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 TumulkaEmail author
  • 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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
    Email author
  • 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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