Communications in Mathematical Physics

, Volume 342, Issue 3, pp 965–988 | Cite as

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


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.


Wave Function Hilbert Space Density Matrix Entangle State Gaussian Measure 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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).
  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. 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

Personalised recommendations