Relative entropy of entanglement of Haar random states

Abstract

We determine the relative entropy of entanglement of a bipartite mixed state ${\rho }_{AB}$ obtained by tracing out one subsystem of a tripartite Haar-random pure state $|\psi {⟩}_{ABC}$, finding $E_R({\rho }_{AB})=\log \frac{d_A{d}_B}{\max (d_A,d_B,d_C)}+O(1)$. Equivalently, the relative entropy of entanglement nearly saturates the smaller of the entanglement of formation $E_F({\rho }_{AB})$ and the mutual information $I(A:B)$. The upper bound is achieved by an explicit separable state obtained through one-sided Schmidt dephasing, which is therefore approximately closest.

11 pages, comments are welcome