Lie-Semigroup Structures for Reachability and Control of Open Quantum Systems: Viewing Markovian Quantum Channels as Lie Semigroups and GKS-Lindblad Generators as Lie Wedge

TL;DR

This paper introduces a Lie-semigroup framework for analyzing open quantum systems, linking quantum channel properties with controllability and optimal control strategies, and proposes a new numerical approach based on Lie wedges.

Contribution

It unifies the structure of quantum channels using Lie semigroup theory, connecting Markovian properties with controllability and optimal control in open quantum systems.

Findings

Identifies Kossakowski-Lindblad generators as Lie wedges

Links divisibility of quantum channels to Markov properties

Suggests a new numerical approach leveraging Lie wedge structure

Abstract

In view of controlling finite dimensional open quantum systems, we provide a unified Lie-semigroup framework describing the structure of completely positive trace-preserving maps. It allows (i) to identify the Kossakowski-Lindblad generators as the Lie wedge of a subsemigroup, (ii) to link properties of Lie semigroups such as divisibility with Markov properties of quantum channels, and (iii) to characterise reachable sets and controllability in open systems. We elucidate when time-optimal controls derived for the analogous closed system already give good fidelities in open systems and when a more detailed knowledge of the open system (e.g., in terms of the parameters of its Kossakowski-Lindblad master equation) is actually required for state-of-the-art optimal-control algorithms. -- As an outlook, we sketch the structure of a new, potentially more efficient numerical approach explicitly making use of the corresponding Lie wedge.