  We review the fundamental properties of the quantum relative entropy for
finite-dimensional Hilbert spaces. In particular, we focus on several
inequalities that are related to the second law of thermodynamics, where the
positivity and the monotonicity of the quantum relative entropy play key roles;
these properties are directly applicable to derivations of the second law
(e.g., the Clausius inequality). Moreover, the positivity is closely related to
the quantum fluctuation theorem, while the monotonicity leads to a quantum
version of the Hatano-Sasa inequality for nonequilibrium steady states. Based
on the monotonicity, we also discuss the data processing inequality for the
quantum mutual information, which has a similar mathematical structure to that
of the second law. Moreover, we derive a generalized second law with quantum
feedback control. In addition, we review a proof of the monotonicity in line
with Petz.
