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arXiv:2605.13774

Affiliated operators for classical and quantum control

Dimitrios Giannakis, Gage Hoefer

math.OCmath.DSmath.OAquant-phSubmitted May 13, 2026• Announced May 14, 2026• Revised Jul 3, 2026 (v2)
v1 · May 13, 2026v2 · Jul 3, 2026
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Abstract

Using techniques from the theory of von Neumann algebras, we propose a framework for addressing questions of controllability of bilinear systems on infinite dimensional Hilbert spaces. In the setup, we assume only that the drift and control terms arising in a bilinear control system are affiliated with a von Neumann algebra of finite type acting on the same Hilbert space. When the control terms satisfy basic norm bound conditions, we prove existence of time-optimal controls. In the more general setting where all operators may be unbounded, we show how the dynamical Lie algebra for the system is still well-defined and may be used to check approximate controllability of the system in question. We discuss how this approach can be applied to classical dynamical systems through the Koopman operator formalism, and investigate potential candidates for the von Neumann algebra which may guide the choice of controls. We illustrate how an affiliation relation naturally arises in both classical and quantum control systems with a few examples.

Comments
34 pages, no figures. Added results involving control problems over reproducing kernel Hilbert algebras

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Using techniques from the theory of von Neumann algebras, we propose a framework for addressing questions of controllability of bilinear systems on infinite dimensional Hilbert spaces. In the setup, we assume only that the drift and control terms arising in a bilinear control system are affiliated with a von Neumann algebra of finite type acting on the same Hilbert space. When the control terms satisfy basic norm bound conditions, we prove existence of time-optimal controls. In the more general setting where all operators may be unbounded, we show how the dynamical Lie algebra for the system is still well-defined and may be used to check approximate controllability of the system in question. We discuss how this approach can be applied to classical dynamical systems through the Koopman operator formalism, and investigate potential candidates for the von Neumann algebra which may guide the choice of controls. We illustrate how an affiliation relation naturally arises in both classical and quantum control systems with a few 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.
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 the framework of bilinear control of the Schrödinger equation with bounded control operators, it has been proved that the reachable set has a dense complemement in S∩H2{\cal S}\cap {\cal H}^{2}S∩H2. Hence, in this setting, exact quantum control in infinite dimensions is not possible. On the other hand it is known that there is a simple choice of operators which, when applied to an arbitrary state, generate dense orbits in Hilbert space. Compatibility of these two results is established in this paper and, in particular, it is proved that the closure of the reachable set of bilinear control by bounded operators is dense in S∩H2{\cal S}\cap {\cal H}^{2}S∩H2. The requirements for controllability in infinite dimensions are also related to the properties of the infinite dimensional unitary group.
The study of mathematical connections between operator-theoretic formulations of classical dynamics and quantum mechanics began at least as early as the 1930s in work of Koopman and von Neumann and was developed in later decades by many authors, often independently, into a framework now broadly known as Koopman-von Neumann representation of classical dynamics. This article surveys aspects of this framework for measure-preserving ergodic dynamical systems and connects it with recent approximation techniques for Koopman and transfer operators that are amenable to data-driven numerical implementation. In broad terms, these methods are based on representations of (i) classical observables as elements of an algebra of operators acting on a Hilbert space; and (ii) classical probability measures as elements of the state space of that algebra, with lifted versions of the Koopman and transfer operators inducing dynamical evolution of observables and states, respectively. A common theme underlying the techniques surveyed here is the use of reproducing kernel Hilbert spaces with coalgebra structure (so-called "reproducing kernel Hilbert algebras'') that aids the quantum representation of classical objects, as well as the use of Fock spaces to build approximation schemes with high expressivity and structure preservation properties (notably, preservation of positivity and multiplicativity of composition operators). Applications to quantum algorithms for approximating the Koopman evolution of observables in systems with pure point spectra are also discussed.
Given a compact, connected Lie group G with Lie algebra � . We discuss time-optimal control of bilinear systems of the form (I) where H d , H j ∈ � , U ∈ G , and the v j act as control variables. The case G = SU (2 n ) has found interesting appli- cations to questions of time-optimal control of spin systems. In this context Eq. (I) describes the dynamics of an n -particle system with fixed drift Hamiltonian H d , which is to be controlled by a number of exterior magnetic fields of variable strength, proportional to the parameters v j . The question of interest here is to transfer the sys- tem from a given initial state U 0 to a prescribed final state U 1 in least possible time. Denote bythe Lie algebra spanned by H 1 , …, H m , and by K the corresponding Lie subgroup of G. After reformulating the optimal control problem for system (I) in terms of an equivalent problem on the homogeneous space G / K we discuss in detail time-optimal control strategies for system (I) in the case where G / K carries the structure of a Riemannian sym- metric space.