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

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.

arXiv:2610.00046 [quant-ph]

(or arXiv:2610.00046v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.00046 Focus to learn more arXiv-issued DOI via DataCite

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