On the generic increase of observational entropy in isolated systems

• URL: http://arxiv.org/abs/2404.11985v1
• Date: Thu, 18 Apr 2024 08:27:04 GMT
• Title: On the generic increase of observational entropy in isolated systems
• Authors: Teruaki Nagasawa, Kohtaro Kato, Eyuri Wakakuwa, Francesco Buscemi,
• Abstract summary: We show how observational entropy of a system undergoing a unitary evolution chosen at random tends to increase with overwhelming probability. We show that for any observation that is sufficiently coarse with respect to the size of the system, regardless of the initial state of the system, random evolution renders its state practically indistinguishable from the microcanonical distribution.
• Score: 6.874745415692133
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Observational entropy - a quantity that unifies Boltzmann's entropy, Gibbs' entropy, von Neumann's macroscopic entropy, and the diagonal entropy - has recently been argued to play a key role in a modern formulation of statistical mechanics. Here, relying on algebraic techniques taken from Petz's theory of statistical sufficiency and on a Levy-type concentration bound, we prove rigorous theorems showing how the observational entropy of a system undergoing a unitary evolution chosen at random tends to increase with overwhelming probability and to reach its maximum very quickly. More precisely, we show that for any observation that is sufficiently coarse with respect to the size of the system, regardless of the initial state of the system (be it pure or mixed), random evolution renders its state practically indistinguishable from the microcanonical distribution with a probability approaching one as the size of the system grows. The same conclusion holds not only for random evolutions sampled according to the unitarily invariant Haar distribution, but also for approximate 2-designs, which are thought to provide a more physically reasonable way to model random evolutions.

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