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arXiv:2610.00046

Beyond Lie Algebras: Lie-Wedge Stratification and Pure-State Stabilizability in Single-Channel Qubit Control

Yibin Wang

quant-phSubmitted Sep 3, 2026• Announced Sep 4, 2026• 0 citations
v1 · Sep 3, 2026v2 · Oct 3, 2026v3 · Oct 27, 2026
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Abstract

Lie algebras describe how control generators combine, but in open systems they erase the distinction between reversible control and irreversible dissipation. Lie wedges retain this information. We study drift-free single-qubit systems with bidirectional Hamiltonian controls and at most one independently actuated dissipator generated by a traceless jump operator. In this work, we prove that they fall into thirteen structural strata defined by the generated algebra and its Lie-wedge geometry. The control algebra and four channel properties determine the stratum; in each wedge, the reversible directions are precisely the coherent controls. We then use the stratification to determine whether a wedge contains a generator with a unique globally attractive pure state. In five strata, stabilizability depends on additional channel data. In one exceptional stratum, no single control-rotated dissipative direction suffices, but averaging two such directions in the wedge yields stabilization. Lie wedges therefore reveal control capabilities that the generated Lie algebra misses.

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Lie algebras describe how control generators combine, but in open systems they erase the distinction between reversible control and irreversible dissipation. Lie wedges retain this information. We study drift-free single-qubit systems with bidirectional Hamiltonian controls and at most one independently actuated dissipator generated by a traceless jump operator. In this work, we prove that they fall into thirteen structural strata defined by the generated algebra and its Lie-wedge geometry. The control algebra and four channel properties determine the stratum; in each wedge, the reversible directions are precisely the coherent controls. We then use the stratification to determine whether a wedge contains a generator with a unique globally attractive pure state. In five strata, stabilizability depends on additional channel data. In one exceptional stratum, no single control-rotated dissipative direction suffices, but averaging two such directions in the wedge yields stabilization. Lie wedges therefore reveal control capabilities that the generated Lie algebra misses.
This thesis focuses on the Lie-theoretic foundations of controlled open quantum systems. We describe Markovian open quantum system evolutions by Lie semigroups, whose corresponding infinitesimal generators lie in a special type of convex cone - a Lie wedge. The Lie wedge associated to a given control system therefore consists of all generators of the quantum dynamical semigroup that are physically realisable as a result of the interplay between the coherent and incoherent processes the quantum system is subject to. For nnn-qubit open quantum systems, we provide a parametrisation of the largest physically relevant Lie algebra (the system algebra), in which these Lie wedges are contained: the Lindblad-Kossakowski Lie algebra. This parametrisation provides several useful benefits. First, it allows us to construct explicit forms of these system Lie wedges and their respective system Lie algebras. Second, we analyse which control scenarios yield Lie wedges that are closed under Baker-Campbell-Hausdorff (BCH) multiplication and therefore generate Markovian semigroups of time-independent quantum channels. Lie wedges of this form are called Lie semialgebras, and we completely solve this open problem by proving that Lie wedges specialise to this form only when the coherent controls have no effect on both the inherent drift Hamiltonian and the incoherent part of the dynamics. Finally, this parametrisation of the Lindblad-Kossakowski Lie algebra points to an intuitive separation between unital and non-unital dissipative dynamics, where the non-unital component of the dynamics is described by affine translation operations. These translation operators are then exploited to construct purely dissipative fixed-point engineering schemes to obtain either pure or mixed states as a system's unique fixed point.
