Localised Davies generators for unbounded operators

TL;DR

This paper extends the construction of localised Davies generators, originally for finite-dimensional systems, to unbounded operators like pseudodifferential operators, broadening their applicability in quantum dynamics.

Contribution

It demonstrates that the localisation-based Davies generator construction applies to classes of unbounded operators, including pseudodifferential operators.

Findings

Construction works for unbounded operators including pseudodifferential operators.

The generators are expected to settle to the Gibbs state in the unbounded case.

Extension of quantum Gibbs samplers to broader operator classes.

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.