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Long-Time Behavior of Typical Pure States from Thermal Equilibrium Ensembles

Cornelia Vogel

TL;DR

The paper extends the concept of normal typicality from the uniform micro-canonical sphere to Gaussian adjusted projected (GAP) measures, showing that for GAP$(\\rho)$-typical initial states, long-time observables $\\langle\\psi_t|B|\\psi_t\\rangle$ concentrate around a fixed value $M_{\\rho B}$ for suitable $B$ under reasonable spectral and smallness conditions. It provides finite-time and infinite-time bounds that scale with the observable norm, the density matrix norm, and spectral degeneracies, and it demonstrates a quantum-ensemble equivalence perspective by connecting GAP measures to canonical-like ensembles. The results hold on both finite and infinite-dimensional Hilbert spaces, leveraging an improved GAP-variance bound and Lévy's Lemma adapted to GAP measures. Together, these findings generalize normal typicality beyond the micro-canonical setting and highlight the robustness of equilibration phenomena under broader quantum statistical ensembles.

Abstract

We consider an isolated macroscopic quantum system in a pure state $ψ_t$ evolving unitarily in a separable Hilbert space $\mathcal{H}$ and take for granted that different macro states $ν$ correspond to mutually orthogonal subspaces $\mathcal{H}_ν\subset\mathcal{H}$. Let $P_ν$ be the projection to $\mathcal{H}_ν$. It was recently shown that for all Hamiltonians with no highly degenerate eigenvalues and gaps most $ψ_0\in\mathcal{H}_μ$ are such that for most $t\geq 0$, $\|P_νψ_t\|^2$ is close to a $t$- and $ψ_0$-independent value $M_{μν}$ provided that $M_{μν}$ is not too small. Here, ``most'' refers to the uniform distribution on the sphere $\mathbb{S}(\mathcal{H}_μ)$. In the present work, we generalize this result from the uniform distribution, corresponding to the micro-canonical ensemble, to the much more general class of Gaussian adjusted projected (GAP) measures. For any density matrix $ρ$ on $\mathcal{H}$, $\mathrm{GAP}(ρ)$ is the most spread out distribution on $\mathbb{S}(\mathcal{H})$ with density matrix $ρ$. We show that also for $\mathrm{GAP}(ρ)$-most $ψ_0\in\mathcal{H}$ for most $t\geq 0$, $\|P_νψ_t\|^2$ is close to a fixed value $M_{ρP_ν}$ (which must not be too small). Moreover, we prove a generalization for certain operators $B$ instead of $P_ν$ and for finite times. Since certain GAP measures are quantum analogs of the (grand-)canonical ensemble, our result expresses a version of equivalence of ensembles.

Long-Time Behavior of Typical Pure States from Thermal Equilibrium Ensembles

TL;DR

The paper extends the concept of normal typicality from the uniform micro-canonical sphere to Gaussian adjusted projected (GAP) measures, showing that for GAP-typical initial states, long-time observables concentrate around a fixed value for suitable under reasonable spectral and smallness conditions. It provides finite-time and infinite-time bounds that scale with the observable norm, the density matrix norm, and spectral degeneracies, and it demonstrates a quantum-ensemble equivalence perspective by connecting GAP measures to canonical-like ensembles. The results hold on both finite and infinite-dimensional Hilbert spaces, leveraging an improved GAP-variance bound and Lévy's Lemma adapted to GAP measures. Together, these findings generalize normal typicality beyond the micro-canonical setting and highlight the robustness of equilibration phenomena under broader quantum statistical ensembles.

Abstract

We consider an isolated macroscopic quantum system in a pure state evolving unitarily in a separable Hilbert space and take for granted that different macro states correspond to mutually orthogonal subspaces . Let be the projection to . It was recently shown that for all Hamiltonians with no highly degenerate eigenvalues and gaps most are such that for most , is close to a - and -independent value provided that is not too small. Here, ``most'' refers to the uniform distribution on the sphere . In the present work, we generalize this result from the uniform distribution, corresponding to the micro-canonical ensemble, to the much more general class of Gaussian adjusted projected (GAP) measures. For any density matrix on , is the most spread out distribution on with density matrix . We show that also for -most for most , is close to a fixed value (which must not be too small). Moreover, we prove a generalization for certain operators instead of and for finite times. Since certain GAP measures are quantum analogs of the (grand-)canonical ensemble, our result expresses a version of equivalence of ensembles.

Paper Structure

This paper contains 21 sections, 4 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Normal Typicality
  3. A Generalization of Normal Typicality for Arbitrary Hamiltonians
  4. Brief Discussion of the Main Result
  5. Structure of the Paper
  6. Background and Mathematical Setup
  7. Measures of "most".
  8. Time averages.
  9. Norms.
  10. Density matrix.
  11. GAP measure.
  12. The Hamiltonian.
  13. Main Result
  14. Statement
  15. Discussion
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\mathcal{H}$ be a separable Hilbert space of dimension $\geq 4$. Let $B$ be a bounded operator with $d_{E,B}<\infty$ and let $\rho$ be a density matrix with $\|\rho\|<1/4$. Let $\varepsilon,\delta,\kappa,T>0$ and define Then, w.r.t. $\textup{GAP}(\rho)$, $(1-\varepsilon)$-most $\psi_0\in\mathbb{S}(\mathcal{H})$ are such that for $(1-\delta)$-most $t\in [0,T]$ where $C=\frac{1}{288\pi^2}$. Mo

Theorems & Definitions (13)

  • Theorem 1: Normal Typicality for $\textup{GAP}(\rho)$
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 2: Lévy's Lemma for GAP measures TTV24
  • Proposition 1
  • proof : Proof of Theorem \ref{['thm: NT GAP']}
  • ...and 3 more