I will introduce and discuss various entropies in quantum theory and their meaning. I will start by introducing various entropies, both in classical and quantum physics and then specifically focus on the classical thermodynamic entropy. I will then generalize this notion to quantum physics to arrive at the notion called observational entropy. Observational entropy is a concept originally introduced by von Neumann [1], long forgotten but in recent years revived [2] and studied deeply. It acts as a bridge between various types of entropies, Shannon entropy on one hand, Boltzmann entropy on the other, while being bounded by the von Neumann and the maximal entropy. I will also discuss this entropy in the relation to quantum chaos and possible future research directions in this regard.
[1] J. von Neumann, “Proof of the ergodic theorem and the H-theorem in quantum mechanics. Translation of: Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik”, European Physical Journal H 35, 201–237 (2010),available at arXiv:1003.2133
[2] D Šafránek, A Aguirre, J Schindler, JM Deutsch Foundations of Physics 51 (5), 101 (2021)