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Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems

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Abstract

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.

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Authors and Affiliations

  1. Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 19, 69120, Heidelberg, Germany

    Markus P. Müller

  2. Department of Applied Mathematics, Department of Philosophy, University of Western Ontario, 1151 Richmond Street, London, ON, N6A 5B7, Canada

    Markus P. Müller

  3. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada

    Markus P. Müller

  4. Centre for Quantum Information and Foundations, DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Emily Adlam

  5. H. H. Wills Physics Laboratory, University of Bristol, Bristol, BS8 1TL, UK

    Lluís Masanes

  6. Department of Physics and Astronomy, University College London, London, WC1E 6BT, UK

    Lluís Masanes

  7. Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA, 98052, USA

    Nathan Wiebe

  8. Institute for Quantum Computing, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Nathan Wiebe

  9. Department of Combinatorics and Opt., University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Nathan Wiebe

Authors
  1. Markus P. Müller
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  2. Emily Adlam
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  3. Lluís Masanes
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  4. Nathan Wiebe
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Correspondence to Markus P. Müller.

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Communicated by M. M. Wolf

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Müller, M.P., Adlam, E., Masanes, L. et al. Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems. Commun. Math. Phys. 340, 499–561 (2015). https://doi.org/10.1007/s00220-015-2473-y

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  • DOI: https://doi.org/10.1007/s00220-015-2473-y

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