  For classical systems driven out of equilibrium, Crooks derived a relation
(the Crooks-Jarzynski relation), whose special cases include a relation (the
Crooks relation) equivalent to the Kawasaki non-linear response relation. We
derive a quantum extension of the Crooks-Jarzynski relation without explicitly
using the principle of microreversibility. Its special cases lead to the
Jarzynski equality and the standard linear response theory with a Green-Kubo
formula with a canonical correlation function. We also derive a quantum
extension of the Crooks relation using the principle of microreversibility. Its
special cases lead to the Jarzynski equality, the Crooks transient fluctuation
theorem, and the fluctuation theorem for current or shear stress, which leads
to a Green-Kubo formula with a symmetrized correlation function. For each
quantum Crooks relation, there exists a corresponding quantum Crooks-Jarzynski
relation. Using either relation, we can derive the Jarzynski equality, the
fluctuation theorems mentioned above, and the standard linear response theory.
