Zero discord implies classicality

TL;DR

This paper provides a straightforward proof that zero quantum discord in bipartite states is equivalent to the state being classical on one subsystem, linking it to Petz's theorem on relative entropy preservation.

Contribution

It offers a simple, direct proof connecting zero discord to classical-quantum states using Petz's theorem, simplifying previous complex proofs.

Findings

Zero discord states are exactly classical-quantum states.

The proof uses Petz's theorem on channels preserving relative entropy.

Clarifies the relationship between discord and classicality in quantum states.

Abstract

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.