  Entanglement, and quantum correlation, are precious resources for quantum
technologies implementation based on quantum information science, such as, for
instance, quantum communication, quantum computing, and quantum interferometry.
Nevertheless, to our best knowledge, a directly computable measure for the
entanglement of multipartite mixed-states is still lacking. In this work, {\it
i)} we derive from a minimum distance principle, an explicit measure able to
quantify the degree of quantum correlation for pure or mixed multipartite
states; {\it ii)} through a regularization process of the density matrix, we
derive an entanglement measure from such quantum correlation measure; {\it
iii)} we prove that our entanglement measure is \textit{faithful} in the sense
that it vanishes only on the set of separable states. Then, a comparison of the
proposed measures, of quantum correlation and entanglement, allows one to
distinguish between quantum correlation detached from entanglement and the one
induced by entanglement, hence to define the set of separable but non-classical
states.
  Since all the relevant quantities in our approach, descend from the geometry
structure of the projective Hilbert space, the proposed method is of general
application.
  Finally, we apply the derived measures as an example to a general Bell
diagonal state and to the Werner states, for which our regularization procedure
is easily tractable.
