  We report two results complementing the second law of thermodynamics for
Markovian open quantum systems coupled to multiple reservoirs with different
temperatures and chemical potentials. First, we derive a nonequilibrium free
energy inequality providing an upper bound for a maximum power output, which
for systems with inhomogeneous temperature is not equivalent to the Clausius
inequality. Secondly, we derive local Clausius and free energy inequalities for
subsystems of a composite system. These inequalities differ from the total
system one by the presence of an information-related contribution and build the
ground for thermodynamics of quantum information processing. Our theory is used
to study an autonomous Maxwell demon.
