  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 measurements, which lead to an entropy that is defined
outside of equilibrium, is in accord with the thermodynamic entropy for
equilibrium systems, and reaches the thermodynamic entropy in the long-time
limit, even in genuinely isolated quantum systems. This answers the
long-standing conceptual problem, as to which entropy is relevant for the
formulation of the second thermodynamic law in closed quantum systems. This
entropy could be in principle measured, especially now that experiments on such
systems are becoming feasible.
