Thermodynamic accounting of control costs for non-degenerate Hamiltonians

Establish the physical control cost associated with implementing sequential quantum work-extraction protocols for systems governed by non-degenerate Hamiltonians, thereby completing the generalization beyond the energy-degenerate case.

Background

The paper develops time-ordered free energy and an optimal dynamic-programming policy under the simplifying assumption that the Hamiltonian of each quantum system is energy-degenerate. Under this assumption, the extraction protocol can access arbitrary bases as energy eigenbases and extract the relevant non-equilibrium free energy without separately accounting for coherence-related control costs.

Appendix sec:nondegen analyzes the non-degenerate case and shows that implementing the required basis rotation can inject additional energy, particularly when the unknown input state is not diagonal in the target-state eigenbasis. Although the authors recover the expected work expression after subtracting this rotation cost, they do not establish whether the rotation can be implemented at its minimal energetic cost or provide a complete accounting of the required external resources. Resolving this issue is necessary for a physically complete extension of the framework to non-degenerate Hamiltonians.