Lie algebras describe how control Hamiltonians combine in closed quantum systems. In open Markovian systems, however, dissipation introduces irreversible directions that the Lie algebra alone does not retain. A dynamical Lie wedge preserves this information as a convex cone of locally admissible generators. The classical Levi decomposition separates a finite-dimensional Lie algebra into semisimple and solvable parts. In this work, we prove a corresponding decomposition and reconstruction theorem for dynamical Lie wedges. The theorem decomposes the wedge into semisimple and solvable data, records how these components are coupled, and provides a converse reconstruction of the wedge. The framework yields four structural types of the generated algebra. We further prove that the total dissipation strength of every admissible generator depends only on its coordinate in the solvable radical. In particular, if the generated Lie algebra is semisimple, every admissible generator is Hamiltonian. Together, these results extend Lie-algebraic structural methods to irreversible Markovian control and clarify how dissipation is encoded in the generator geometry.
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.
Determining the unitary dynamics accessible from finite Hamiltonian resources is a central problem in Hamiltonian engineering and quantum control. Dynamical Lie algebras (DLAs) connect available control Hamiltonians with the reachable dynamics, but their use as a design tool for modifying Hamiltonian generator sets remains less developed. In this work, we develop a finite-dimensional DLA framework for three generator-set operations: composition, invariance, and reduction. For composition, we construct direct sums of component DLAs using spectral projectors on an auxiliary register. For invariance, we analyze when modifications of Pauli-string generating sets preserve the generated Lie algebra, and introduce algebraic diagnostics for added generators. For reduction, we consider compact reductive DLAs and show how projection onto selected simple ideals gives reduced generating sets whose Lie closures are the corresponding ideal sums. We illustrate these results with finite-dimensional examples and numerical checks, including direct-sum dimension addition, central-spin invariance diagnostics, and DLA-based ansatz reduction for block-local Hamiltonians. The results show how DLA structure can be used to diagnose controllability and guide Hamiltonian generator design under explicit algebraic assumptions.
Dissipative dynamics can stabilize a quantum target without providing a reliable sequence of finite control operations. We derive checkable conditions for implementing such stabilization with imperfect controls, with bounds on convergence, errors during operations, and physical cost. The analysis includes pulses, idle intervals, and all simultaneously acting background dynamics. Using a common Lyapunov measure of deviation from the target, we require a contraction bound that remains uniformly below one over the prescribed uncertainty set after accounting for implementation errors. If all allowed paths preserve the target subspace throughout each operation, repeated cycles converge to that subspace from every initial state, even when errors vary between cycles. Comparison with these certified target-preserving paths gives an explicit long-time error bound under bounded leakage. Two single-qubit counterexamples show why a stabilizing average or a stabilizing generator obtained by unitary conjugation does not ensure these path conditions. The certification approach also supports a finite-library search. For a specified four-level cascade model with finite pulses, calibration and timing errors, dephasing, and bounded pumping, this search yields a sequence with an explicit preparation-time bound. For every initial state, it guarantees trace distance at most 0.10.10.1 from the target at the end of preparation and throughout subsequent cycles.
We study controllability of finite-dimensional open quantum systems under a general Markovian control model combining full coherent (unitary) control with tunable dissipative channels. Assuming the Hamiltonian controls is a Hörmander system that generate su(n)\mathfrak{su}(n)su(n), we ask how little dissipation suffices to make the full state space D(H)\mathcal{D}(\mathcal{H})D(H) controllable. We show that minimal non-unital noise can break unitary-orbit invariants and, in many cases, a very small set of jump operators yields transitivity on D(H)\mathcal{D}(\mathcal{H})D(H). For multi-qubit systems we prove explicit transitivity results for natural resources such as a single-qubit amplitude-damping jump together with a dephasing channel, and we identify obstructions when only self-adjoint jump operators are available (yielding only unital evolutions). We further develop a geometric viewpoint and ask the ``lifting'' question: when can a path of densities be obtained from applying a time-dependent family of Lindbladian to an initial state? For this, we have to analyze the tangent structure of the ``manifold with corners'' and how this tangent structure reflects Lindbldian evolution. Building on this framework, we derive reachability criteria and no-go results based on a norm-decrease alignment condition, including a geometric obstruction arising from the incompatibility between admissible tangent directions and dissipative contraction.