# Reachable Sets from Toy Models to Controlled Markovian Quantum Systems

## TL;DR

This paper characterizes the reachable states of controlled open quantum systems, especially at zero temperature, using a framework involving permutations and dissipative contractions, advancing understanding of quantum controllability.

## Contribution

It introduces a novel approach to analyze reachable sets in controlled Markovian quantum systems using simple control operations and provides a complete characterization at zero temperature.

## Key findings

- Complete characterization of reachable sets at T=0.
- Inclusions for reachable sets at T>0.
- Framework connecting toy models to complex quantum systems.

## Abstract

In the framework of bilinear control systems, we present reachable sets of coherently controllable open quantum systems with switchable coupling to a thermal bath of arbitrary temperature $T geq 0$. The core problem boils down to studying points in the standard simplex amenable to two types of controls that can be used interleaved: (i) permutations within the simplex, (ii) contractions by a dissipative one-parameter semigroup. Our work illustrates how the solutions of the core problem pertain to the reachable set of the original controlled Markovian quantum system. We completely characterize the case $T=0$ and present inclusions for $T>0$.