Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE Conference on Decision and Control (CDC)

A note on time-zero controllability and density of orbits for quantum systems

Andrei Alexandrovich Agrachev, Ugo Boscain, Jean-Paul Gauthier, Mario Sigalotti

Published Dec 1, 2017• 7 citations
Publisher

Abstract

In this paper we give a necessary and sufficient condition for the controllability in arbitrarily small time of a finite-dimensional control-affine quantum system. The condition is expressed in terms of the Lie algebra generated by the controlled Hamiltonians of the system.
DOI
10.1109/cdc.2017.8264480

Related papers

Geometric quantum control

J. W. Clark, TZYH JONG TARN, Dennis G. Lucarelli

The property of controllability of quantum systems is explored, and conditions for controllability based on Lie-algebraic properties are enunciated. Due consideration is given to the unbounded character of some of the operators that arise and to the generally noncompact nature of the operator algebras associated with the system. Results for both finite and infinite-dimensional systems are provided; in the latter case the results are not limited to cases having purely discrete spectra. The applicability of the results to systems with both bound and scattering states is demonstrated through examples, namely the Poschl-Teller potential and the repulsive radial oscillator, which reveal some of the subtleties of the controllability issue. Attention is also directed to transitivity of the pertinent Lie algebra of a system without drift as a necessary and sufficient condition for its controllability.

IEEE International Workshop on Workload Characterization (WWC)•Apr 23, 2004•3 citations

Analytic controllability of time-dependent quantum control systems

Chunhua Lan, Tzyh‐Jong Tarn, Quo-Shin Chi +1 more

The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert space—and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonian have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples.

arXiv:quant-ph/0409147•Journal of Mathematical Physics•Apr 4, 2005•37 citations

Quantum control in infinite dimensions and Banach-Lie algebras

Michael Keyl

In finite dimensions, controllability of bilinear quantum control systems can be decided quite easily in terms of the "Lie algebra rank condition" (LARC), such that only the systems Lie algebra has to be determined from a set of generators. In this paper we study how this idea can be lifted to infinite dimensions. To this end we look at control systems on an infinite dimensional Hilbert space which are given by an unbounded drift Hamiltonian H0and bounded control Hamiltonians H1,...,HN. The drift H0is assumed to have empty continuous spectrum. We use recurrence methods and the theory of Abelian von Neumann algebras to develop a scheme, which allows us to use an approximate version of LARC, in order to check approximate controllability of the control sysPtem in question.

IEEE Conference on Decision and Control (CDC)•Dec 1, 2019•3 citations

A complete classification of operator controllability for closed three-level quantum systems driven by a single control

Sergey Kuznetsov, Alexander Pechen

In this work, we study operator controllability of finite-dimensional closed quantum systems driven by a single coherent control. Two general Lie-algebraic results on controllability of such systems are proved. The first gives sufficient conditions under which controllability of the system is preserved when controlled energy-level shifts are added. The second gives a sufficient basis-element condition under which the dynamical Lie algebra contains $\mathfrak{su}(N)$ and the system is therefore operator controllable. These results, together with known controllability conditions, are applied to obtain a complete classification of operator controllability of all three-level quantum systems in terms of their free and interaction Hamiltonian matrix elements. This classification covers all combinations of energy-level and level-spacing degeneracies, two or three allowed transitions, as well as the presence or absence of controlled energy-level shifts, and gives necessary and sufficient conditions for operator controllability in each case.

quant-pharXiv:2610.14938•Oct 13, 2026•0 citations

Controllability of quantum mechanical systems with continuous spectra

TZYH JONG TARN, J. W. Clark, Dennis G. Lucarelli

The property of controllability of quantum systems is revisited. In the case of a system having this property, couplings to external agents are available whose adjustment can guide the state to any chosen target on a suitably defined manifold (or arbitrarily close to any such target) at any chosen time. With due consideration to unbounded operators corresponding to physical observables possessing continuous spectra, sufficient conditions for controllability based on Lie-algebraic arguments are obtained. The results are not limited to finite-dimensional systems, nor to infinite-dimensional systems with discrete spectra. The applicability of the results to systems with both bound and scattering states is demonstrated for the case of the one-dimensional Poschl-Teller potential. Attention is also directed to transitivity of the pertinent Lie algebra of a system without drift as a necessary and sufficient condition for controllability.

IEEE Conference on Decision and Control (CDC)•Nov 11, 2002•11 citations

The Lie Algebra Rank Condition for Non-Bilinear Quantum System

Domenico D’Alessandro

The controllability condition for finite dimensional quantum systems, the Lie Algebra Rank Condition, has been stated assuming that the right invariant differential system under consideration is bilinear. We remark that this assumption is not necessary and discuss the extension to the general case.

arXiv:quant-ph/0301144•Jan 27, 2003•4 citations
The property of controllability of quantum systems is explored, and conditions for controllability based on Lie-algebraic properties are enunciated. Due consideration is given to the unbounded character of some of the operators that arise and to the generally noncompact nature of the operator algebras associated with the system. Results for both finite and infinite-dimensional systems are provided; in the latter case the results are not limited to cases having purely discrete spectra. The applicability of the results to systems with both bound and scattering states is demonstrated through examples, namely the Poschl-Teller potential and the repulsive radial oscillator, which reveal some of the subtleties of the controllability issue. Attention is also directed to transitivity of the pertinent Lie algebra of a system without drift as a necessary and sufficient condition for its controllability.
The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert space—and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonian have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples.
In finite dimensions, controllability of bilinear quantum control systems can be decided quite easily in terms of the "Lie algebra rank condition" (LARC), such that only the systems Lie algebra has to be determined from a set of generators. In this paper we study how this idea can be lifted to infinite dimensions. To this end we look at control systems on an infinite dimensional Hilbert space which are given by an unbounded drift Hamiltonian H0and bounded control Hamiltonians H1,...,HN. The drift H0is assumed to have empty continuous spectrum. We use recurrence methods and the theory of Abelian von Neumann algebras to develop a scheme, which allows us to use an approximate version of LARC, in order to check approximate controllability of the control sysPtem in question.
In this work, we study operator controllability of finite-dimensional closed quantum systems driven by a single coherent control. Two general Lie-algebraic results on controllability of such systems are proved. The first gives sufficient conditions under which controllability of the system is preserved when controlled energy-level shifts are added. The second gives a sufficient basis-element condition under which the dynamical Lie algebra contains su(N)\mathfrak{su}(N)su(N) and the system is therefore operator controllable. These results, together with known controllability conditions, are applied to obtain a complete classification of operator controllability of all three-level quantum systems in terms of their free and interaction Hamiltonian matrix elements. This classification covers all combinations of energy-level and level-spacing degeneracies, two or three allowed transitions, as well as the presence or absence of controlled energy-level shifts, and gives necessary and sufficient conditions for operator controllability in each case.
The property of controllability of quantum systems is revisited. In the case of a system having this property, couplings to external agents are available whose adjustment can guide the state to any chosen target on a suitably defined manifold (or arbitrarily close to any such target) at any chosen time. With due consideration to unbounded operators corresponding to physical observables possessing continuous spectra, sufficient conditions for controllability based on Lie-algebraic arguments are obtained. The results are not limited to finite-dimensional systems, nor to infinite-dimensional systems with discrete spectra. The applicability of the results to systems with both bound and scattering states is demonstrated for the case of the one-dimensional Poschl-Teller potential. Attention is also directed to transitivity of the pertinent Lie algebra of a system without drift as a necessary and sufficient condition for controllability.