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Localised Davies generators for unbounded operators

Jeffrey Galkowski, Maciej Zworski

Abstract

A classical Davies generator provides a Lindbladian for which the Gibbs state is stationary. Its construction involves precise knowledge of the Bohr spectrum or equivalently state evolution for all times. Recently Chen, Kastoryano and Gilyen proposed a construction involving localisation in time and carried out it out in the case of finite dimensional Hilbert spaces. The resulting generators are called quantum Gibbs samplers as the corresponding Lindblad is expected to settle to the Gibbs state. In this note, we show that the construction also works for classes of unbounded operators, including pseudodifferential operators used in the study of classical/quantum correspondence in Lindblad evolution.

Localised Davies generators for unbounded operators

Abstract

A classical Davies generator provides a Lindbladian for which the Gibbs state is stationary. Its construction involves precise knowledge of the Bohr spectrum or equivalently state evolution for all times. Recently Chen, Kastoryano and Gilyen proposed a construction involving localisation in time and carried out it out in the case of finite dimensional Hilbert spaces. The resulting generators are called quantum Gibbs samplers as the corresponding Lindblad is expected to settle to the Gibbs state. In this note, we show that the construction also works for classes of unbounded operators, including pseudodifferential operators used in the study of classical/quantum correspondence in Lindblad evolution.

Paper Structure

This paper contains 13 sections, 12 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Localised generators
  3. Statements of the results
  4. Differential and pseudodifferential operators
  5. Examples for Theorem \ref{['t:2']}
  6. Examples for Theorem \ref{['t:1']}
  7. The matrix case revisited
  8. The Davies generator
  9. Davies generator as a limit of localised generators
  10. A localised version
  11. Solving functional equation \ref{['eq:funct']} in the Gaussian case
  12. Unbounded operators
  13. Pseudodifferential operators

Key Result

Theorem 1

Assume eq:assP, eq:assA1, $f = f_\sigma$ in eq:defsigma, $\mathcal{A}$ is closed under taking adjoints, and If $B$ is defined by eq:defB0 with (see eq:b1b2FT for $b_j (t)$'s) then, in the notation of eq:Lind, $\mathcal{L}_f ( e^{-P } ) = 0$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • proof : Proof of Theorem \ref{['t:2']}
  • ...and 12 more