State--Generator Geometry of Open Quantum Systems: Compatibility and Covariant Transport

arXiv:2608.19175v1 Announce Type: new Abstract: We develop a geometry for transporting stationary-state response across the control space of an open quantum system. A physical model is represented by the ordered pair of its stationary state and dynamical generator. Embedding these pairs in a common ambient space induces a metric, a response one-form, and a closed two-form on the control manifold. The ambient space admits a canonical complex structure that exchanges state and generator directions, but the Liouvillian null-state condition restricts physical models to a submanifold that need not preserve this structure. For an amplitude-damped optical Bloch model, the metric and response two-form are compatible at the single point $\omega=0$ and $g/\Gamma=1/\sqrt{2}$; the physical manifold is non-K\"ahler elsewhere. The induced metric defines the Levi--Civita connection, geodesics, and parallel transport without requiring K\"ahler compatibility. We compute the connection by automatic differentiation through the stationary Liouvillian solve and recover the symbolic result to machine precision. A two-point calculation then shows that the resulting geodesic differs from linear interpolation in control space and follows a shorter path through the family of state--generator models. This geometry supplies the intrinsic derivative and transport structure needed to carry observable response, including multidimensional spectra, between admissible stationary models.

Abstract:We develop a geometry for transporting stationary-state response across the control space of an open quantum system. A physical model is represented by the ordered pair of its stationary state and dynamical generator. Embedding these pairs in a common ambient space induces a metric, a response one-form, and a closed two-form on the control manifold. The ambient space admits a canonical complex structure that exchanges state and generator directions, but the Liouvillian null-state condition restricts physical models to a submanifold that need not preserve this structure. For an amplitude-damped optical Bloch model, the metric and response two-form are compatible at the single point $\omega=0$ and $g/\Gamma=1/\sqrt{2}$; the physical manifold is non-Kähler elsewhere. The induced metric defines the Levi--Civita connection, geodesics, and parallel transport without requiring Kähler compatibility. We compute the connection by automatic differentiation through the stationary Liouvillian solve and recover the symbolic result to machine precision. A two-point calculation then shows that the resulting geodesic differs from linear interpolation in control space and follows a shorter path through the family of state--generator models. This geometry supplies the intrinsic derivative and transport structure needed to carry observable response, including multidimensional spectra, between admissible stationary models.

arXiv:2608.19175 [quant-ph]

(or arXiv:2608.19175v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.19175 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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State--Generator Geometry of Open Quantum Systems: Compatibility and Covariant Transport

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