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We investigate the detailed properties of Observational entropy, introduced by v{S}afr{a}nek et al. [Phys. Rev. A 99, 010101 (2019)] as a generalization of Boltzmann entropy to quantum mechanics. This quantity can involve multiple coarse-grainings, even those that do not commute with each other, without losing any of its properties. It is well-defined out of equilibrium, and for some coarse-grainings it generically rises to the correct thermodynamic value even in a genuinely isolated quantum system. The quantity contains several other entropy definitions as special cases, it has interesting information-theoretic interpretations, and mathematical properties -- such as extensivity and upper and lower bounds -- suitable for an entropy. Here we describe and provide proofs for many of its properties, discuss its interpretation and connection to other quantities, and provide numerous simulations and analytic arguments supporting the claims of its relationship to thermodynamic entropy. This quantity may thus provide a clear and well-defined foundation on which to build a satisfactory understanding of the second thermodynamical law in quantum mechanics.
We extend classical coarse-grained entropy, commonly used in many branches of physics, to the quantum realm. We find two coarse-grainings, one using measurements of local particle numbers and then total energy, and the second using local energy measu
In statistical mechanics, a small system exchanges conserved quantities---heat, particles, electric charge, etc.---with a bath. The small system thermalizes to the canonical ensemble, or the grand canonical ensemble, etc., depending on the conserved
Relaxation of few-body quantum systems can strongly depend on the initial state when the systems semiclassical phase space is mixed, i.e., regions of chaotic motion coexist with regular islands. In recent years, there has been much effort to understa
Thermalization has been shown to occur in a number of closed quantum many-body systems, but the description of the actual thermalization dynamics is prohibitively complex. Here, we present a model - in one and two dimensions - for which we can analyt
We develop the framework of classical Observational entropy, which is a mathematically rigorous and precise framework for non-equilibrium thermodynamics, explicitly defined in terms of a set of observables. Observational entropy can be seen as a gene