Illustrating the Geometry of Coherently Controlled Quantum Channels

TL;DR

This paper explores the geometric structure of controlled quantum channels using Lie semigroups, providing insights into how quantum systems can be steered through Hamiltonian controls and their implications for quantum dynamics.

Contribution

It introduces a differential geometric framework for controlled quantum channels using Lie wedges, extending the understanding of quantum control beyond standard dissipative interactions.

Findings

Characterization of Lie wedges for controlled quantum channels

Insight into the steerability of open quantum systems

Potential applications in approximating reachable quantum states

Abstract

We extend standard Markovian open quantum systems (quantum channels) by allowing for Hamiltonian controls and elucidate their geometry in terms of Lie semigroups. For standard dissipative interactions with the environment and different coherent controls, we particularly specify the tangent cones (Lie wedges) of the respective Lie semigroups of quantum channels. These cones are the counterpart of the infinitesimal generator of a single one-parameter semigroup. They comprise all directions the underlying open quantum system can be steered to and thus give insight into the geometry of controlled open quantum dynamics. Such a differential characterisation is highly valuable for approximating reachable sets of given initial quantum states in a plethora of experimental implementations.