Unified View of Quantum and Classical Correlations

We discuss the problem of the separation of total correlations in a given quantum state into entanglement, dissonance, and classical correlations using the concept of relative entropy as a distance measure of correlations. This allows us to put all correlations on an equal footing. Entanglement and dissonance, whose definition is introduced here, jointly belong to what is known as quantum discord. Our methods are completely applicable for multipartite systems of arbitrary dimensions. We investigate additivity relations between different correlations and show that dissonance may be present in pure multipartite states.

Figure 1

Correlations in a quantum state. An arrow from $x$ to $y$, $x\to y$, indicates that $y$ is the closest state to $x$ as measured by the relative entropy $S(x\parallel y)$. The state $\rho \in \mathcal{E}$ (the set of entangled states), $\sigma \in \mathcal{S}$ (the set of separable states), $\chi \in \mathcal{C}$ (the set of classical states), and $\pi \in \mathcal{P}$ (the set of product states). The distances are entanglement, $E$, quantum discord, $D$, quantum dissonance, $Q$, total mutual information, $T_{\rho }$ and $T_{\sigma }$, and classical correlations, $C_{\sigma }$ and $C_{\rho }$. All relative entropies, except for entanglement, reduce to the differences in entropies of $y$ and $x$, $S(x\parallel y)=S(y)-S(x)$. With the aid of $L_{\rho }$ and $L_{\sigma }$ the closed paths are additive, i.e., Eq. (5).