Finite-Dimensional Modular and Entropic Methods for Quantum State Optimization

We develop a nite-dimensional framework for modular and entropic methods in quantum state optimization. For a faithful Gibbs state and a nite Lindblad generator satisfying anexplicit paired detailed-balance condition, we prove stationarity, derive the relative-entropy production identity, and record the consequence of a separately assumed modied logarithmic Sobolev inequality. We also derive the relative-entropy mirror step and a conditional trace-norm perturbationestimate. The algorithm is formulated with faithful-state regularization so that every logarithm isdened. The results are deliberately conditional: contraction constants, optimization convergence,and hardware performance are not inferred without additional hypotheses or experiments.