# Quantum renewal processes

## TL;DR

This paper introduces quantum renewal processes, a class of non-Markovian quantum dynamics modeled by memory kernels and renewal theory, highlighting the importance of operator ordering in their behavior.

## Contribution

It presents a general construction of quantum renewal processes with memory kernels, extending beyond Lindblad dynamics, and explores the impact of operator ordering on their properties.

## Key findings

- Quantum renewal processes are described by completely positive trace preserving maps.
- Operator ordering significantly influences the resulting quantum dynamics.
- Modified processes with different initial event distributions are naturally incorporated.

## Abstract

We introduce a general construction of master equations with memory kernel whose solutions are given by completely positive trace preserving maps. These dynamics going beyond the Lindblad paradigm are obtained with reference to classical renewal processes, so that they are termed quantum renewal processes. They can be described by means of semigroup dynamics interrupted by jumps, separated by independently distributed time intervals, following suitable waiting time distributions. In this framework, one can further introduce modified processes, in which the first few events follow different distributions. A crucial role, marking an important difference with respect to the classical case, is played by operator ordering. Indeed, for the same choice of basic quantum transformations, different quantum dynamics arise. In particular, for the case of modified processes, it is natural to consider the time inverted operator ordering, in which the last few events are distributed differently.