I'm trying to prove that $-\log{d} \leq H(A|B) \leq \log{d}$ for von Neumann entropy.
Now, for this to make sense I should give some definitions. System $A$ lives in Hilbert space $\mathcal{H}_A$, system $B$ in $\mathcal{H}_B$, and these are finite dimensional systems. Moreover, system A has dimension $d$. $H(A|B)$ is the conditional entropy, which I here take to be $H(A|B) = H(AB) - H(B)$, where $H(A) = -Tr(\rho_A \log{\rho_A})$, $\rho_A$ being the density operator of system $A$, so that $H(AB)$ is the density operator of the joint system.
Now, I can prove the upperbound quite easily:
$H(A|B) = H(AB) - H(B) \leq H(A) + H(B) - H(B) \leq \log{d}$, where both inequalities follow from the positivity of the relative entropy $H(A||B) = Tr(\rho_A\log{\rho_A}-\rho_A\log{\rho_B})$ quite easily. If needed, I can write those two line proofs in an update.
However, for the lowerbound I'm quite lost. I'm pretty sure I need to make some clever choice of $A$ and $B$ for the relative entropy again and abuse its positivity, but I can't figure out what to do. Could anyone give me a hint?