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Provable quantum thermalization without statistical averages

Amit Vikram

Abstract

We develop a rigorous system-agnostic method to predict quantum thermalization in an overwhelming fraction of accessible pure states in a many-body system, entirely in terms of certain out-of-time-ordered correlators of few-body observables. In contrast to previous rigorous results on thermalization with semiclassical counterparts, our method is not limited to statistical averages of observables, such as time averages in ergodicity or state averages in mixing. Moreover, consistent with such approaches, we retain the advantage of not requiring a detailed knowledge of energy eigenstate structure or thermodynamically large times, which can become intractable for systems with more than a handful of particles. Our approach is centered on a geometric result that connects thermalization to the alignment of high dimensional subspaces in a Hilbert space, which is determined by the saturation of "controllably nonlocal" out-of-time-ordered correlators. This formalism reduces the problem of establishing pure state quantum thermalization at finite times in almost all complex many-body states to a theoretically or experimentally accessible study of few-body correlators, even in thermodynamically large systems.

Provable quantum thermalization without statistical averages

Abstract

We develop a rigorous system-agnostic method to predict quantum thermalization in an overwhelming fraction of accessible pure states in a many-body system, entirely in terms of certain out-of-time-ordered correlators of few-body observables. In contrast to previous rigorous results on thermalization with semiclassical counterparts, our method is not limited to statistical averages of observables, such as time averages in ergodicity or state averages in mixing. Moreover, consistent with such approaches, we retain the advantage of not requiring a detailed knowledge of energy eigenstate structure or thermodynamically large times, which can become intractable for systems with more than a handful of particles. Our approach is centered on a geometric result that connects thermalization to the alignment of high dimensional subspaces in a Hilbert space, which is determined by the saturation of "controllably nonlocal" out-of-time-ordered correlators. This formalism reduces the problem of establishing pure state quantum thermalization at finite times in almost all complex many-body states to a theoretically or experimentally accessible study of few-body correlators, even in thermodynamically large systems.

Paper Structure

This paper contains 29 sections, 6 theorems, 127 equations.

Table of Contents

  1. Introduction: Provability in quantum statistical mechanics
  2. Quantum statistical mechanics without energy eigenstates
  3. Statistical averages are necessary in classical statistical mechanics
  4. Overview and organization of this paper
  5. Synopsis
  6. Previous work: Predicting thermalization with statistical averages
  7. This work: OTOCs establish thermalization without averages
  8. Geometry: Quantum thermalization and aligned subspaces
  9. Operator products as probes of subspace geometry
  10. Instantaneous quantum thermalization in pure states
  11. Dynamics: Subspace alignment in many-body systems
  12. Typicality of instantaneous subspace alignment
  13. Dynamics of subspace alignment
  14. Application to many-body systems
  15. Discussion: Experiments, timescales, and forecasts
  16. ...and 14 more sections

Key Result

Lemma 3.1

There exist $(D_R - D_{\rho})$ vectors $\lvert a_j\rangle \in \mathcal{H}$, $D_{\rho}$ vectors $\lvert u_k\rangle \in \mathcal{H}$ and $D_{\rho}$ vectors $\lvert v_k\rangle \in \mathcal{H} \oplus \mathcal{H}_{\text{aux}}$, all mutually orthonormal: with $D_{\rho}$ corresponding principal angles $\theta_k \in [0,\pi/2]$, such that $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (13)

  • proof : Justification
  • Lemma 3.1: Halmos' decomposition of two subspaces Halmos2subIntro2sub
  • proof
  • Theorem 3.2: Quantum thermalization in pure states in nearly aligned subspaces
  • proof
  • Proposition 3.3: Pure state thermalization in all bases implies a near-alignment of subspaces
  • proof
  • Corollary 4.1: Quantum thermalization from correlators for pure states of a bath
  • proof
  • Theorem A.1: Autocorrelator smallness predicts correlator smallness
  • ...and 3 more