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 no􀆟on to quantum physics to arrive at the no􀆟on called observa􀆟onal entropy. Observa􀆟onal entropy is a concept originally introduced by von Neumann [1], long forgoten 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. It has found applica􀆟ons in describing thermaliza􀆟on of isolated quantum systems [3], work extrac􀆟on from unknown quantum sources [4], as a signifier of chaos [5], and many others [6].

References:

1. J. von Neumann, “Proof of the ergodic theorem and the H-theorem in quantum mechanics. Transla􀆟on of: Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik”, European Physical Journal H 35, 201–237 (2010). arXiv:1003.2133
2. D Šafránek, A Aguirre, J Schindler, JM Deutsch Founda􀆟ons of Physics 51 (5), 101 (2021)
3. D Šafránek, JM Deutsch, A Aguirre, Physical Review A 99 (1), 012103 (2019)
4. D. Šafránek, D. Rosa, F. Binder, "Work extrac􀆟on from unknown quantum sources", Phys. Rev. Let. 130, 210401 (2023)
5. Sreeram PG, R. Modak, S. Aravinda, "Witnessing quantum chaos using observa􀆟onal entropy", Phys. Rev. E 107, 064204 (2023)
6. J. Schindler, P. Strasberg, N. Galke, A. Winter, M. G. Jabbour "Unifica􀆟on of observa􀆟onal entropy with maximum entropy principles", arXiv:2503.15612 (2025)

*corresponding author: e-mail: dominik.safranek@matfyz.cuni.cz