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Relative entropy of entanglement (REE) is an entanglement measure ofbipartite mixed states, defined by the minimum of the relative entropy$S(\rho_{AB}|| \sigma_{AB})$ between a given mixed state $\rho_{AB}$ and anarbitrary separable state $\sigma_{AB}$. The REE is always bounded by themutual information $I_{AB}=S(\rho_{AB} || \rho_{A}\otimes \rho_{B})$ becausethe latter measures not only quantum entanglement but also classicalcorrelations. In this paper we address the question of to what extent REE canbe small compared to the mutual information in conformal field theories (CFTs).For this purpose, we perturbatively compute the relative entropy between thevacuum reduced density matrix $\rho^{0}_{AB}$ on disjoint subsystems $A \cup B$and arbitrarily separable state $\sigma_{AB}$ in the limit where two subsystemsA and B are well separated, then minimize the relative entropy with respect tothe separable states. We argue that the result highly depends on the spectrumof CFT on the subsystems. When we have a few low energy spectrum of operatorsas in the case where the subsystems consist of a finite number of spins in spinchain models, the REE is considerably smaller than the mutual information.However in general our perturbative scheme breaks down, and the REE can be aslarge as the mutual information.

Towards an Entanglement Measure for Mixed States in CFTs Based on Relative Entropy