state due to the abundance of constants of the motion in the Schrödinger dynamics. On the other hand the study of the probability distributions of some functions, such as the entropy or the equilibrium state of a subsystem, in statistical ensembles of pure states reveals the crucial role of typicality as the bridge between macroscopic thermodynamics and microscopic quantum dynamics. We shall consider two particular ensembles: the random pure state ensemble and the fixed expectation energy ensemble. The relation between the introduced ensembles, the properties of a given isolated system, and the standard quantum statistical description are discussed throughout the presentation. Finally we point out the conditions which should be satisfied by an ensemble in order to get meaningful thermodynamical characterization of an isolated quantum system. ©2010 American Institute of Physics
History: Received 16 March 2010; accepted 2 June 2010; published 20 July 2010
Permalink: http://link.aip.org/link/?JCPSA6/133/034509/1

REFERENCES (58)

For access to fully linked references, you need to log in. For access to fully linked references, you need to Log in.
  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, New York, 2000).
  2. V. Vedral, Nature (London) 453, 1004 (2008).
  3. S. Popescu, A. J. Short, and A. Winter, Nature (London) 2, 754 (2006)
  4. S. Popescu, A. J. Short, and A. Winter, e-print arXiv:quant-ph/org/abs/0511225, 2005.
  5. W. H. Zurek, Phys. Rev. D 26, 1862 (1982)
    6. Rev. Mod. Phys. 75, 715 (2003).
  6. M. Schlosshauer, Rev. Mod. Phys. 76, 1267 (2005).
  7. R. Baer and R. Kosloff, J. Chem. Phys. 106, 8862 (1997).
  8. G. Katz, D. Gelman, M. A. Ratner, and R. Kosloff, J. Chem. Phys. 129, 034108 (2008).
  9. C. M. Goletz and F. Grossmann, J. Chem. Phys. 130, 244107 (2009).
  10. E. Schrödinger, Phys. Rev. 28, 1049 (1926).
  11. R. C. Tolman, The Principles of Statistical Mechanics (Dover, New York, 1980).
  12. K. Huang, Statistical Mechanics (Wiley, New York, 1987).
  13. S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghì, Phys. Rev. Lett. 96, 050403 (2006).
  14. P. Reimann, Phys. Rev. Lett. 99, 160404 (2007).
  15. G. Mahler, J. Gemmer, and M. Michel, Physica E (Amsterdam) 29, 53 (2005).
  16. R. D. Levine, J. Stat. Phys. 52, 1203 (1988).
  17. J. Brickmann, Y. M. Engel, and R. D. Levine, Chem. Phys. Lett. 137, 441 (1987).
  18. V. Buch, R. B. Gerber, and M. A. Ratner, J. Chem. Phys. 76, 5397 (1982).
  19. V. Buch, R. B. Gerber, and M. A. Ratner, J. Chem. Phys. 81, 3393 (1984).
  20. A. Gross and R. D. Levine, J. Chem. Phys. 125, 144516 (2006).
  21. B. V. Fine, Phys. Rev. E 80, 051130 (2009).
  22. D. C. Brody, D. W. Hook, L. P. Hughston, Proc. R. Soc. London, Ser. A 463, 2021 (2007)
    23. e-print arXiv:quant-ph/05061632005, 2005.
  23. C. M. Bender, D. C. Brody, and D. W. Hook, J. Phys. A 38, L607 (2005).
  24. R. D. Levine, J. Phys. Chem. 99, 2561 (1995).
  25. B. d'Espagnat, Found. Phys. 20, 1147 (1990).
  26. J. Gemmer and G. Mahler, Eur. Phys. J. D 17, 385 (2001).
  27. J. Gemmer and G. Mahler, Europhys. Lett. 59, 159 (2002).
  28. J. von Neumann, Gött. Nachr. 1927, 273;
    29. see also Collected Works of John von Neumann, A. H. Taub, Ed. (Pergamon, Oxford, 1961), Vol. 1, 236.
  29. C. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois, Urbana, 1949).
  30. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, Cambridge, 1996).
