For an isolated quantum system, one can study the time evolution of entanglement entropy after a quantum quench (always a pure state), which has a rich behaviour in various different models.
However, one can study the classical limit of the quantum system, which also has non-trivial time evolution of other physical observables, such as magnetisation in a spin chain. Sometimes one can find a classical-quantum correspondence of the dynamics of some physical observables after the quench.
I'm wondering if there exists a similar classical concept that links to the quantum entanglement entropy. It seems to me that it cannot be classical thermodynamic entropy, since the classical counterpart doesn't have a well defined temperature.