  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.