  31. Notice that the uniform probability distribution on the phases is equivalent to the postulate of random phases of standard quantum statistical mechanics (Ref. 11) but in the present framework there are no reasons to postulate it since it emerges naturally from the definition of the PSD itself.
  32. N. Linden, S. Popescu, A. J. Short, and A. Winter, Phys. Rev. E 79, 061103 (2009).
  33. T. Guhr, A. Muller–Groeling, and H. A. Weidenmuller, Phys. Rep. 299, 189 (1998).
  34. T. Zimmermann, L. S. Cederbaum, H. D. Meyer, and H. Koeppel, J. Phys. Chem. 4446, 91 (1987).
  35. J. L. Lebowitz and L. Pastur, J. Phys. A 37, 1517 (2004).
  36. D. C. Brody and L. P. Hughston, J. Geom. Phys. 38, 19 (2001).
  37. G. H. Walker and J. Ford, Phys. Rev. 188, 416 (1969).
  38. Ya. G. Sinai, Dokl. Akad. Nauk SSSR 153, 1261 (1963)
    39. Sov. Math. Dokl. 4, 1818 (1963).
  39. O. Penrose, Foundations of Statistical Mechanics. A Deductive Treatment (Dover, New York, 2005).
  40. G. Gallavotti, Eur. Phys. J. B 64, 315 (2008).
  41. It should be mentioned that, while the justification of the microcanonical ensemble in equilibrium classical mechanics has been the object of a lively debate for almost one century, in quantum mechanics the microcanonical ensemble is simply assumed, and the only justification remains an analogy with classical ergodicity and the agreement with experimental observation.
  42. J. Naudts and E. Van der Straeten, J. Stat. Mech.: Theory Exp. 2006, P06015 (2006).
  43. J. W. Gibbs, Elementary Principles in Statistical Mechanics. Developed with Especial Reference to the Foundation of Thermodynamics (Yale Univ. Press, New Haven, CT, 1902).
  44. E. T. Jaynes, in Delaware Seminar in the Foundations of Physics, edited by M. Bunge (Springer-Verlag, Berlin, 1967).
  45. J. L. Lebowitz and O. Penrose, Phys. Today 26, 23 (1973).
  46. G. Gallavotti, F. Bonetto, and G. Gentile, Aspects of the Ergodic, Qualitative and Statistical Theory of Motion (Springer-Verlag, Berlin, 2004).
  47. For the relevance in quantum information protocols see: A. Harrow, P. Hayden, and D. Leung, Phys. Rev. Lett. 92, 187901 (2004)
    48. C. H. Bennett, P. Hayden, D. Leung, P. Shor, and A. Winter, IEEE Trans. Inf. Theory 51, 56 (2005)
    With reference to their entanglement properties see: E. Lubkin, J. Math. Phys. 19, 1028 (1978)
    D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)
    A. J. Scott and C. M. Caves, J. Phys. A 36, 9553 (2003)
    O. Giraud, ibid. 40, 2793 (2007)
    A scheme to generate them is presented in J. Emerson, Y. Weinstein, M. Saraceno, S. Lloyd, and D. Cory, Science 302, 2098 (2003).
  48. B. Fresch and G. J. Moro, J. Phys. Chem. A 113, 14502 (2009).
  49. A. Peres, Phys. Rev. A 30, 504 (1984).
  50. J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).
  51. H. Tasaki, Phys. Rev. Lett. 80, 1373 (1998).
  52. G. Casati, Nuovo Cimento B 114, 11 (1999).
  53. M. Srednicki, Phys. Rev. E 50, 888 (1994).
  54. M. Srednicki, J. Phys. A 29, L75 (1996).
  55. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).
  56. B. Galvan, Found. Phys. 37, 1540 (2007).
  57. S. B. Volchan, Stud. Hist. Philos. Mod. Phys. 38, 801 (2007).
  58. P. Hayden, D. W. Leung, and A. Winter, Commun. Math. Phys. 265, 95 (2006).
  59. A. Y. Khinchin, Mathematical Foundations of Statistical Mechanics (Dover, New York, 1949).

CITING ARTICLES
For access to citing articles, you need to log in.
For access to citing articles, you need to Log in.
ADVERTISEMENT