Optimal dissipative pathways in open quantum systems: a variational framework for environmental heat dissipation minimization

Optimal control in open quantum systems is considered, with a focus on a two-level logical qubit system whose time evolution is governed by Markovian dynamics described within the Lindblad (GKSL) formalism. The objective is to minimize the net heat transferred to the environment during the transition while explicitly accounting for the work performed by the external control field. In this context, we formulate a variational optimal control framework for minimizing dissipated heat under fidelity constraints, which establishes existence under standard assumptions of bounded controls and lower semi-continuity of at least one bounded control that minimizes the environmental heat dissipation functional while maintaining the final-state fidelity above a designated threshold. The analysis combines the direct method of the calculus of variations with the Pontryagin Maximum Principle. The direct method in the calculus of variations (in L ∞ spaces) guarantees existence by demonstrating the compactness of admissible controls and the lower semi-continuity of the cost functional. Second, the Pontryagin Maximum Principle for open quantum systems establishes essential optimality conditions, resulting in coupled state–costate equations and switching functions that define optimal controls. We establish a physical foundation for the optimization problem by defining internal energy, heat flow, and work within the Lindblad framework, resulting in a thermodynamically coherent expression for dissipated energy. This formulation allows for both analytical understanding together with a practical numerical approximation based on the MDR-GRAPE algorithm. To demonstrate numerical consistency, we present MDR-GRAPE, a modified gradient-based algorithm that extends GRAPE to dissipative environments, integrating positivity-preserving techniques for density matrices. Numerical experiments demonstrate reproducible convergence across the five independent initializations considered in this study, providing empirical evidence of consistent convergence toward feasible low-dissipation solutions.