Zero discord implies classicality

The "classical-quantum" ($\mathit{cq}$) discord of a bipartite state $\rho^{AB}$ is the smallest difference between the mutual information $S(\rho^{A:B})$ of $\rho$ and that of $\rho$ after a measurement channel is applied on the $A$ system. Relating zero discord to the strong subadditivity of the Von Neumann entropy, Datta proved that a state has zero $\mathit{cq}$ discord iff and only if it can be written in the form $\sum_i p_i\, |i\rangle\langle i|\otimes \rho^B_i$ for $p_i$ a probability distribution, $|i\rangle$ a basis of the $A$ system and $\rho^B_i$ states of the $B$ system. We provide a simple proof of that same result using directly a theorem of Petz on channels that leave unchanged the relative entropy of two given states.