The work cost of keeping states with coherences out of thermal equilibrium

We consider the problem of keeping an arbitrary state $\rho_s$ out of thermal equilibrium. We find that counter-acting thermalisation using only a resource system which is in a stationary state at the initial time and a system-resource interaction that preserves the global energy is possible if and only if the target state $\rho_s$ is block-diagonal in the eigenbasis of the system's Hamiltonian $H_s.$ As a consequence, we compute the extra work the operator must provide by tuning the resource-system interaction to overcome this constraint. This quantity, which is interpreted as the work needed to preserve the coherences in the state, can be expressed in terms of the target state $\rho_s$ and the thermal equilibrium state $\rho_{\beta},$ and it is proportional to the symmetrized relative entropy between $\rho_s$ and $\rho_{\beta}.$