Second Law-Like Inequalities with Quantum Relative Entropy: An Introduction

Understand

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.