Relative entropy of entanglement and tripartite minimal surface

Abstract: We propose a holographic dual of the relative entropy of entanglement for a boundary tripartition $A:B:C$. Our proposal is that, at the leading order, \begin{align} E_R(ρ{AB}) = \frac{1}{4G_N}\Big[\mathrm{Area}(Γ{\min}) - \mathrm{Area}(γ{AB})\Big] \notag \end{align} where $Γ</em>{\min}$ is the minimal bulk tripartition surface and $γ<em>{AB}$ is the minimal surface homologous to $AB$. For a general random tensor network, we prove the corresponding upper bound by constructing a separable state associated with $Γ</em>{\min}$ and evaluating its relative entropy. We also establish the matching lower bound rigorously for networks of one, two, and three Haar random tensors by bounding the maximal tripartite product-state overlap. In appropriate geometries, the minimal tripartition surface can form a nontrivial tri-junction resembling the Mercedes logo. The proof proceeds by sequential optimization over candidate product states, which has a natural geometric interpretation as a local search for the minimal tripartition surface.