Applications of Lie systems in Quantum Mechanics and Control Theory

A "minimal" generalization of Quantum Mechanics is proposed, where the Lagrangian or the action functional is a mapping from the (classical) states of a system to the Lie algebra of a general compact Lie group, and the wave function takes values in the corresponding group algebra. This formalism admits a probability interpretation and a suitable dynamics, but has no obvious classical correspondence. Allowing the Lagrangian or the action functional to take values in a general ...

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 g...

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.

We discuss some aspects and examples of applications of dual algebraic pairs $({\cal G}_1,{\cal G}_2)$ in quantum many-body physics. They arise in models whose Hamiltonians $H$ have invariance groups $G_i$. Then one can take ${\cal G}_1 = G_i$ whereas another dual partner ${\cal G}_2= g^D$ is generated by $G_i$ invariants, possesses a Lie-algebraic structure and describes dynamic symmetry of models; herewith polynomial Lie algebras $\hat g = g^D$ appear in models with essenti...

We propose a technique to design control algorithms for a class of finite dimensional quantum systems so that the control law does not present discontinuities. The class of models considered admits a group of symmetries which allows us to reduce the problem of control to a quotient space where the control system is `fully actuated'. As a result we can prescribe a desired trajectory which is, to some extent, arbitrary and derive the corresponding control. We discuss the appl...

We review the geometrical formulation of Quantum Mechanics to identify, according to Klein's programme, the corresponding group of transformations. For closed systems, it is the unitary group. For open quantum systems, the semigroup of Kraus maps contains, as a maximal subgroup, the general linear group. The same group emerges as the exponentiation of the $C^{*}$--algebra associated with the quantum system, when thought of as a Lie algebra. Thus, open quantum systems seem to ...

The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have bee...

In this paper, we define four different notions of controllability of physical interest for multilevel quantum mechanical systems. These notions involve the possibility of driving the evolution operator as well as the state of the system. We establish the connections among these different notions as well as methods to verify controllability. The paper also contains results on the relation between the controllability in arbitrary small time of a system varying on a compact t...

The classical Hamilton equations of motion yield a structure sufficiently general to handle an almost arbitrary set of ordinary differential equations. Employing elementary algebraic methods, it is possible within the Hamiltonian structure to describe many physical systems exhibiting Lie group symmetries. Elementary examples include magnetic moment precession and the mechanical orbits of color charged particles in classical non-abelian chromodynamics.

Classical simulation of quantum dynamics plays an important role in our understanding of quantum complexity, and in the development of quantum technologies. Compared to other techniques for efficient classical simulations, methods relying on the Lie-algebraic structure of quantum dynamics have received relatively little attention. At their core, these simulations leverage the underlying Lie algebra - and the associated Lie group - of a dynamical process. As such, rather than ...