A three-level
Λ -atom (resp.,
V -atom) is a three-level quantum system with a forbidden direct transition between the two lowest (resp., highest) energy states. These systems are particular examples of three-level systems with only two allowed transitions and are important in quantum optics, e.g., for modeling electromagnetically induced transparency, etc. For generality, we consider not only
Λ - and
V -atoms, but all other types of three-level systems with only two allowed transitions between basis states. For the most general case of arbitrary non-zero complex values of the interaction Hamiltonian, we combine in a unified way the description of the structure of the dynamical Lie algebra as well as controllability properties for such systems. It turns out that for non-degenerate transition frequencies the system is completely controllable for any non-zero interaction Hamiltonian, while for the degenerate case the system is either uncontrollable or controllable, depending on the exact values of matrix elements of the free and interaction Hamiltonians.
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) 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.
We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra
A^1, defined as the real subalgebra of the Weyl algebra
A1 consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for
A^1, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in
A^1 generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in
A^1 that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in
A^1? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of
A^1 that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in
A^1, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.
Abstract Last decades have faced considerable interest in various problems of quantum control. One of them is to determine which quantum systems are controllable, meaning that all possible states of such systems can be reached from any initial one under available controls. This paper focuses on controllability exploration of a specific four-level quantum system with three levels having the same energy and driven by an electromagnetic control field which allows transitions only between adjacent basis states. Such systems appear in the analysis of higher order traps in quantum control landscapes. This paper explores a few properties of the system, including a proof it is completely controllable for any complex values of the non-zero matrix elements of the interaction Hamiltonian.
Determining whether quantum states can be interconverted by a prescribed set of transformations is a fundamental problem in quantum information. Here we develop a general classification framework based on conserved operators (COs), i.e., not necessarily hermitian matrices having the state of interest as an eigenstate. We first show that COs belonging to the algebra generating the allowed transformations can be obtained from the kernel of a quantum covariance matrix. Their number determines the dimension of the accessible orbit, while their algebraic structure and eigenvalues provide further constraints on state convertibility. We establish general bounds on the number of COs and prove that highest-weight states, like separable states in the case of
n qudits, attain the maximum. We also show how these operators generate parent Hamiltonians having the characterized state as an exact ground state. We then also consider positive zero-eigenvalue COs, which encode forbidden directions of reduced density matrices and incorporate rank-based constraints into the same framework. Applying this approach to multiqubit SLOCC, we obtain complementary classifications from algebraic and rank COs. While for
n≤3 qubits both lead to the usual classes, for
n=4 qubits they distinguish 17 and 20 types separately, and 29 when combined. The combined classification yields 287 fixed types and
4 continuous families for
n=5 qubits, and 4926 fixed types and 147 continuous families for
n=6 qubits, providing, to our knowledge, the most refined exhaustive classification of SLOCC-convertibility classes based on such data.
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.