Upper bounds for relative entropy of entanglement based on convex hull approximation

Quantifying entanglement for multipartite quantum state is a crucial task in many aspects of quantum information theory. Among all the entanglement measures, relative entropy of entanglement $E_{R}$ is an important quantity due to its clear geometric meaning, easy compatibility with different system sizes, and various applications in many other related quantity calculations. Lower bounds of $E_R$ were previously found based on distance to the set of positive partial transpose states. We propose a method to calculate upper bounds of $E_R$ based on the convex hull approximation of the set of separable states. We apply our method to calculate $E_R$ for composite systems of various sizes, and compare with the previous known lower bounds, obtaining promising results. Our method adds a new tool for entanglement measure calculation and deepens our understanding for the structure of separable states.