  We introduce an axiomatic approach for characterizing quantum conditional
entropy. Our approach relies on two physically motivated axioms: monotonicity
under conditional majorization and additivity. We show that these two axioms
provide sufficient structure that enable us to derive several key properties
applicable to all quantum conditional entropies studied in the literature.
Specifically, we prove that any quantum conditional entropy must be negative on
certain entangled states and must equal -log(d) on dxd maximally entangled
states. We also prove the non-negativity of conditional entropy on separable
states, and we provide a generic definition for the dual of a quantum
conditional entropy. Finally, we develop an operational approach for
characterizing quantum conditional entropy via games of chance, and we show
that, for the classical case, this complementary approach yields the same
ordering as the axiomatic approach.
