Entropy production bounds under Hamiltonian and rate matrix constraints

Entropy production (EP) is a fundamental measure of the thermodynamic inefficiency of a physical process. If there are no constraints on the rate matrices and Hamiltonians available to a driving protocol, one can transform a system from any initial Hamiltonian and state distribution to any final Hamiltonian and state distribution with zero EP. We investigate the minimal EP that must be incurred to implement such a transformation, if there are constraints on the set of allowed Hamiltonians and rate matrices. Our first result is that zero EP can be achieved even when the Hamiltonian has only a single controllable degree of freedom, as long as there are no constraints on the rate matrix (beyond detailed balance). We then derive non-trivial bounds on the EP that arise from the presence of simultaneous constraints on the Hamiltonian and the rate matrix. These bounds are determined by an effective non-equilibrium free energy, which reflects the work value of a distribution and Hamiltonian under a set of constraints.