Unifying relation for quantum systems driven out of equilibrium

We extend a classical relation by Crooks to quantum systems and show that it unifies the Crooks transient fluctuation theorem and the Kawasaki non-linear response relation, which leads to the standard linear response theory. We also show that the Green-Kubo formula for a system in a steady state driven by a time-independent driving force is always expressed in terms of the symmetrized correlation function for a quantity induced by the driving force and that a generalized entropy for systems in steady heat conduction states satisfies extensions of the Clausius and the Gibbs relations exactly.