  A conjecture -- \emph{the modified super-additivity inequality} of relative
entropy -- was proposed in \cite{Zhang2012}: There exist three unitary
operators $U_A\in \unitary{\cH_A},U_B\in \unitary{\cH_B}$, and $U_{AB}\in
\unitary{\cH_A\ot \cH_B}$ such that $$
\rS(U_{AB}\rho_{AB}U^\dagger_{AB}||\sigma_{AB}) \geqslant
\rS(U_A\rho_AU^\dagger_A||\sigma_A) + \rS(U_B\rho_BU^\dagger_B||\sigma_B), $$
where the reference state $\sigma$ is required to be full-ranked. A numerical
study on the conjectured inequality is conducted in this note. The results
obtained indicate that the modified super-additivity inequality of relative
entropy seems to hold for all qubit pairs.
