  Quantum correlations in a physical system are usually studied with respect to
a unique (fixed) decomposition of the system into subsystems, without fully
exploiting the rich structure of the state-space. Here, we show several
examples in which the consideration of different ways to decompose a physical
system enhances the quantum resources and accounts for a more flexible
definition of quantumness-measures. Furthermore, we give a new perspective
regarding how to reassess the fact that local operations play a key role in
general quantumness-measures that go beyond entanglement --as discord-like
ones. We propose a way to quantify the maximum quantumness of a given state.
Applying our definition to low-dimensional bipartite states, we show that
different behaviours are reported for separable and entangled states than those
corresponding to the usual measures of quantum correlations. We show that there
is a close link between our proposal and the criterion to witness quantum
correlations based on the rank of the correlation matrix, proposed by Daki\'c,
Vedral and Brukner.
