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


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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© 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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