  We consider a driven quantum particle in the strong friction regime described
by the quantum Smoluchowski equation. We derive Crooks and Jarzynski type
relations for the reduced quantum system by properly generalizing the entropy
production to take into account the non-Gibbsian character of the equilibrium
distribution. In the case of a nonequilibrium steady state, we obtain a quantum
version of the Hatano-Sasa relation. We, further, propose an experiment with
driven Josephson junctions that would allow to investigate nonequilibrium
entropy fluctuations in overdamped quantum systems.
