  We analyze the general nonclassicality of correlations of a composite quantum
systems as measured by the negativity of quantumness. The latter corresponds to
the minimum entanglement, as quantified by the negativity, that is created
between the system and an apparatus that is performing local measurements on a
selection of subsystems. The negativity of quantumness thus quantifies the
degree of nonclassicality on the measured subsystems. We demonstrate a number
of possible different interpretations for this measure, and for the concept of
quantumness of correlations in general. In particular, for general bipartite
states in which the measured subsystem is a qubit, the negativity of
quantumness acquires a geometric interpretation as the minimum trace distance
from the set of classically correlated states. This can be further
reinterpreted as minimum disturbance, with respect to trace norm, due to a
local measurement or a nontrivial local unitary operation. We calculate the
negativity of quantumness in closed form for Werner and isotropic states, and
for all two-qubit states for which the reduced state of the system that is
locally measured is maximally mixed---this includes all Bell diagonal states.
We discuss the operational significance and potential role of the negativity of
quantumness in quantum information processing.
