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