A geometric approach to quantum control in projective Hilbert spaces

TL;DR

This paper explores a geometric Hamiltonian framework for quantum control in finite-dimensional Hilbert spaces, utilizing the projective space structure and Killing vector fields to analyze controllability.

Contribution

It introduces a novel geometric approach to quantum control, applying classical control concepts like accessibility algebra and Killing vector fields to quantum systems.

Findings

Characterization of quantum controllability via Killing vector fields

Application of accessibility algebra in geometric quantum control

Insights into controllability conditions in projective Hilbert spaces

Abstract

A quantum theory in a finite-dimensional Hilbert space can be geometrically formulated as a proper Hamiltonian theory as explained in [2, 3, 7, 8]. From this point of view a quantum system can be described in a classical-like framework where quantum dynamics is represented by a Hamiltonian flow in the phase space given by projective Hilbert space. This paper is devoted to investigate how the notion of accessibility algebra from classical control theory can be applied within geometric Hamiltonian formulation of Quanum Mechanics to study controllability of a quantum system. A new characterization of quantum controllability in terms of Killing vector fields w.r.t. Fubini-Study metric on projective space is also discussed.