We study quantum coarse-grained entropy and demonstrate that the gap in entropy between local and global coarse-grainings is a natural generalization of entanglement entropy to mixed states and multipartite systems. This ``quantum correlation entropy''
Sqc is additive over independent systems, is invariant under local unitary operations, measures total nonclassical correlations (vanishing on states with strictly classical correlation), and reduces to the entanglement entropy for bipartite pure states. It quantifies how well a quantum system can be understood via local measurements and ties directly to nonequilibrium thermodynamics, including representing a lower bound on the quantum part of thermodynamic entropy production. We discuss two other measures of nonclassical correlation to which this entropy is equivalent and argue that together they provide a unique thermodynamically distinguished measure.
The entanglement entropy is related to quantum phase transitions. It behaves differently when a quantum system is in a phase with a finite correlation length (such as an insulator phase), compared to a system in a critical phase in which the correlation length is infinite such as a metallic phase. The entanglement entropy of disordered systems does not depend only on the length of the relevant segment and the size of the system but also on the specific realization of disorder. Thus, the entanglement entropy of a disordered system exhibits mesoscopic fluctuations. Here we shall show numerical results that exhibit a crossover in the distribution of the entanglement entropy from a normal distribution at the weak localization regime to a log-normal distribution at the strong localization regime.