Communications in Mathematical Physics

, Volume 340, Issue 2, pp 499–561 | Cite as

Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems

  • Markus P. MüllerEmail author
  • Emily Adlam
  • Lluís Masanes
  • Nathan Wiebe


It has previously been suggested that small subsystems of closed quantum systems thermalize under some assumptions; however, this has been rigorously shown so far only for systems with very weak interaction between subsystems. In this work, we give rigorous analytic results on thermalization for translation-invariant quantum lattice systems with finite-range interaction of arbitrary strength, in all cases where there is a unique equilibrium state at the corresponding temperature. We clarify the physical picture by showing that subsystems relax towards the reduction of the global Gibbs state, not the local Gibbs state, if the initial state has close to maximal population entropy and certain non-degeneracy conditions on the spectrumare satisfied.Moreover,we showthat almost all pure states with support on a small energy window are locally thermal in the sense of canonical typicality. We derive our results from a statement on equivalence of ensembles, generalizing earlier results by Lima, and give numerical and analytic finite size bounds, relating the Ising model to the finite de Finetti theorem. Furthermore, we prove that global energy eigenstates are locally close to diagonal in the local energy eigenbasis, which constitutes a part of the eigenstate thermalization hypothesis that is valid regardless of the integrability of the model.


Inverse Temperature Boundary Condition Gibbs State Energy Eigenstates Periodic Boundary Condition 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Markus P. Müller
    • 1
    • 2
    • 3
    Email author
  • Emily Adlam
    • 4
  • Lluís Masanes
    • 5
    • 6
  • Nathan Wiebe
    • 7
    • 8
    • 9
  1. 1.Institut für Theoretische PhysikUniversität HeidelbergHeidelbergGermany
  2. 2.Department of Applied Mathematics, Department of PhilosophyUniversity of Western OntarioLondonCanada
  3. 3.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  4. 4.Centre for Quantum Information and Foundations, DAMTP, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK
  5. 5.H. H. Wills Physics LaboratoryUniversity of BristolBristolUK
  6. 6.Department of Physics and AstronomyUniversity College LondonLondonUK
  7. 7.Quantum Architectures and Computation Group, Microsoft ResearchRedmondUSA
  8. 8.Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada
  9. 9.Department of Combinatorics and Opt.University of WaterlooWaterlooCanada

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