  Regarding the strange properties of quantum entropy and entanglement, e.g.,
the negative quantum conditional entropy, we revisited the foundations of
quantum entropy, namely, von Neumann entropy, and raised the new method of
quantum fine-grained entropy. With the applications in entanglement theory,
quantum information processing, and quantum thermodynamics, we demonstrated the
capability of quantum fine-grained entropy to resolve some notable confusions
and problems, including the measure of entanglement and quantumness, the
additivity conjecture of entanglement of formation etc, and the definition of
temperature for single quantum system.
