Zero discord implies classicality

The "classical-quantum" ($\mathit{cq}$) discord of a bipartite state $ρ^{AB}$ is the smallest difference between the mutual information $S(ρ^{A:B})$ of $ρ$ and that of $ρ$ 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 $\sum_i p_i\, |i\rangle\langle i|\otimes ρ^B_i$ for $p_i$ a probability distribution, $|i\rangle$ a basis of the $A$ system and $ρ^B_i$ 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.

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