A measure of entanglement which distinguishes between classical and quantum correlations

The Von Neumann entanglement entropy is a measure of entanglement. For an overall pure state a non zero entanglement entropy for a subsystem A indicates the degree of quantum correlation between the subsystem and the rest of the system. Also, if the overall state is mixed, then for a subsystem the entanglement entropy may be non zero. But, in this case the entanglement entropy may be due to quantum correlations or classical correlations. Is there a way to distinguish between the two?

Are there other entanglement measures which helps us to distinguish between quantum and classical correlations even though the overall state of the system is mixed?

Most entanglement measures do this. A property that is usually required for entanglement measures is that they are zero for separable states (product states in the case of pure states). Although, note that this is not an iff-condition, i.e. there could be entangled states for which the entanglement measure is zero.

The von Neumann entanglement entropy is in this sense only a entanglement measure for pure states. Entanglement measures that are also defined on mixed states are for example the relative entropy of entanglement or distillable entanglement (which both reduce to the von Neumann entanglement entropy for pure states).

As a side remark, the question of whether a state is entangled (non-separable) or not has be shown to be NP-Hard. Therefore, to decide if a state has quantum or classical correlations is very hard in general.