  We investigate monogamy of correlations and entropy inequalities in the Bloch
representation. Here, both can be understood as direct relations between
different correlation tensor elements and thus appear intimately related. To
that end we introduce the split Bloch basis, that is particularly useful for
representing quantum states with low dimensional support and thus amenable to
purification arguments. Furthermore, we find dimension dependent entropy
inequalities for the Tsallis 2-entropy. In particular, we present an analogue
of the strong subadditivity and a quadratic entropy inequality. These relations
are shown to be stronger than subadditivity for finite dimensional cases.
