Quantum thermodynamics with coherence: Covariant Gibbs-preserving operation is characterized by the free energy

The resource theory with covariant Gibbs-preserving operations, also called enhanced thermal operations, is investigated. We prove that with the help of a correlated catalyst, the state convertibility for any coherent state is fully characterized by the free energy defined with the quantum relative entropy. We can extend this result to general resource theories in the form that imposing the covariant condition to a general resource theory does not change the state convertibility as long as the initial state is coherent and distillable. This means that the additional constraint from the law of energy conservation is irrelevant in the correlated-catalytic framework.