  Generalized measurement schemes on one part of bipartite states, which would
leave the set of all separable states insensitive are explored here to
understand quantumness of correlations in a more general perspecitve. This is
done by employing linear maps associated with generalized projective
measurements. A generalized measurement corresponds to a quantum operation
mapping a density matrix to another density matrix, preserving its positivity,
hermiticity and traceclass. The Positive Operator Valued Measure (POVM) --
employed earlier in the literature to optimize the measures of
classical/quatnum correlations -- correspond to completely positive (CP) maps.
The other class, the not completely positive (NCP) maps, are investigated here,
in the context of measurements, for the first time. It is shown that that such
NCP projective maps provide a new clue to the understanding the quantumness of
correlations in a general setting. Especially, the separability-classicality
dichotomy gets resolved only when both the classes of projective maps (CP and
NCP) are incorporated as optimizing measurements. An explicit example of a
separable state -- exhibiting non-zero quantumn discord when possible
optimizing measurements are restricted to POVMs -- is re-examined with this
extended scheme incorporating NCP projective maps to elucidate the power of
this approach.
