  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" $S^{\rm QC}$ 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 non-equilibrium 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.
