# On Howland time-independent formulation of CP-divisible quantum evolutions

## TL;DR

This paper extends the Howland time-independent formalism to periodic quantum dynamics governed by Lindbladians, providing a new framework for analyzing open quantum systems with time-dependent behavior.

## Contribution

It introduces a generalized space of states using Bochner spaces and constructs time-independent Lindbladians for periodic quantum evolutions.

## Key findings

- Constructed a generalized space of states for periodic quantum systems.
- Formulated time-independent Lindbladians that generate Markovian dynamics.
- Proved the resulting semigroups are completely positive, trace-preserving, and contractions.

## Abstract

We extend Howland time-independent formalism to the case of completely positive and trace preserving dynamics of finite dimensional open quantum systems governed by periodic, time dependent Lindbladian in Weak Coupling Limit, expanding our result from previous papers. We propose the Bochner space of periodic, square integrable matrix valued functions, as well as its tensor product representation, as the generalized space of states within the time-independent formalism. We examine some densely defined operators on this space, together with their Fourier-like expansions and address some problems related to their convergence by employing general results on Banach-space valued Fourier series, such as the generalized Carleson-Hunt theorem. We formulate Markovian dynamics in the generalized space of states by constructing appropriate time-independent Lindbladian in standard (Lindblad-Gorini-Kossakowski-Sudarshan) form, as well as one-parameter semigroup of bounded evolution maps. We show their similarity with Markovian generators and dynamical maps defined on matrix space, i.e. the generator still possesses a standard form (extended by closed perturbation) and the resulting semigroup is also completely positive, trace preserving and a contraction.