Abstract: 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...
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Topics: 
Statistical physics
Theoretical physics
Mathematical physics