Quantum conditional entropy ($H(A|B{)}_{\rho }$)

The quantum conditional entropy is $H(A|B{)}_{\rho }=S({\rho }_{AB})-S({\rho }_B)$. Unlike its classical counterpart, it may be negative.

For bipartite density operators,This is a consequence of Strong subadditivity of Von Neumann entropy.

If $\epsilon =\Vert {\rho }_{AB}-{\sigma }_{AB}{\Vert }_1/2$ and $d_A=\dim {\mathcal{H}}_A$, thenThe proof couples the two states through their positive and negative differences, then combines concavity of quantum conditional entropy with the entropy bound for a binary mixture.

For every state on ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$,The upper bound follows from Subadditivity of Von Neumann entropy and the lower bound from the Araki–Lieb inequality.