Rare Fluctuations of Entropy in Quantum Systems

The notion of entropy has been at the core of thermodynamics and statistical physics since the 19th century, originally mentioned in the statement of the second law of thermodynamics as ``entropy is non-decreasing in an isolated system.'' In the context of both classical and quantum mechanics, there are cases where the second law is violated. In other words, entropy of the system may fluctuate downward. The question of how low or high the entropy of a quantum system can get is of interest in the context of small systems that can maintain quantum coherence for short times, and in the context of our universe, assuming it is an isolated quantum system. We explore the rare and extreme fluctuations of two well-developed notions of entropy that are relevant and interesting in isolated thermodynamic quantum systems, namely the observational and entanglement entropy. We find several fundamental differences in the behavior of the two when fluctuating to their extremes. We also provide an analytical proof of a tight upper bound on entanglement entropy for systems constrained to conservation laws