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Control theory & applications

Calculations of the controllability Lie algebra for spin quantum systems

Wang Men

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

Controllability of dynamical systems is a fundamental problem for both classical control as well as quantum control.This paper focuses on investigating the controllability Lie algebra of single-spin and two spin quantum systems.Firstly,based on the equivalent controllability conditions of quantum systems,some general necessary and sufficient conditions related to the basis coefficients are obtained by computing Lie brackets of Hamiltonian operators for single-spin quantum systems.Secondly,the matrix basis of Lie algebra su(4) is generated by Cartan decomposition.Using the matrix basis and employing the controllability theorem of quantum system,a calculation method of multiple Lie brackets for the Hamiltonian operators and a controllability criterion of quantum systems are also put forward in this article.

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Controllability of dynamical systems is a fundamental problem for both classical control as well as quantum control.This paper focuses on investigating the controllability Lie algebra of single-spin and two spin quantum systems.Firstly,based on the equivalent controllability conditions of quantum systems,some general necessary and sufficient conditions related to the basis coefficients are obtained by computing Lie brackets of Hamiltonian operators for single-spin quantum systems.Secondly,the matrix basis of Lie algebra su(4) is generated by Cartan decomposition.Using the matrix basis and employing the controllability theorem of quantum system,a calculation method of multiple Lie brackets for the Hamiltonian operators and a controllability criterion of quantum systems are also put forward in this article.
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.
Bilinear systems, matrix systems and right-invariant systems are distinguished first. Then controllability theorems of such three different systems are summarized, the key points focus on the necessary and sufficient controllability conditions for right-invariant systems, and the relation with bilinear systems. Controllability theorems of these three different systems are especially emphasized by using Lie groups and Lie algebra. Different controllability theorems of quantum systems using similar method are analyzed in detail. Corresponding relations of controllability between quantum systems with bilinear systems are pointed out by means of comparison, and each type of controllability applied to different situations are also given. Relationships among different notions of controllability of quantum systems are established.
In this paper, we present a controllability analysis of the quantum Ising periodic chain of n spin 1/2 particles where the interpolating parameter between the two Hamiltonians plays the role of the control. A fundamental result in the control theory of quantum systems states that the set of achievable evolutions is (dense in) the Lie group corresponding to the Lie algebra generated by the Hamiltonians of the system. Such a dynamical Lie algebra therefore characterizes all the state transitions available for a given system. For the Ising spin periodic chain we characterize such a dynamical Lie algebra and therefore the set of all reachable states. In particular, we prove that the dynamical Lie algebra is a (3n-1)-dimensional Lie sub-algebra of su(2^n) which is a direct sum of a two dimensional center and a (3n-3)-dimensional semisimple Lie subalgebra. This in turn is the direct sum of n-1 Lie algebras isomorphic to su(2) parametrized by the eigenvalues of a fixed matrix. We display the basis for each of these Lie subalgebras. Therefore the problem of control for the Ising spin periodic chain is, modulo the two dimensional center, a problem of simultaneous control of n-1 spin 1/2 particles. In the process of proving this result, we develop some tools which are of general interest for the controllability analysis of quantum systems with symmetry.
In this paper, we study the controllability properties and the Lie algebra structure of networks of particles with spin immersed in an electro-magnetic field. We relate the Lie algebra structure to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active. For networks with different gyromagnetic ratios, we provide a necessary and sufficient condition of controllability in terms of the properties of the above mentioned graph and describe the Lie algebra structure in every case. For these systems all the controllability notions, including the possibility of driving the evolution operator and/or the state, are equivalent. For general networks (with possibly equal gyromagnetic ratios), we give a sufficient condition of controllability. A general form of interaction among the particles is assumed which includes both Ising and Heisenberg models as special cases. Assuming Heisenberg interaction we provide an analysis of low dimensional cases (number of particles less then or equal to three) which include necessary and sufficient controllability conditions as well as a study of their Lie algebra structure. This also, provides an example of quantum mechanical systems where controllability of the state is verified while controllability of the evolution operator is not.
The ability to navigate and control the state space of a quantum system is fundamental to quantum computing, quantum information processing, and advanced spectroscopy. This paper explores a rigorous mathematical framework for quantifying state space accessibility by leveraging the theory of operator commutation relations, particularly within the context of Lie algebras. We demonstrate that the span of iterated commutators of system Hamiltonians and control operators forms a dynamical Lie algebra, whose dimension and structure directly dictate the reachable states and unitary transformations. For closed quantum systems, the Lie algebra rank condition provides a powerful criterion for complete controllability, asserting that if the dynamical Lie algebra spans the entire space of anti-Hermitian operators, then any state can be reached from any other, and any unitary operation can be implemented. For open quantum systems, while environmental interactions introduce decoherence and dissipation, the Lie algebraic approach can still offer insights into the attainable subspaces and the limits of control, often involving Lie semigroups and Lie wedges. We present methods to analyze the accessibility of both pure and mixed states, illustrating how the algebraic properties encoded in commutation relations provide a precise measure of the manipulability of quantum systems. The implications extend to the design of robust quantum control protocols, optimized state preparation, and the development of error-resilient quantum technologies.
In this paper, we consider the problem of model equivalence for quantum systems. Two models are said to be (input-output) equivalent if they give the same output for every admissible input. In the case of quantum systems, the output is the expectation value of a given observable or, more in general, a probability distribution for the result of a quantum measurement. We link the input-output equivalence of two models to the existence of a homomorphism of the underlying Lie algebra. In several cases, a Cartan decomposition of the Lie algebra su(n) is useful to find such a homomorphism and to determine the classes of equivalent models. We consider in detail the important cases of two level systems with a Cartan structure and of spin networks. In the latter case, complete results are given generalizing previous results to the case of networks of spin particles with any value of the spin. In treating this problem, we prove some instrumental results on the subalgebras of su(n) which are of independent interest.