Communications in Mathematical Physics

, Volume 342, Issue 3, pp 965–988

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment

  • Sheldon Goldstein
  • Joel L. Lebowitz
  • Christian Mastrodonato
  • Roderich Tumulka
  • Nino Zanghì
Article
  • 112 Downloads

Abstract

A quantum system (with Hilbert space \({\mathcal {H}_{1}}\)) entangled with its environment (with Hilbert space \({\mathcal {H}_{2}}\)) is usually not attributed to a wave function but only to a reduced density matrix \({\rho_{1}}\). Nevertheless, there is a precise way of attributing to it a random wave function \({\psi_{1}}\), called its conditional wave function, whose probability distribution \({\mu_{1}}\) depends on the entangled wave function \({\psi \in \mathcal {H}_{1} \otimes \mathcal {H}_{2}}\) in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of \({\mathcal {H}_{2}}\) but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about \({\mu_{1}}\), e.g., that if the environment is sufficiently large then for every orthonormal basis of \({\mathcal {H}_{2}}\), most entangled states \({\psi}\) with given reduced density matrix \({\rho_{1}}\) are such that \({\mu_{1}}\) is close to one of the so-called GAP (Gaussian adjusted projected) measures, \({GAP(\rho_{1})}\). We also show that, for most entangled states \({\psi}\) from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval \({[E, E+ \delta E]}\)) and most orthonormal bases of \({\mathcal {H}_{2}}\), \({\mu_{1}}\) is close to \({GAP(\rm {tr}_{2} \rho_{mc})}\) with \({\rho_{mc}}\) the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then \({\mu_{1}}\) is close to \({GAP(\rho_\beta)}\) with \({\rho_\beta}\) the canonical density matrix on \({\mathcal {H}_{1}}\) at inverse temperature \({\beta=\beta(E)}\). This provides the mathematical justification of our claim in Goldstein et al. (J Stat Phys 125: 1193–1221, 2006) that GAP measures describe the thermal equilibrium distribution of the wave function.

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References

  1. 1.
    Billingsley P.: Probability and measure. Wiley, New York (1986)MATHGoogle Scholar
  2. 2.
    Collins, B.: Intégrales matricielles et Probabilités Non-Commutatives. Ph. D. thesis, Department of Mathematics, Université Paris 6 (2003). http://tel.archives-ouvertes.fr/docs/00/04/59/88/PDF/tel-00004306
  3. 3.
    Dürr, D., Goldstein, S., Zanghì, N.: Quantum equilibrium and the origin of absolute uncertainty. J. Stat. Phys. 67, 843–907 (1992). arXiv:quant-ph/0308039
  4. 4.
    Gaspard P., Nagaoka M.: Non-Markovian stochastic Schrödinger equation. J. Chem. Phys. 111(13), 5676–5690 (1999)CrossRefADSGoogle Scholar
  5. 5.
    Gemmer, J., Mahler, G., Michel, M.: Quantum thermodynamics: emergence of thermodynamic behavior within composite quantum systems. Lecture notes in physics, vol. 657. Springer, Berlin (2004)Google Scholar
  6. 6.
    Georgii H.-O.: The equivalence of ensembles for classical systems of particles. J. Stat. Phys. 80, 1341–1378 (1995)CrossRefADSMathSciNetMATHGoogle Scholar
  7. 7.
    Goldstein, S., Lebowitz, J.L., Mastrodonato, C., Tumulka, R., Zanghì, N.: Approach to thermal equilibrium of macroscopic quantum systems. Phys. Rev. E 81, 011109 (2010). arXiv:0911.1724
  8. 8.
    Goldstein, S., Lebowitz, J.L., Tumulka, R., Zanghì, N.: On the distribution of the wave function for systems in thermal equilibrium. J. Stat. Phys. 125, 1193–1221 (2006). arXiv:quant-ph/0309021
  9. 9.
    Goldstein, S., Lebowitz, J.L., Tumulka, R., Zanghì, N.: Canonical typicality. Phys. Rev. Lett. 96, 050403 (2006). arXiv:cond-mat/0511091
  10. 10.
    Goldstein, S., Lebowitz, J.L., Tumulka, R., Zanghì, N.: Any orthonormal basis in high dimension is uniformly distributed over the sphere. Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, in print (2016). arXiv:1406.2576
  11. 11.
    Jozsa R., Robb D., Wootters W.K.: Lower bound for accessible information in quantum mechanics. Phys. Rev. A 49, 668–677 (1994)CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Linden, N., Popescu, S., Short, A.J., Winter, A.: Quantum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103 (2009). arXiv:0812.2385
  13. 13.
    Martin-Löf, A.: Statistical mechanics and the foundations of thermodynamics, Lecture notes in physics, vol. 101. Springer, Berlin (1979)Google Scholar
  14. 14.
    Milman, V.D., Schechtman, G.: Asymptotic theory of finite dimensional normed spaces, Lecture notes in mathematics vol. 1200. Springer, Berlin (1986)Google Scholar
  15. 15.
    Pandya, V., Tumulka, R.: Spin and the thermal equilibrium distribution of wave functions. J. Stat. Phys. 154, 491–502 (2014). arXiv:1306.1659
  16. 16.
    Popescu, S., Short, A.J., Winter, A.: The foundations of statistical mechanics from entanglement: Individual states vs. averages. (2005, Preprint). arXiv:quant-ph/0511225
  17. 17.
    Popescu S., Short A.J., Winter A.: Entanglement and the foundation of statistical mechanics. Nat. Phys. 21(11), 754–758 (2006)CrossRefGoogle Scholar
  18. 18.
    Reimann, P.: Typicality for generalized microcanonical ensembles. Phys. Rev. Lett. 99, 160404 (2007). arXiv:0710.4214
  19. 19.
    Reimann P.: Typicality of pure states randomly sampled according to the Gaussian adjusted projected measure. J. Stat. Phys. 132(5), 921–935 (2008). arXiv:0805.3102
  20. 20.
    Schmidt decomposition. In: Wikipedia, the free encyclopedia. http://en.wikipedia.org/wiki/Schmidt_decomposition. Accessed 19 Dec 2009
  21. 21.
    Schrödinger E.: Statistical thermodynamics, 2nd edn. University Press, Cambridge (1952)Google Scholar
  22. 22.
    Tumulka, R., Zanghì, N.: Smoothness of wave functions in thermal equilibrium. J. Math. Phys. 46, 112104 (2005). arXiv:math-ph/0509028
  23. 23.
    von Neumann, J.: Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik. Zeitschrift für Physik 57, 30–70 (1929). English translation Eur. Phys. J. H 35, 201–237 (2010). arXiv:1003.2133

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Sheldon Goldstein
    • 1
  • Joel L. Lebowitz
    • 1
  • Christian Mastrodonato
    • 2
  • Roderich Tumulka
    • 3
  • Nino Zanghì
    • 2
  1. 1.Departments of Mathematics and PhysicsRutgers University, Hill CenterPiscatawayUSA
  2. 2.Dipartimento di FisicaUniversità di Genova and INFN sezione di GenovaGenoaItaly
  3. 3.Department of MathematicsRutgers University, Hill CenterPiscatawayUSA

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