  The notion of quantum discord introduced by Ollivier and Zurek [Phys. Rev.
Lett 88, 017901 (2001)] (see also Henderson and Vedral [J. Phys. A 34, 6899
(2001)]) has attracted increasing attention, in recent years, as an entropic
quantifier of non-classical features pertaining to the correlations exhibited
by bipartite quantum systems. Here we generalize the notion so as to encompass
power-law q-entropies (that reduce to the standard Shannon entropy in the limit
$q \to 1$) and study the concomitant consequences. The ensuing, new
discord-like measures we advance describe aspects of non-classicality that are
different from those associated with the standard quantum discord. A particular
manifestation of this difference concerns a feature related to order. Let $D_1$
stand for the standard, Shannon-based discord measure and $D_q$ for the $q \ne
1$ one. If two quantum states $A$, $B$ are such that $D_1(A) > D_1(B)$, this
order-relation does not remain invariant under a change from $D_1$ to $D_q$.
