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arXiv:2610.00046

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• Revised Oct 3, 2026 (v2)
v1 · Sep 3, 2026v2 · Oct 3, 2026v3 · Oct 27, 2026
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Abstract

We ask when qubit controls can produce an effective generator that drives every initial state to the same pure state. Lie-algebra closure alone cannot answer this question because it loses the distinction between reversible control and irreversible dissipation. Lie wedges retain this distinction by enforcing nonnegative strengths for dissipative generators. For drift-free qubits with Hamiltonian controls available with either sign and at most one independently actuated dissipator generated by a traceless jump operator, we classify all nontrivial systems into thirteen structural types, or strata. Explicit tests assign each system to a stratum, and necessary and sufficient conditions determine whether its Lie wedge contains a stabilizing generator. These criteria distinguish stabilizable and non-stabilizable systems even when they share the same generated Lie algebra and stratum. In one exceptional stratum, a single dissipative orientation cannot stabilize a pure state even with the available Hamiltonian controls. Averaging two orientations of the same dissipative channel, related by those controls, produces a stabilizing generator.

Comments
29 pages, 1 figure. Revised title and abstract; added worked examples and an explicit convergence bound; reorganized proofs into appendices

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We ask when qubit controls can produce an effective generator that drives every initial state to the same pure state. Lie-algebra closure alone cannot answer this question because it loses the distinction between reversible control and irreversible dissipation. Lie wedges retain this distinction by enforcing nonnegative strengths for dissipative generators. For drift-free qubits with Hamiltonian controls available with either sign and at most one independently actuated dissipator generated by a traceless jump operator, we classify all nontrivial systems into thirteen structural types, or strata. Explicit tests assign each system to a stratum, and necessary and sufficient conditions determine whether its Lie wedge contains a stabilizing generator. These criteria distinguish stabilizable and non-stabilizable systems even when they share the same generated Lie algebra and stratum. In one exceptional stratum, a single dissipative orientation cannot stabilize a pure state even with the available Hamiltonian controls. Averaging two orientations of the same dissipative channel, related by those controls, produces a stabilizing generator.
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.
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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.