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

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.