Quantitative Convergence and Spectral-Projector Stability for Finite-Dimensional GKSL Semigroups, with Illustrations on Truncated Bosonic Models

We develop a quantitative framework for analyzing convergence and spectral-projector stability in finite-dimensional open quantum systems governed by Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) dynamics. Our focus is on computable bounds for the approach to the asymptotic sector and on perturbative stability estimates for the associated spectral projection. We distinguish two geometrically different cases: (i) primitive semigroups with a unique steady state, and (ii) non-primitive semigroups with multidimensional steady-state structure relevant to encoded quantum information. We then illustrate the framework on truncated bosonic models motivated by dissipatively stabilized cat-qubit memories. Validation code and numerical results available at https://doi.org/10.17605/OSF.IO/GP3Z9